Set Theory and Logic

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Recent submissions

Any replacements are listed farther down

[259] viXra:1811.0269 [pdf] submitted on 2018-11-17 13:14:25

Refutation of Superposition as Glue in Matita Theorem Prover

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the substitution lemma for the successor function, smart application of inductive hypotheses, and proof traces of a complex example in the Matita standard library. Results are not tautologous, hence refuting superposition.
Category: Set Theory and Logic

[258] viXra:1811.0264 [pdf] submitted on 2018-11-17 18:17:31

Refutation of Metamath Theorem Prover

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate six conjectures and one theorem, as proffered by Metamath staff. The conjectures are not tautologous. The Tarski-Grothendieck theorem is also not tautologous. Metamath fails our analysis.
Category: Set Theory and Logic

[257] viXra:1811.0220 [pdf] submitted on 2018-11-14 18:40:03

Shorter Refutation of the Löb Theorem and Gödel Incompleteness by Substitution of Contradiction

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Löb’s theorem □(□X→X)→□X and Gödel’s incompleteness as □(□⊥ →⊥)→□⊥ are refuted.
Category: Set Theory and Logic

[256] viXra:1811.0154 [pdf] submitted on 2018-11-09 08:30:47

Find the extra shape (In Russian)

Authors: V. A. Kasimov
Comments: 2 Pages. in Russian

First, the task will put you in a dead end. No figure clearly stands out from the total number more than others. It's not as easy as it may seem in the first seconds.
Category: Set Theory and Logic

[255] viXra:1811.0148 [pdf] submitted on 2018-11-09 19:18:36

Refutation of the Alleged Łukasiewicz Nightmare in Ł4 Logic: (◇p&◇q)→◇(p&q)

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

In Prover9 the alleged Łukasiewicz nightmare of (◇p&◇q)→◇(p&q) is not tautologous. However, in Prover9 the nightmare recast in one variable as (◇p&◇~p)→◇(p&~p) is tautologous. In Meth8/VŁ4, both propositions are tautologous. This speaks to Meth8/VŁ4, based on the corrected modern Square of Opposition, as an exact bivalent system, as opposed to Prover9, based on the uncorrected modern Square of Opposition, as an inexact probabilistic vector space.
Category: Set Theory and Logic

[254] viXra:1811.0078 [pdf] submitted on 2018-11-05 15:44:40

Refutation of Behavioral Mereology

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

If P≤P′ and Q′≤Q, proposition <>P'P<>QP' = <>QP is equivalent to []P'P[]QP' = []QP and respectively not tautologies.
Category: Set Theory and Logic

[253] viXra:1811.0075 [pdf] submitted on 2018-11-05 20:42:06

Refutation of the Blok-Esakia Theorem for Universal Classes

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Grzegorczyk (gzr) algebras as used for support and the Blok-Esakia theorems are not confirmed as tautologies and hence refuted.
Category: Set Theory and Logic

[252] viXra:1811.0059 [pdf] submitted on 2018-11-04 19:11:29

Refutation of First-Order Continuous Induction on Real Closed Fields

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

By mapping definitions, theorems, and propositions into Meth8/VŁ4, we refute the first-order continuous induction principle on real closed fields.
Category: Set Theory and Logic

[251] viXra:1811.0020 [pdf] submitted on 2018-11-01 10:35:55

New Axioms in the Set Theory

Authors: Antoine Balan
Comments: 2 pages, written in english

We propose three new axioms in set theory, axiomatising the measure theory of the Hilbert spaces.
Category: Set Theory and Logic

[250] viXra:1811.0018 [pdf] submitted on 2018-11-01 11:11:48

Elementary Set Theory Can Be Used to Prove Fermat's Last Theorem (FLT)

Authors: Phil A. Bloom
Comments: 3 Pages.

An open problem is proving FLT simply for each integral $n>2$. Our proof of FLT is based on our algebraic identity, denoted, {for convenience}, as $r^n+s^n=t^n$. For $n\geq1$ we relate $r,s,t>0$, each a different function of variables comprising $r^n+s^n=t^n$, with $x,y,z>0$ for which $x^n+y^n=z^n$ holds. We infer as true by \emph{direct argument} (not BWOC), for any given $n>2$, that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}$. In addition, we show, for $n>2$, that $\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}=\varnothing$. Thus, for $n\in\mathbb{Z},n>2$, it is true that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\varnothing$.
Category: Set Theory and Logic

[249] viXra:1810.0511 [pdf] submitted on 2018-10-30 08:19:57

Recent Advances in Refutation and Confirmation Using Meth8 Modal Logic Model Checker

Authors: Colin James III
Comments: 461 Pages. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com; email: info@cec-services dot com

We evaluated 338 artifacts in 1705 assertions to confirm 348 as tautology and 1357 as not (79.6%). We resuscitated the four-valued logic of Łukasiewicz in Meth8/VŁ4 on the 2-tuple {00, 10, 01, 11} as respectively {False for contradiction; Contingent for falsity; Non contingent for truthity; Tautology for proof}. The designated proof value is T for tautology. Meth8 contains recent advances in parsing technology named sliding window.
Category: Set Theory and Logic

[248] viXra:1810.0350 [pdf] submitted on 2018-10-21 12:28:45

First Cryptocurrency Project Based on Strict Mathematical Rules

Authors: Linda Wiklund
Comments: 9 Pages.

First Cryptocurrency project based on strict mathematical rules
Category: Set Theory and Logic

[247] viXra:1810.0328 [pdf] submitted on 2018-10-20 21:05:52

Refutation of Set Theory by Supremum and Infimum

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Set theory is refuted by the supremum and infimum artifacts.
Category: Set Theory and Logic

[246] viXra:1810.0309 [pdf] submitted on 2018-10-19 12:19:11

Refutation of the Second Incompleteness Theorem by Gödel Logic

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Gödel's second incompleteness theorem as based on the minimal modal logic to express the Löb axiom is not tautologous. Subsequent substitutions into the Löb axiom along with Hájek's earlier lemma raise further suspicion about Gödel-justification logic.
Category: Set Theory and Logic

[245] viXra:1810.0082 [pdf] submitted on 2018-10-07 04:14:05

Refutation of the Sliding Scale Theorem in Law

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

The sliding scale theorem, and as implemented in fuzzy logic, is refuted.
Category: Set Theory and Logic

[244] viXra:1810.0072 [pdf] submitted on 2018-10-05 19:34:31

Refutation of Existentially Closed De Morgan Algebras

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Existentially closed De Morgan algebras are refuted.
Category: Set Theory and Logic

[243] viXra:1810.0067 [pdf] submitted on 2018-10-06 02:40:03

Refutation of Short Circuit Evaluation for Propositional Logic by Commutative Variants

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

The short circuit evaluation for propositional logic by commutative variants is not tautologous, and thereby refuted.
Category: Set Theory and Logic

[242] viXra:1810.0047 [pdf] submitted on 2018-10-04 22:33:02

Visualizing the Distribution of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 3 Pages.

The distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[241] viXra:1809.0263 [pdf] submitted on 2018-09-12 06:11:47

Refutation of Mapping mu-Calculus Onto Second-Order Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

When mapping mu-calculus onto second-order logic, we show use of the fixpoint operator as untenable. What follows is the effective refutation of mapping mu-calculus onto second-order logic.
Category: Set Theory and Logic

[240] viXra:1809.0257 [pdf] submitted on 2018-09-12 11:52:54

Refutation of Completeness for Non-Deterministic Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We show that the four-valued, non-deterministic semantics for modal logic are not complete. The demonstration uses contradictions based on Carnielli's paraconsistent logic. What follows is that infinite non-deterministic matrices are by definition incomplete.
Category: Set Theory and Logic

[239] viXra:1808.0090 [pdf] submitted on 2018-08-07 15:06:59

Refutation of Lean Theorem Prover from Microsoft

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Respond to this author's email address: info@ersatz-systems dot com. (We instruct troll Mikko at Disqus to read the entire article twice before she starts typing.)

These examples were found not tautologous: (∀x,px→r)↔(∃x,px)→r; (∃x,px→r)↔(∀x,px)→r; (∃x,r→px)↔(r→∃x,px). Hence Lean prover from Microsoft is not bivalent and refuted.
Category: Set Theory and Logic

[238] viXra:1808.0047 [pdf] submitted on 2018-08-03 22:08:52

Shortest Refutations of the Zermelo-Fraenkel (ZF) Axioms

Authors: Colin James III
Comments: 4 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

The nine Zermelo-Fraenkel (ZF) axioms evaluated are not tautologous.
Category: Set Theory and Logic

[237] viXra:1807.0514 [pdf] submitted on 2018-07-30 18:56:24

Confirmation that Every Straight Line Through a Point Inside a Circle Intersects the Circumference

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

"An interior point inside the circle and an exterior point outside the circle imply that every straight line intersecting both points intersects a point on the circle." This conjecture is comfirmed.
Category: Set Theory and Logic

[236] viXra:1807.0509 [pdf] submitted on 2018-07-29 06:56:29

Confirmation of Playfair's Axiom

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

Per "De Morgan that this proposition is logically equivalent to Playfair’s axiom. ... Let X be the set of pairs of distinct lines which meet and Y the set of distinct pairs of lines each of which is parallel to a single common line. If z represents a pair of distinct lines, then the statement, For all z, if z is in X then z is not in Y, is Playfair's axiom, and its logically equivalent contrapositive, For all z, if z is in Y then z is not in X, is Euclid I.30, the transitivity of parallelism." This is confirmed as tautologous.
Category: Set Theory and Logic

[235] viXra:1807.0499 [pdf] submitted on 2018-07-29 12:12:31

Confirmation of the Triangle Inequality

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

"[T]he triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side." We rewrite this as contradiction F implies tautology T using absolute values of distances: "If qr is not greater than pq with pr, then both qr is greater than pq and qr is greater than pr." Hence the triangle inequality is confirmed as tautologous. Remark: This exercise indirectly speaks to the fact that the vector space is not bivalent.
Category: Set Theory and Logic

[234] viXra:1807.0478 [pdf] submitted on 2018-07-28 16:50:58

Refutation of the Heisenberg Principle as a no-go Axiom, and Its Trivial Replacement Theorem

Authors: Colin James III
Comments: 2 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

The Heisenberg uncertainty principle "states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa". This is refuted as not tautologous. In terms of the statistic of deviation, a trivial replacement theorem is: "If position is greater than momentum, then position and momentum are greater than position or momentum."
Category: Set Theory and Logic

[233] viXra:1807.0472 [pdf] submitted on 2018-07-27 15:42:55

Refutation of Relevance Logic Via Routely and Meyer

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

The conjecture "A → B is true at a world a if and only if for all worlds b and c such that Rabc (R is the accessibility relation) either A is false at b or B is true at c" is not tautologous.
Category: Set Theory and Logic

[232] viXra:1807.0452 [pdf] submitted on 2018-07-26 23:41:32

Implication Combinations Derived from (P>q)>r

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

12 tautologous equations are named the general forms of (p>q)>r by implication on Meth8/VŁ4. (They are not listed here because the abstract block mungs line spacing.)
Category: Set Theory and Logic

[231] viXra:1807.0431 [pdf] submitted on 2018-07-25 18:48:36

Trivial Proofs

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

For all p, p+p=p (1.1). #p>((p+p)=p) ; TTTT TTTT TTTT TTTT (1.2). For one p, p+p=p (2.1). %p>((p+p)=p) ; TTTT TTTT TTTT TTTT (2.2). Axiom of associativity (3.1). ((p+q)+r)=(p+(q+r)) ; TTTT TTTT TTTT TTTT (3.2).
Category: Set Theory and Logic

[230] viXra:1807.0418 [pdf] submitted on 2018-07-24 20:38:58

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. Source code is provided.
Category: Set Theory and Logic

[229] viXra:1807.0417 [pdf] submitted on 2018-07-24 22:55:24

A Dual Identity Based Symbolic Understanding of the Gödel’s Incompleteness Theorems, P-NP Problem, Zeno’s Paradox and Continuum Hypothesis

Authors: Arun Uday
Comments: 29 Pages.

A semantic analysis of formal systems is undertaken, wherein the duality of their symbolic definition based on the “State of Doing” and “State of Being” is brought out. We demonstrate that when these states are defined in a way that opposes each other, it leads to contradictions. This results in the incompleteness of formal systems as captured in the Gödel’s theorems. We then proceed to resolve the P-NP problem, which we show to be a manifestation of Gödel’s theorem itself. We then discuss the Zeno’s paradox and relate it to the same aforementioned duality, but as pertaining to discrete and continuous spaces. We prove an important theorem regarding representations of irrational numbers in continuous space. We extend the result to touch upon the Continuum Hypothesis and present a new symbolic conceptualization of space, which can address both discrete and continuous requirements. We term this new mathematical framework as “hybrid space”.
Category: Set Theory and Logic

[228] viXra:1807.0410 [pdf] submitted on 2018-07-23 22:31:40

Refutation of Stit Logic (Sees to it That)

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

From four papers in the relevant literature for 2017-2018, 13 equations are not tautologous. For example, an equation claimed for necessitation as p>[]p is TNTN TNTN TNTN TNTN (not all T proof values). Hence Stit logic is refuted
Category: Set Theory and Logic

[227] viXra:1807.0392 [pdf] submitted on 2018-07-24 11:54:04

Refutation of Levi-Identity and Agm Postulates

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The Levi-identity as rendered is not tautologous, as a basis for the subsequent AGM expressions. Four of the eight AGM equations are not tautologous, hence refuting them as a postulate system for fictional logic.
Category: Set Theory and Logic

[226] viXra:1807.0380 [pdf] submitted on 2018-07-22 08:28:58

Refutation of Connexive Logic Based on Wansing's Nightmare

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Connexive logic turns on Wansing's nightmare of ∼(A→B)↔(A→∼B), or the falsity-weakend ∼(A→B)↔(∼A→B), both of which are not tautologous.
Category: Set Theory and Logic

[225] viXra:1807.0373 [pdf] submitted on 2018-07-22 17:06:11

Refutation of Aristotle's and Boethius' Theses

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com

Aristotle's Theses and Boethius' Theses are below. ~(~A> A)=(A=A) ; TNCF TNCF TNCF TNCF (AT) . ~( A>~A)=(A=A) ; FCNT FCNT FCNT FCNT (AT)'. (A> B)>~(A>~B) ; FCNT FCNT FCNT FCNT (BT) . (A>~B)>~(A> B) ; FCNT FCNT FCNT FCNT (BT)'. Eqs. AT, AT', BT, and BT' are not tautologous, hence refuting those theses of Aristotle and Boethius. Because ~AT = AT' = BT = BT', what follows is that using conditionals to justify connexive logic makes Wansing's nightmare worse.
Category: Set Theory and Logic

[224] viXra:1807.0365 [pdf] submitted on 2018-07-21 09:51:40

Refutation of the Improved Adams Hypothesis of Conditional Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The seven equations as rendered are not tautologous. This means the probability axioms of Popper and McGee and the improved hypothesis of Adams are refuted.
Category: Set Theory and Logic

[223] viXra:1807.0357 [pdf] submitted on 2018-07-21 23:55:29

Refutation of Conditional Events in Quantum Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The following conjectures are refuted: Definition of simultaneous verifiability; Definition of simultaneous falsifiability; Corollary on simultaneous verifiability and falsifiability; and Theorem on uniqueness of relative negation. The concluding conjecture to map superposition, while tautologous, as (((A∩B)∪(C∩D))|(B∪D))⊃((A|B)∪(C|D)) is replaced with this non-tautologous equation: (((A∩B)∪(C∩D))|(B∩D))⊃((A|B)∩(C|D)).
Category: Set Theory and Logic

[222] viXra:1807.0300 [pdf] submitted on 2018-07-17 15:04:21

Refutation of the Postulate of Contradiction Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The conjecture that a postulate of contradiction for superposition implies contradiction to map quantum logic is refuted.
Category: Set Theory and Logic

[221] viXra:1807.0294 [pdf] submitted on 2018-07-18 00:57:44

Refutation that Classical Logic is a Completion of Quantum Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The distributivity of classical conjunction and disjunction does not fail. The conclusion to translate quantum negation as not necessity is not tautologous. This refutes quantum logic as a fragment of classical logic (or vice versa, as others write).
Category: Set Theory and Logic

[220] viXra:1807.0288 [pdf] submitted on 2018-07-16 21:41:03

Refutation of Quantum Logic as Tautologous Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 5 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

We evaluate quantum logic as described from two sources with different readership levels. We find quantum logic is not tautologous.
Category: Set Theory and Logic

[219] viXra:1807.0279 [pdf] submitted on 2018-07-15 15:01:34

Refutation of Whewell's Axiom of Causality Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

"Everything that becomes or changes must do so owing to some cause; for nothing can come to be without a cause." — Plato in Timaeus William Whewell [1794-1866] refined this as the concept of causality to depend on three axioms: nothing takes place without a cause; the magnitude of an effect is proportional to the magnitude of its cause; and to every action there is an equal and opposed reaction. The second axiom is tautologous, but the others are not, hence refuting the conjecture. From a metaphysical view, the axiom of causality is a bar to miracle because first cause is always assumed. This is overcome with rewriting the conjecture as "The necessity of effect implies the possibility of cause or no cause".
Category: Set Theory and Logic

[218] viXra:1807.0231 [pdf] submitted on 2018-07-12 23:29:05

If at Least One Question Implies Any Answer, Then Any Question Implies at Least One Answer. Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Is it true that any question implies at least one answer? No. Is it true that at least one question implies any answer? No. Is it true that if at least one question implies any answer, then any question implies at least one answer? Yes.
Category: Set Theory and Logic

[217] viXra:1807.0157 [pdf] submitted on 2018-07-09 06:47:49

Shortest Refutation of Gödel's Completeness Theorem Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

"By Gödel's completeness result, the formula (∀x.R(x,x))→(∀x∃y.R(x,y)) holds in all structures, and hence must have a natural deduction proof." We prove the formula is not tautologous, meaning it does not hold in all structures and serves as a contra-example. Hence Gödel's completeness theorem is refuted.
Category: Set Theory and Logic

[216] viXra:1807.0051 [pdf] submitted on 2018-07-02 23:08:38

Definition of Nothing in Mathematical Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Nothing is defined as "not necessarily a thing". This leads to how to collect not everything as nothing in multiple variables into a larger nothing variable, implying a set of nothing as a null set. We write this as nothing in p and nothing in q and nothing in r are all greater than nothing in s: ((~#p&~#q)&~#r)>~#s TTTT TTTT CTTT TTTT. As rendered, this is not tautologous, although nearly so with one deviant C contingency (falsity) value. Hence a collection of nothing does not imply anything outside itself. By extension, the null set is not logically feasible and cannot exist: a collection must contain something even though it is nothing.
Category: Set Theory and Logic

[215] viXra:1806.0450 [pdf] submitted on 2018-06-29 14:56:00

Refutation of the Free Will Hypothesis Based on Its Defective Fin Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Axioms for SPIN and TWIN are tautologous, but the axiom for FIN is not tautologous. Because the assumption of axiom FIN is essential to the authors' proof, the Free Will hypothesis is also not tautologous and refuted by its own derivation. This means the Free Will theorem can not be reasserted by resurrection as such.
Category: Set Theory and Logic

[214] viXra:1806.0449 [pdf] submitted on 2018-06-30 00:31:58

Refutation of the Strong Free Will Hypothesis Based on Its Defective Min' Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

"MIN′: In an A-first frame, B can freely choose any one of the 33 directions w, and a’s prior response is independent of B’s choice. Similarly, in a B-first frame, A can independently freely choose any one of the 40 triples x, y, z, and b’s prior response is independent of A’s choice." The equation as rendered is not tautologous. This means axiom MIN', as replacement for the previous FIN in the Free Will theorem, is not tautologous. Because the assumption of axiom MIN' is essential to the authors' proof, the Strong Free Will theorem is also not tautologous and refuted by its own derivation. This also means the Strong Free Will theorem can not be reasserted by resurrection as such.
Category: Set Theory and Logic

[213] viXra:1806.0338 [pdf] submitted on 2018-06-22 06:37:14

Independency of the Provability of Cliques

Authors: Holger H. Hoo
Comments: 27 Pages.

In this paper, we introduce DLS-MC, a new stochastic local search algorithm for the maximum clique problem. DLS-MC alternates between phases of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, during which vertices of the current clique are swapped with vertices not contained in the current clique. The selection of vertices is solely based on vertex penalties that are dynamically adjusted during the search, and a perturbation mechanism is used to overcome search stagnation. The behaviour of DLS-MC is controlled by a single parameter, penalty delay, which controls the frequency at which vertex penalties are reduced. We show empirically that DLSMC achieves substantial performance improvements over state-of-the-art algorithms for the maximum clique problem over a large range of the commonly used DIMACS benchmark instances.
Category: Set Theory and Logic

[212] viXra:1806.0213 [pdf] submitted on 2018-06-19 14:16:30

Refutation of the Solovay Theorem Copyright © 2008, 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Solovay’s arithmetical completeness theorem for provability logic is refuted by showing the following are not tautologous: Löb's rule as an inference; Gödel's logic system (GL); Gödel's second incompleteness theorem; inconsistency claims of Peano arithmetic (PA); and inability to apply semantical completeness to results which are not contradictory and which are not tautologous.
Category: Set Theory and Logic

[211] viXra:1806.0162 [pdf] submitted on 2018-06-12 23:21:45

The Converse Implication Operator Eqt as a Tense Connective Copyright © 2008, 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 4 Pages. © Copyright 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The converse implication operator named EQT arises to study the inequality in the tense of time for Past > Present > Future. EQT is symmetrically bivalent with the 8-bit pattern {1101 1101} as decimal 187. It is shown that: Past in terms of Present is a falsity; Present in terms of Present is a tautology; and Future in terms of Present is a tautology. Derivations are by Peirce NOR and a 2-tuple truth table. What follows is that tense is not tautologous but non contingent and a truthity. Hence the assumption that time is a theerem is mistaken.
Category: Set Theory and Logic

[210] viXra:1806.0113 [pdf] submitted on 2018-06-09 22:49:48

Refutation of Constructive Brouwer Fixed Point Theorem © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-servcies dot com

We construct the Brouwer fixed point theorem (BFPT) as implications of four variables as the antecedent. Because the consequent is composed of disjunctions of ordered pairs, the totality of ordered combinations requires that the argument connective is equivalence. The result is not tautologous and refutes BFPT using a constructive proof. (If the consequent is taken as a multiplicity of ordered combinations, the equivalence connective and the implication connective share the same table result which deviates further from tautology.) We conclude that BFPT is mislabeled as a theorem, as non constructively based on set theory, and correctly named as the Brouwer fixed point conjecture (BFPC).
Category: Set Theory and Logic

[209] viXra:1806.0109 [pdf] submitted on 2018-06-08 12:57:27

Evaluating f(x) = C for Infinite Set Domains

Authors: Ron Ragusa
Comments: 5 Pages. email: ron.ragusa@gmail.com

In a previous paper, The Function f(x) = C and the Continuum Hypothesis, posted on viXra.org (viXra:1806.0030), I demonstrated that the set of natural numbers can be put into a one to one correspondence with the set of real numbers, f : N → R. In that paper I used the function f(x) = C to create an indexed array of the function’s real number domain d, the constant range, C, and the index value of each iteration of the function’s evaluation, i, for each member of the domain di. The purpose of the exercise was to provide constructive proof of Cantor’s continuum hypothesis which has been shown to be independent of the ZFC axioms of set theory. Because the domain of f(x) = C contains all real numbers, evaluating and indexing the function over the entire domain leads naturally to the bijective function f : N → R. In this paper I’ll demonstrate how the set of natural numbers N can be put into a one to one correspondence the power set of natural numbers, P(N). From this I will derive the bijective function f : N → P(N). Lastly, I’ll propose a conjecture asserting that f(x) = C can be employed to construct a one to one correspondence between the natural numbers and any infinite set that can be cast as the domain of the function.
Category: Set Theory and Logic

[208] viXra:1806.0108 [pdf] submitted on 2018-06-08 15:43:03

Refutation of Two Modern Modal Logics: Jyb4 and Follow-on Ar4 © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

JYB4, a logic system named after its writer Jean-Yves Béziau, is less a logic system and more of a model checking system based on 13 axioms for which p and q are assigned ±1 to evaluate models by arithmetic. AR4, described by its writer Fabian Schang as a doxistic "deviant" logic system, is a follow-on of six more axioms. Of the 19 axioms, 8 or 42% are not tautologous. We conclude that those statistics remove JYB4 and AR4 from further serious consideration as viable modern modal four-valued logics.
Category: Set Theory and Logic

[207] viXra:1806.0030 [pdf] submitted on 2018-06-03 07:41:10

The Function f(x) = C and the Continuum Hypothesis

Authors: Ron Ragusa
Comments: 5 Pages.

This paper examines whether or not an analysis of the behavior of the continuous function f(x) = C, where C is any constant, on the interval (a, b) where a and b are real numbers and a < b, will provide a method of proving the truth or falsity of the CH. The argument will be presented in three theorems and one corollary. The first theorem proves, by construction, the countability of the domain d of f(x) = C on the interval (a, b) where a and b are real numbers. The second theorem proves, by substitution, that the set of natural numbers N has the same cardinality as the subset of real numbers S on the given interval. The corollary extends the proof of theorem 2 to show that N and R are of the same cardinality. The third theorem proves, by logical inference, that the CH is true.
Category: Set Theory and Logic

[206] viXra:1805.0543 [pdf] submitted on 2018-05-31 13:58:46

Refutation of the Löwenheim–Skolem Theorem © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services.com

Refuted in each segment and in toto.
Category: Set Theory and Logic

[205] viXra:1805.0332 [pdf] submitted on 2018-05-19 00:21:07

Set Theory

Authors: Salvatore Gerard Micheal
Comments: 4 Pages.

Objective: to create a precise axiomatic system of exact connected meaningful statements which we can use to build set theory from. The purpose is to make explicit our notion of elements before that notion is used in set equivalence. It is required in a dependency framework. We will find that this system is most economical in terms of repetition and symbol usage. We will state and prove a theorem regarding this.
Category: Set Theory and Logic

[204] viXra:1805.0327 [pdf] submitted on 2018-05-17 13:04:37

On The Mistake That Is Riemann's Prime Zeta Function

Authors: Max Davidović
Comments: 1 Page.

I have succesfully disproved Riemann's hypothesis. It has taken me years of dedicated reseach to realise that such and endeavor is frankly impossible without changing the axioms upon which math is built. The file that I have attached is a shitpost born from my frustration and anger.
Category: Set Theory and Logic

[203] viXra:1805.0302 [pdf] submitted on 2018-05-15 22:37:07

Refutation of Gettier Problem of Justified True/false Belief © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

We find justified true belief is not a theorem. We find justified false belief is also not a theorem. This means the Gettier problem as the superset of the justified belief arguments is refuted as a problem and resolved as a non-problem.
Category: Set Theory and Logic

[202] viXra:1805.0266 [pdf] submitted on 2018-05-13 22:32:32

Many Questions and Many Answers © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services.com

If for some questions there are many answers, then for many questions there are some answers, this is a theorem. If for many questions there are some answers, then for some questions there are many answers, this is not a theorem but a truthity.
Category: Set Theory and Logic

[201] viXra:1805.0249 [pdf] submitted on 2018-05-12 13:16:02

Refutation of the Continuum Hypothesis © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

"The continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers." That equation is not tautologous, and the briefest known refutation of the continuum hypothesis.
Category: Set Theory and Logic

[200] viXra:1805.0228 [pdf] submitted on 2018-05-11 17:01:14

Refutation of Craig's Interpolation Theorem © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services.com

Craig’s theorem (initially referred to by him as a lemma) is not tautologous, hence refuting itself.
Category: Set Theory and Logic

[199] viXra:1805.0223 [pdf] submitted on 2018-05-12 04:55:58

Refutation of the Molyneux Problem © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The equation as rendered is not tautologous. This means the Molyneux problem is resolved as two unrelated states, and hence not a problem.
Category: Set Theory and Logic

[198] viXra:1805.0215 [pdf] submitted on 2018-05-10 23:50:55

Refutation of the Hilbert Grand Hotel Paradox © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The equation as rendered is not contradictory but rather falsity. Hence this refutes the Hilbert Grand Hotel paradox. Remark: We could not reduce this paradox to one variable because rooms and guests are distinctly counted.
Category: Set Theory and Logic

[197] viXra:1805.0189 [pdf] submitted on 2018-05-09 12:10:44

Method of Reducing Paradox to not Contradictory and in One Variable © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

This demonstration serves as two examples to confirm the method that a paradox refuted as not contradictory is also reducible to one variable. The examples are the paradox of Zhuangzi known as the butterfly dream and the paradox of Maimonides known as the argument from freewill.
Category: Set Theory and Logic

[196] viXra:1805.0183 [pdf] submitted on 2018-05-09 20:19:28

Refutation of Fredkin Paradox in One Variable © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

"The more equally attractive two alternatives seem, the harder it can be to choose between them —- no matter that, to the same degree, the choice can only matter less." This is not contradictory, and not tautologous, thereby refuting the Fredkin paradox, and in one variable p as chosen state and alternative state.
Category: Set Theory and Logic

[195] viXra:1805.0158 [pdf] submitted on 2018-05-06 21:33:10

Refutation of Fitch's Paradox of Knowability © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

We find the Fitch paradox to be tautologous, that is, not contradictory, and hence a theorem. We state the theorem as "every truth is discernible", and we add, "by the instant modal logic model checker". A subsequent invocation of Gödel incompleteness and with jettison of the knowability rule to generalize and solve the paradox is contradictory, proving nothing.
Category: Set Theory and Logic

[194] viXra:1805.0155 [pdf] submitted on 2018-05-06 23:17:06

Refutation of Drinker's Paradox © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The drinker's paradox is not tautologous and also not contradictory. Therefore, the drinker's paradox is refuted as a paradox.
Category: Set Theory and Logic

[193] viXra:1805.0150 [pdf] submitted on 2018-05-07 07:40:33

Refutation of the New Riddle of Induction © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Both states of the argument are tautologous, not contradictory, and hence theorems. Therefore the new riddle of induction is refuted as a riddle or paradox.
Category: Set Theory and Logic

[192] viXra:1805.0146 [pdf] submitted on 2018-05-07 10:09:55

Refutation of the Paradox of Epimenides the Cretan © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The argument is: "Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie?" It is not tautologous and not contradictory. Therefore the paradox of Epimenides is refuted as a paradox. The answer to the question "Was this a lie" is neither contradiction nor proof.
Category: Set Theory and Logic

[191] viXra:1805.0138 [pdf] submitted on 2018-05-07 23:05:53

Refutation of the Paradox of Moses Maimonides © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Does God know or does He not know that a certain individual will be good or bad? If thou sayest 'He knows', then it necessarily follows that the man is compelled to act as God knew beforehand he would to act, otherwise God's knowledge would be imperfect. The question is tautologous. The additional sentence with the first is not tautologous, not contradictory. Therefore the paradox of Maimonides is refuted as a paradox.
Category: Set Theory and Logic

[190] viXra:1805.0124 [pdf] submitted on 2018-05-05 23:01:52

Refutation of the Prisoner Paradox© Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The three states of the prisoner paradox are not contradictory. This means the prisoner paradox is not a paradox.
Category: Set Theory and Logic

[189] viXra:1805.0110 [pdf] submitted on 2018-05-05 08:48:59

Refutation of Carroll's Tortoise and Achilles as a Paradox © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

A: "Things that are equal to the same are equal to each other" B: "The two sides of this triangle are things that are equal to the same" Therefore Z: "The two sides of this triangle are equal to each other" The argument as rendered is tautologous and hence a theorem. Because it is not contradictory: this refutes Carroll's tortoise and Achilles as a paradox.
Category: Set Theory and Logic

[188] viXra:1805.0106 [pdf] submitted on 2018-05-05 11:23:45

Refutation of the Paradox of the Concept of an Analysis as Both Correct and Informative © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The concept of an analysis as both correct and informative is not a paradox.
Category: Set Theory and Logic

[187] viXra:1805.0105 [pdf] submitted on 2018-05-05 14:58:45

Refutation of the Newcomb Paradox© Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The two separate states of the Newcomb paradox are not contradictory (and not tautologous), so the paradox is refuted. However interestingly if the states are tested as one system (either or), it is tautologous.
Category: Set Theory and Logic

[186] viXra:1804.0351 [pdf] submitted on 2018-04-25 09:48:21

Refutation of Additive Arithmetic Operations in the Riemann Sphere © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The definition of multiplication for extended complex numbers and undefined values of ∞–∞ and 0×∞ are theorems. The definition of addition for extended complex numbers is not a theorem. The custom of forcing a field definition for extended complex numbers is mistaken as are the undefined values of the quotients 0/0 and ∞/∞.
Category: Set Theory and Logic

[185] viXra:1804.0174 [pdf] submitted on 2018-04-12 13:53:22

The Universe Equation

Authors: Seamus McCelt
Comments: 2 Pages.

If you claim there are particles: there would actually have to be particles.
And that would mean there are about 18 different microscopic things that work flawlessly together -- just like clockwork to make even just one basic atom "gear" set.
If you have larger sized atoms: it would be like throwing more and more gear sets into the clockwork -- but that is ok because no matter what you throw in -- it will still work just fine.
How can an infinity of 18 different things (infinity times 18 different things) just happen to be here, know how work together as a group and also successfully work together as a group(s)?
How is that possible? It isn't...

Stuff cannot be made from what they call "particles."

If there are particles; this is equation of the universe:
Universe = Infinity × {a,b,c,d,f,g,h,j,k,l,m,o,p,q,t,w,x,y,z}
Category: Set Theory and Logic

[184] viXra:1804.0067 [pdf] submitted on 2018-04-04 15:38:26

Particular Solutions for Boolean Algebra Systems

Authors: Franco Sabino Stoianoff Lindstron
Comments: 22 Pages.

Any system of 'big' Boolean equations can be reduced to a single Boolean equation {í µí±”(í µí²) = 1}. We propose a novel method for producing a general parametric solution for such a Boolean equation without attempting to minimize the number of parameters used, but instead using independent parameters belonging to the two-valued Boolean algebra B2 for each asserted atom that appears in the discriminants of the function í µí±”(í µí²). We sacrifice minimality of parameters and algebraic expressions for ease, compactness and efficiency in listing all particular solutions. These solutions are given by additive formulas expressing a weighted sum of the asserted atoms of í µí±”(í µí²), with the weight of every atom (called its contribution) having a number of alternative possible values equal to the number of appearances of the atom in the discriminants of í µí±”(í µí²). This allows listing a huge number of particular solutions within a very small space and the possibility of constructing solutions of desirable features. The new method is demonstrated via three examples over the 'big' Boolean algebras, í µí°µ 4 , í µí°µ 16 , and í µí°µ 256 , respectively. The examples demonstrate a variety of pertinent issues such as complementation, algebra collapse, incremental solution, and handling of equations separately or jointly.
Category: Set Theory and Logic

[183] viXra:1803.0318 [pdf] submitted on 2018-03-19 18:43:19

Refutation of Abductive Reasoning © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Abductive logic of C.S. Peirce is refuted as not tautologous.
Category: Set Theory and Logic

[182] viXra:1803.0180 [pdf] submitted on 2018-03-12 15:52:51

Refutation of the Euathlus Paradox: Neither Pay© Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved.

Regardless of who wins the lawsuit of Portagoras, Euathlus does not pay. Hence the Euathlus paradox is refuted and resolved by default in favor of Euathlus.
Category: Set Theory and Logic

[181] viXra:1803.0094 [pdf] submitted on 2018-03-07 11:14:13

Refutation of Cantor's Original Continuum Hypothesis Via Injection and Binary Trees © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

This is the briefest known such refutation of Cantor's continuum conjecture.
Category: Set Theory and Logic

[180] viXra:1803.0088 [pdf] submitted on 2018-03-07 03:33:06

The Continuum Hypothesis

Authors: Chris Pindsle
Comments: 12 Pages.

A proof of the Continuum Hypothesis as originally posed by Georg Cantor in 1878; that an uncountable set of real numbers has the same cardinality as the set of all real numbers. Any set of real numbers can be encoded by the infinite paths of a binary tree. If the binary tree has an uncountable node it must have a descendant with 2 uncountable successors. Each of those will have descendants with 2 uncountable successors, recursively. As a result the infinite paths of an uncountable binary tree will have the same cardinality as the set of all real numbers, as will the uncountable set of real numbers encoded by the tree.
Category: Set Theory and Logic

[179] viXra:1803.0034 [pdf] submitted on 2018-03-02 17:28:55

Meth8/VŁ4 on Complex Numbers (ℂ) © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

Meth8/VŁ4 maps complex numbers (ℂ) using implication and not equivalence which serves to reason since complex numbers are imaginary and not real.
Category: Set Theory and Logic

[178] viXra:1802.0357 [pdf] submitted on 2018-02-25 07:56:34

P=NP Resolution, with 3-Sat not Tautologous © Copyright 2017-2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. © Copyright 2017-2018 by Colin James III All rights reserved.

We use the definition of NP as “nondeterministic polynomial time” from Stephen Cook at claymath.org/sites/default/files/pvsnp.pdf. It is not tautologous and is presented in a truth table on 1024-values. An example from the same source for the 3-SAT test as NP-complete for the expression (p∨q∨r)∧(~p∨q∨~r)∧(p∨~q∨s)∧(~p∨~r∨~s), with τ(P)=τ(Q)=Tautologous and τ(R)=τ(S)=contradictory, is not tautologous.
Category: Set Theory and Logic

[177] viXra:1802.0329 [pdf] submitted on 2018-02-22 14:00:52

Refutation of the ef-Axiom © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

The EF-axiom describes the Efremovifč proximity δ by V.A. Efremovič from 1934 and published in Russian in 1951. We interpret the operator δ to mean "nearby" or "in proximity". The size of an antecedent or consequent is not stated for the operator, so we determine that the operator applies to unrelated literals as as ((A∈ B) Nor (B∈ A)). The proof result is for 256-values because four theorems are evaluated as variables. We find the EF-axiom is not tautologous. The implications to topology are legion.
Category: Set Theory and Logic

[176] viXra:1802.0306 [pdf] submitted on 2018-02-21 22:19:21

Refutation of Higher-Order Logic as Bivalent © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

We evaluate higher-order logic based on the principle of mathematical induction. Meth8/VŁ4 treats sets and variables as variables. The quantification over quantification is not bivalent. We alleviate this constraining condition by distributing the quantified expression over nested expressions. At each nested level, the quantification is explicitly distributed for clarity. We conclude that higher-order logic is not bivalent and that nested quantification is better expressed as explicitly distributed.
Category: Set Theory and Logic

[175] viXra:1802.0172 [pdf] submitted on 2018-02-14 06:16:03

Refutation of Tarski's Undefinability of Truth Theorem © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

This refutes Tarski's theorem for the undefinability of truth as: "no definable and sound extension of Peano Arithmetic can be complete"; or in abstract terms, "the proof of a system cannot be demonstrated by itself". Tarski's theorem is an arguable equivalent to Godel's incompleteness theorem, as based on the liar's paradox. [Remark added later: Tarski's theorem as used since about 1936 is an underpinning of quantum theory and a universal justification for atheism.]
Category: Set Theory and Logic

[174] viXra:1801.0380 [pdf] submitted on 2018-01-27 22:55:45

Refutation of the Direct Correspondence of Quantum Gates to Reversible Classical Gates © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

These quantum gates do not directly correspond to reversible classical gates: CNOT (XOR, Feynman); Toffoli (AND); X (NOT); and n-qubit Toffoli (AND). Hence quantum gates cannot map to bivalent logic. This paper demonstrates the shortest refutation.
Category: Set Theory and Logic

[173] viXra:1801.0253 [pdf] submitted on 2018-01-19 16:14:43

Refutation of Neutrosophic Soft Lattice Theory © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. Papers by and relating to Florentin Smaradnache span many areas at viXra, but the appropriate field is Set Theory and Logic. Many of them are translated as crude ruse to garner publicity. The study neutrosohic logic is mostly specious.

We evaluate the neutrosophic logic based on its most atomic level of soft latices, as published by Springer-Verlag in 2016. We refute the theorem "Every neutrosophic soft lattice is a one-sided distributive neutrosophic soft lattice." This brief evaluation implies that the field of soft set theory as originally introduced by D. Molodtsov is suspicious and specifically that the field of neutrosophic logic, as evidenced in its basis of soft set theory, is unworkable. This conclusion is multitudinal because of the plethora of duplicated papers as translations in multiple fields at vixra.org regarding the neutrosophic logic system of Florentin Smarandache.
Category: Set Theory and Logic

[172] viXra:1801.0122 [pdf] submitted on 2018-01-11 01:52:00

The Incalculable Continuum

Authors: Miguel A. Sanchez-Rey
Comments: 1 Page.

A continuum in sight.
Category: Set Theory and Logic

[171] viXra:1712.0600 [pdf] submitted on 2017-12-26 09:29:23

ZF Law of Excluded Middle on Infinite Sets (Lemi) © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved

Eqs. 1.2 and 1.4 as rendered are not tautologous. Hence Meth8/VL4 finds LEMI suspicious.
Category: Set Theory and Logic

[170] viXra:1712.0595 [pdf] submitted on 2017-12-26 00:23:47

The Formal-Logical Analysis of the Foundation of Set Theory

Authors: Temur Z. Kalanov
Comments: 17 Pages.

The critical analysis of the foundation of set theory is proposed. The unity of formal logic and rational dialectics is the correct methodological basis of the analysis. The analysis leads to the following results: (1) the mathematical concept of set should be analyzed on the basis of the formal-logical clauses “Definition of concept”, “Logical class”, “Division of concept”, “Basis of division”, “Rules of division”; (2) the standard mathematical theory of sets is an erroneous theory because it does not contain definition of the concept “element (object) of set”; (3) the concept of empty set (class) is a meaningless, erroneous, and inadmissible one because the definition of the concept “empty set (class)” contradicts to the definition of the logical class. (If the set (class) does not contain a single element (object), then there is no feature (sign) of the element (object). This implies that the concept of empty set (class) has no content and volume (scope). Therefore, this concept is inadmissible one); (4) the standard mathematical operations of union, intersection and difference of sets (classes) are meaningless, erroneous and inadmissible operations because they do not satisfy the following formal-logical condition: every separate element (object) of the set (class) must be in only one some set (class) and cannot be in two sets (classes). Thus, the results of formal-logical analysis prove that the standard mathematical theory of sets is an erroneous theory because it does not satisfy the criterion of truth.
Category: Set Theory and Logic

[169] viXra:1712.0403 [pdf] submitted on 2017-12-13 04:09:51

There is no Standard Model of ZFC

Authors: Jaykov Foukzon
Comments: 15 Pages.

Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st), (ii) let k be an inaccessible cardinal then  ~Con(ZFC+∃k),[10]-[11].
Category: Set Theory and Logic

[168] viXra:1712.0386 [pdf] submitted on 2017-12-12 01:26:41

On Multi-Criteria Pythagorean Fuzzy Decision-Making

Authors: Liguo Fei, Yong Deng
Comments: 21 Pages.

Pythagorean fuzzy set (PFS) initially extended by Yager from intuitionistic fuzzy set (IFS), which can model uncertain information with more general conditions in the process of multi-criteria decision making (MCDM). The fuzzy decision analysis of this paper is mainly based on two expressions in Pythagorean fuzzy environment, namely, Pythagorean fuzzy number (PFN) and interval-valued Pythagorean fuzzy number (IVPFN). We initiate a novel axiomatic definition of Pythagorean fuzzy distance measure, including PFNs and IVPFNs, and put forward the corresponding theorems and prove them. Based on the defined distance measures, the closeness indexes are developed for PFNs and IVPFNs inspired by the idea of technique for order preference by similarity to ideal solution (TOPSIS) approach. After these basic definitions have been established, the hierarchical decision approach is presented to handle MCDM problems under Pythagorean fuzzy environment. To address hierarchical decision issues, the closeness index-based score function is defined to calculate the score of each permutation for the optimal alternative. To determine criterion weights, a new method based on the proposed similarity measure and aggregation operator of PFNs and IVPFNs is presented according to Pythagorean fuzzy information from decision matrix, rather than being provided in advance by decision makers, which can effectively reduce human subjectivity. An experimental case is conducted to demonstrate the applicability and flexibility of the proposed decision approach. Finally, the extension forms of Pythagorean fuzzy decision approach for heterogeneous information are briefly introduced as the further application in other uncertain information processing fields.
Category: Set Theory and Logic

[167] viXra:1712.0368 [pdf] submitted on 2017-12-09 22:30:01

The Brain Simulator Reply (BSR) of the Chinese Room Argument (Cra) is Confirmed. © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

"[O]one cannot infer from X simulates Y, and Y has property P, to the conclusion that therefore X has Y's property P for arbitrary P." is tautologous. "The contrapositive of the inference is logically equivalent—X simulates Y, X does not have P therefore Y does not [have P]" is not tautologous. The two conjectuses are not logically equivalent. Hence the brain Simulator reply of the Chinese room argument is confirmed and validated.
Category: Set Theory and Logic

[166] viXra:1712.0205 [pdf] submitted on 2017-12-06 16:42:00

Refutation of Cantor's Diagonal Argument © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We map the argument for Cantor's diagonal argument into the Meth8 modal logic model checker. The two main equations as rendered are not tautologous. Hence Cantor's diagonal argument is not supported.
Category: Set Theory and Logic

[165] viXra:1712.0204 [pdf] submitted on 2017-12-06 17:11:12

Refutation of Axiom of Choice Via Refutation of the Gödel-Löb Axiom © Copyright 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016 by Colin James III All rights reserved.

Refutation of the Gödel-Löb Axiom implies the refutation of the Axiom of Choice. Both axioms as separately rendered are also not tautologous.
Category: Set Theory and Logic

[164] viXra:1712.0139 [pdf] submitted on 2017-12-06 12:44:00

A Proof of the Falsity of the Axiom of Choice.

Authors: Johan Noldus
Comments: 1 Page.

We show that the axiom of choice is false.
Category: Set Theory and Logic

[163] viXra:1711.0473 [pdf] submitted on 2017-11-29 13:25:59

Unification by Neutrosophic Logic not Tautologous © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2017 by Colin James III All rights reserved.

We evaluate the two axioms with attendant rules which define neutrosophic logic. Both produce the same proof table which is not tautologous. What follows is that neutrosophic logic is not bivalent, but a vector space, and cannot unify other logics in a tautology.
Category: Set Theory and Logic

[162] viXra:1711.0462 [pdf] submitted on 2017-11-28 22:51:13

Definition of the Zero Knowledge Proof Refuted © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

The formal definition of the zero-knowledge proof as rendered is not tautologous. What follows is the assumption that in NP all problems and all languages have zero-knowledge proofs is mistaken. What also follows is that one-way functions do not exist.
Category: Set Theory and Logic

[161] viXra:1711.0425 [pdf] submitted on 2017-11-25 13:35:07

Refutation of Realizability Semantics for QML © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages.

We found that the following are not tautologous: Epistemic Church's Thesis; EZF induction schema; and Scedrov's modal foundation. We did not test subsequent axioms. What follows is that Flagg's construction, Goodman's intensional set theory, and epistemic logic are suspicious.
Category: Set Theory and Logic

[160] viXra:1711.0416 [pdf] submitted on 2017-11-25 11:03:31

Stone Space Type Lattice Logic Model © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

Injection of a binary relation does not support it as symmetric and irreflexive.
Category: Set Theory and Logic

[159] viXra:1711.0412 [pdf] submitted on 2017-11-24 21:11:36

Refutation of the Universal Finite Set © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We evaluated two parts of the proof of Lemma 2 (Folklore). The equations as rendered are not tautologous.
Category: Set Theory and Logic

[158] viXra:1711.0378 [pdf] submitted on 2017-11-20 08:04:42

Refutation of Chaitin's Theorem of Incompleteness © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages.

We evaluate Chaitin's incompleteness theorem of 1974. Martin Davis described it as “a dramatic extension of Gödel’s incompleteness theorem” in 1978. We find the approach of the conjecture is moot, refute Chaitin's theorem of incompleteness, and remark that Chaitin's constant is suspicious.
Category: Set Theory and Logic

[157] viXra:1711.0364 [pdf] submitted on 2017-11-19 10:38:24

The Shortest Refutation of Gödel's Theorem of Incompleteness © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2017 by Colin James III All rights reserved.

Our example in the positive and contra example show the shortest known definitive refutation for Gödel's incompleteness theorem.
Category: Set Theory and Logic

[156] viXra:1711.0357 [pdf] submitted on 2017-11-18 08:01:42

Note on Grilliot's Trick and Effective Implication © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We evaluate two of the simpler equations for tautology which relate to Grilliot's trick and the standard extensionality trick via effective implication. Eqs., as rendered, are not tautologous. According to variant sysem VŁ4, this means Grilliot's trick, effective implication, and the subsequent non standard extensionality trick are not bivalent, but rather are a vector space.
Category: Set Theory and Logic

[155] viXra:1711.0323 [pdf] submitted on 2017-11-15 14:07:02

Logic not Tautologous in Neutrosophic Sets © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We test a theorem and two properties from above. Eqs. 1.1, 5.2, and 6.1 should be tautologous, but are not.
Category: Set Theory and Logic

[154] viXra:1711.0320 [pdf] submitted on 2017-11-15 18:23:23

Law of Self-Equilibrium: not Law; not Paradox © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

The law of self-equilibrium sometimes uses this example: Too much work produces sickness; sickness produces less work; therefore, too much work implies less work. We rewrite the sentence to replace the connective verb with "causes" for better meaning and also include a modal operator for clarity: Too much work causes possible sickness; sickness causes less work; therefore, too much work causes less work. ... the law of self-equilibrium is not tautologous, and hence not a theorem and not a paradox.
Category: Set Theory and Logic

[153] viXra:1711.0312 [pdf] submitted on 2017-11-15 06:04:19

Re: Deducibility Theorems in Boolean Logic Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, Usa e-Mail: Smarand@unm.edu Http://vixra.org/abs/1003.0171

Authors: Colin James III
Comments: 1 Page.

As presumably a basis for neutrosophic logic these mistakes were found: Theorems 1 and 2 are not tautologous.
Category: Set Theory and Logic

[152] viXra:1711.0290 [pdf] submitted on 2017-11-12 12:05:19

Rationale of Rendering Quantifiers as Modal Operators © 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 14 Pages. © 2016 by Colin James III All rights reserved.

The rationale for rendering quantifiers as modal operators in Meth8 has arguments from satisfiability and reproducability of invalidating and validating syllogisms. The Square of Opposition (original) produced four combinations for each corner A, I, E, O for 4 ^ 4 = 256 syllogisms. Medieval scholars determined 24 of the 256 syllogisms were valid deductions. Of those, 9 were made valid but only after additional known assumptions were applied as fix ups. Meth8 Tautologous none of the 24 syllogisms before fix ups. Meth8 also discovered correct additional assumptions to render the other 15 syllogisms Tautologous. We use Meth8 to replicate the 24 valid syllogisms derived from the original Square of Opposition. In the process we make three recent advances. 1. A third assumption is needed to fix up Modus Cesare EAE-2 2. The third assumption cannot be removed from Modus Camestros AEO-2 (as in other syllogisms with known third assumptions); and 3. No third assumptions are required for the other 22 syllogisms.
Category: Set Theory and Logic

[151] viXra:1711.0289 [pdf] submitted on 2017-11-12 12:09:05

Square of Opposition Meth8 Corrected © 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © 2016 by Colin James III All rights reserved.

The modern revision of the square of opposition is not validated as tautologous by the Meth8 logic model checker, as based on system variant VŁ4. Consequently we redefine the square so that it is validated as true my Meth8. Instead of definientia using implication for universal terms or conjunction for existential terms, we adopt the equivalent connective for all terms. The modal modifiers necessity and possibility map quantifiers as applying to the entire terms rather than to the antecedent within the terms. We note the validating connectives for the edges on the square are: \ Nand for the Contraries and Contradictories; > Imply for the Subalterns; and + Or for the Subcontraries.
Category: Set Theory and Logic

[150] viXra:1711.0288 [pdf] submitted on 2017-11-12 12:11:20

Square of Opposition Modern Revised: not Validated as Tautologous © 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © 2016 by Colin James III All rights reserved.

The modern, revised square of opposition is not validated as tautologous by the Meth8 logic checker in five models for all expressions. This leads us to consider that any logic system based on the square of opposition is spurious. What follows then is that a first order predicate logic based on the square of opposition is now suspicious.
Category: Set Theory and Logic

[149] viXra:1711.0271 [pdf] submitted on 2017-11-10 16:06:05

Rule of Necessitation: True, But not Tautologous © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages.

1. The axiom or rule of necessitation N states that if p is a theorem, then necessarily p is a theorem: If ⊢ p then ⊢ ◻p. We show this is non-contingent (a truth), but not tautologous (a proof). We evaluate axioms (in bold) of N, K, T, 4, B, D, 5 to derive systems (in italics) of K, M, T, S4, S5, D. We conclude that N the axiom or rule of necessitation is not tautologous Because system M as derived and rendered is not tautologous, system G-M also not tautologous. What follows is that systems derived from using M are tainted, regardless of the tautological status of the result so masking the defect, such as systems S4, B, and S5. We also find that Gentzen-sequent proof is suspicious, perhaps due to its non bi-valent lattice basis in a vector space.
Category: Set Theory and Logic

[148] viXra:1711.0113 [pdf] submitted on 2017-11-03 07:20:31

Tautology Problem and Two Dimensional Formulas

Authors: Deniz Uyar
Comments: 38 Pages.

Finding whether a boolean formula is a tautology or not in a feasible time is an important problem of computer science. Many algorithms have been developed to solve this problem but none of them is a polynomial time algorithm. Our aim is to develop an algorithm that achieve this in polynomial time. In this article, we convert boolean functions to some graph forms in polynomial time. They are called two dimensional formulas and similar to AND-OR graphs except arches on them are bidirectional. Then these graphs are investigated to find properties which can be used to differentiate tautological formulas from non tautological ones.
Category: Set Theory and Logic

[147] viXra:1710.0309 [pdf] submitted on 2017-10-28 12:24:53

Alpha-Conversion for Lambda Terms with Explicit Weakenings

Authors: George Cherevichenko
Comments: 15 Pages.

Using explicit weakenings, we can define alpha-conversion by simple equations without any mention of free variables.
Category: Set Theory and Logic

[146] viXra:1710.0237 [pdf] submitted on 2017-10-21 10:32:06

Fundamental Set Theory

Authors: Anders Lindman
Comments: 3 Pages.

The fundamental set theory (FST) is defined as an axiomatic set theory using nonclassical three-valued logic in the foundation and classical two-valued logic in its applications. In this way the nonclassical logic becomes encapsulated and is only used for resolving inconsistencies such as Russell's paradox.
Category: Set Theory and Logic

[145] viXra:1710.0226 [pdf] submitted on 2017-10-20 08:20:23

Do The Real Numbers form a Set?

Authors: Divyendu Priyadarshi
Comments: 2 Pages.

I have argued that the Real Numbers do not form a set in the sense that they lack any specific character to define them. I am not a professional mathematician but a Physics teacher. So my arguments may lack mathematical precision. All suggestions and criticisms are heartily welcome.
Category: Set Theory and Logic

[144] viXra:1710.0223 [pdf] submitted on 2017-10-19 23:09:13

The Set of Standard Numbers

Authors: Anders Lindman
Comments: 3 Pages.

The set of standard numbers D is constructed from an axiom of infinite fraction sum together with the power set of the set of all rational numbers in the form 2^-n. The power set contains all possible infinite binary sequences who represent the fraction part of the standard numbers together with the integers for the whole number part. The set of standard numbers includes the rational numbers and forms a field (D, +, *) similar to, yet distinct from the set of real numbers R.
Category: Set Theory and Logic

[143] viXra:1710.0035 [pdf] submitted on 2017-10-04 01:50:21

The Statistical Proof of the Riemann Hypothesis

Authors: Dmitri Martila
Comments: 4 Pages.

Derived the Statistics of the un-solved problems (conjectures). The probability, what a conjecture will be solved is 50 %. The probability, that a conjecture is true is p=37 %. The probability, what we get to know the latter is psi=29 %....
Category: Set Theory and Logic

[142] viXra:1709.0391 [pdf] submitted on 2017-09-26 10:31:21

A New Axiom for ZFC Set Theory that Results in a Problem

Authors: Andew Banks
Comments: 5 Pages.

This article adds a new axiom to ZFC that assumes there is a set x which is initially the empty set and thereafter the successor function (S) is instantly applied once in-place to x at each time interval (½ⁿ n>0) in seconds. Next, a very simple question is proposed to ZFC. What is x after one second elapses? By definition, each time S is applied in-place to x, a new element is inserted into x. So, given that S is applied at each time interval (½ⁿ n>0) then an infinite collection of elements is added to x so, x is countable infinite. On the other hand, since x begins as the empty set and only S is applied to x then x cannot be anything other than a finite natural number. Hence, x is finite. Clearly, in-place counting according to the interval timings (½ⁿ n>0) demonstrates a problem in ZFC
Category: Set Theory and Logic

[141] viXra:1709.0076 [pdf] submitted on 2017-09-07 11:08:35

P Versus Np's Answer.

Authors: Gokulakannan.P
Comments: 2 Pages. Please give comments. For further enquiry mail @ sbsp181107@gmail.com.

Always think simple to answer a question which is being seem tough.
Category: Set Theory and Logic

[140] viXra:1708.0156 [pdf] submitted on 2017-08-14 10:23:58

Some Critical Notes on the Cantor Diagonal Argument

Authors: Philip Molyneux
Comments: 10 Pages.

This paper critically examines the Cantor Diagonal Argument (CDA) that is used in set theory to draw a distinction between the cardinality of the natural numbers and that of the real numbers. In the absence of a verified English translation of the original 1891 Cantor paper from which it is said to be derived, the CDA is discussed here using a consensus from the forms found in a range of published sources (from "popular" to "professional"). Some general comments are made on these sources. The discussion then focusses on the CDA as applied to the correspondence between the set of the natural numbers, and the set of real numbers in the open range (0,1) in their expansion from decimal digits (0.123… etc.). Four points critical of the CDA are raised: (1) The conventional presentation of the CDA forms a putative new real number (X) from the "diagonal" of the chosen list of real numbers and which is therefore not on this initial list; however, it omits to consider that it may indeed be on the later part of the list, which is never exhausted however far the "diagonal" list is extended. (2) This aspect, combined with the fact that X is still composed of decimal digits, that is, it is a real number as defined, indicates that it must be on the later part of the list, that is, it is not a "new" number at all. (3) The conventional application of the CDA leads to one putative "new" real number (X); however, the logical extension of this in its "exhaustive" application, that is, by using all possible different methods of alteration of the decimal digits on the "diagonal", and by reordering the list in all possible ways, leads to a list of putative "new" real numbers that become orders of magnitude longer than the chosen "diagonal" list. (4) The CDA is apparently considered to be a method that is applicable generally; however, testing this applicability with the natural numbers themselves leads to a contradiction. Following on from this, it is found that it indeed is possible to set up a one-to-one correspondence between the natural numbers and the real numbers in (0,1), that is, ! ⇔ "; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension of the natural number is intended to be a mirror image of the left hand extension of the real number. It is also shown how this may be extended to real numbers outside the range (0,1). Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998 critical review of "hopeless papers" dealing with the CDA; this form is also examined from the same viewpoints, and to the same conclusions. Finally, some comments are made on the concept of "infinity", pointing out that to consider this as an entity is a category error, since it simply represents an absence, that is, the absence of a termination to a process.
Category: Set Theory and Logic

[139] viXra:1707.0220 [pdf] submitted on 2017-07-16 10:14:11

Proof of ZFC Axioms as Normal Statements No. 2.2

Authors: Thomas Limberg
Comments: 6 Pages. Language: German

We interpret 5 of the 10 axioms of ZFC (Zermelo-Fraenkel set theory with axiom of choice) as normal statements and proof them. So these 5 sentences don't need to be introduced as axioms, but can be used as proven statements.
Category: Set Theory and Logic

[138] viXra:1706.0321 [pdf] submitted on 2017-06-12 04:00:23

Aristotle’s Answer on Russell’s Paradox

Authors: Nikolay Dementev
Comments: 9 Pages.

An attempt of resolving Russell’s paradox with the help of Aristotle’s ideas is presented.
Category: Set Theory and Logic

[137] viXra:1705.0226 [pdf] submitted on 2017-05-14 17:15:50

Is Classical Mathematics Appropriate for Theory of Computation?

Authors: Farzad Didehvar
Comments: 9 Pages.

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”, first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved. .
Category: Set Theory and Logic

[136] viXra:1705.0173 [pdf] submitted on 2017-05-10 12:07:14

Dialectical Logic – Negation Of Classical Logic

Authors: Ilija Barukčić
Comments: 23 Pages. Copyright © 2017 by Ilija Barukčić, Jever, Germany. Published by:

The division of zero by zero turns out to be a long lasting and not ending puzzle in mathematics and physics. An end of this long discussion is not in sight. In particular zero divided by zero is treated as indeterminate thus that a result cannot be found out. It is the purpose of this publication to solve the problem of the division of zero by zero while relying on the general validity of classical logic. According to classical logic, zero divided by zero is one.
Category: Set Theory and Logic

[135] viXra:1704.0161 [pdf] submitted on 2017-04-13 03:51:02

The Number Density Paradox

Authors: Ken Seton
Comments: 6 Pages. Enjoy

In the world of transfinite cardinality, any talk of number density, dart-board hits, proportions or probability is just a pouring from the empty into the void. Transfinite cardinality is unrelated to any normal concept of proportion or density over an interval and no understanding of it can obtained from probabilistic analogies. This is demonstrated by comparing extremely sparse and extremely dense collections of reals which are normally understood as uncountable and countable respectively. Should this incline one to think that Cantor’s equivalence definition is inappropriate for identifying the “size” of an infinite collection ?
Category: Set Theory and Logic

[134] viXra:1704.0115 [pdf] submitted on 2017-04-10 01:13:52

The Simple Infinite Set

Authors: Ken Seton
Comments: 8 Pages. Enjoy

Many have suggested that the infinite set has a fundamental problem. The usual complaint rails against the actually infinite which (to critics of various finitist persuasions) unjustifiably goes beyond the finite. Here we observe the exact opposite. The problem of the infinite set defined to have an identity (content) that is specified and restricted to be forever finite . Set theory is taken at its word. The existence of the infinite set and the representation of irrational reals as infinite sets of terms is accepted. In this context, it is shown that the standard definition of the infinite countable set is inconsistent with the existence of its own classic convergents of construction. If the set is infinite then it must be quite unlike that which set theory asserts it to be. Set theory found itself into some trouble over a century ago trusting an unrestricted anthropic comprehension. But serious doubt is cast on the validity of infinite sets which have been defined by a comprehension which overly-restricts their content.
Category: Set Theory and Logic

[133] viXra:1704.0008 [pdf] submitted on 2017-04-01 18:47:38

The Last Slice of Cake

Authors: Igor Hrncic
Comments: 2 Pages.

This letter is the short continuation of the previous paper titled "The infinitesimal error", available for free at the internet address http://vixra.org/abs/1703.0280. This letter is written just to further clarify the subject of "The infinitesimal error".
Category: Set Theory and Logic

[132] viXra:1703.0280 [pdf] submitted on 2017-03-29 22:47:37

The Infinitesimal Error

Authors: Igor Hrncic
Comments: 6 Pages.

Unfortunately, Cantor was wrong. His notion of transfinite bijection is flawed. Cantor introduced this notion of transfinite bijection as the additional axiom, even though without even realising this. From this error, other errors sprung into the existence. He did all this in the heroic effort to justify the death of infinitesimals, even though he wasn't aware of this either. Cantor went bravely on to defend the established error in higher mathematics before his mentors and peers who banished infinitesimals. Instead, he demonstrated the error of it. He never realised this as well. This paper elucidates this link between Cantor's errors and infinitesimals.
Category: Set Theory and Logic

[131] viXra:1703.0113 [pdf] submitted on 2017-03-13 04:06:52

Failure of the Diagonal Argument

Authors: Wolfgang Mückenheim
Comments: 3 Pages.

It is shown that Cantor's diagonal argument fails because either there is no actual infinity and hence no defined diagonal number or there is actual infinity but the diagonal number cannot be distinguished from all real numbers of the Cantor list. Further it is shown by another argument that there are not uncountably many paths in the complete infinite Binary Tree.
Category: Set Theory and Logic

[130] viXra:1703.0112 [pdf] submitted on 2017-03-13 04:15:07

Set Theory or Slipper Animalcule: Who Wins?

Authors: Wolfgang Mückenheim
Comments: 3 Pages.

Limits of sequences of sets required to define infinite bijections do not only raise paradoxes but cause self-contradictory results.
Category: Set Theory and Logic

[129] viXra:1703.0032 [pdf] submitted on 2017-03-03 16:07:51

The Union is not the Limit.

Authors: W. Mückenheim
Comments: 5 Pages.

Contrary to the assumptions of transfinite set theory, limit and union of infinite sequences of sets differ. We will show this for the set of natural numbers by the newly devised powerful tool of arithmogeometry.
Category: Set Theory and Logic

[128] viXra:1702.0293 [pdf] submitted on 2017-02-24 03:55:26

Not Enumerating All Positive Rational Numbers

Authors: W. Mückenheim
Comments: 2 Pages.

It is shown that the enumeration of rational numbers cannot be complete.
Category: Set Theory and Logic

[127] viXra:1701.0564 [pdf] submitted on 2017-01-22 04:25:16

The Declaration of Unity and Union

Authors: Amir Deljoo
Comments: 11 Pages.

This is a declaration. The identity of mathematics and number theory or arithmetics. I have defined a pattern here that shows consciousness is a pure unique entity that is present everywhere and whole the existence is a graphical manifestation that has been phenomenoned over to enclose it and I hermetically simplify my intuition to transfer it to curious ones. Since explaining the methodology requires in thousands of pages, the final concluded statements and equations are only declared here.
Category: Set Theory and Logic

[126] viXra:1701.0563 [pdf] submitted on 2017-01-22 04:47:06

بیانیه ی یگانه و انجمن

Authors: امیر دلجو
Comments: 12 Pages.

این سند یک منشور است برای بیان ماهیت ریاضیات و نظریه ی اعداد یا حساب. در اینجا من الگویی را تعریف کرده ام که نشان می دهد، خودآگاهی یک وجود واحده بسیط و همه جاحاضر بوده و هستی به مثابه یک کلیّت، یک تجلّی گرافیکی ست که در جهت افشای این خودآگاهی عارض شده و من هرمس وار، شهود خود بر یگانگی و آفرینش را برای انتقال به انسان های کنجکاو ساده سازی و تحریر کرده ام. از آنجا که تبیین روش شناختی این منشور مستلزم هزاران صفحه است، در اینجا تنها عبارات و معادلات منتج شده ی نهایی اعلان می شود.
Category: Set Theory and Logic

[125] viXra:1701.0328 [pdf] submitted on 2017-01-07 23:13:35

A Thing Exists If It Is A Grouping Defining What Is Contained Within: Application to The Russell Paradox and Godel's Incompleteness Theorem

Authors: Roger Granet
Comments: 3 Pages.

    The Russell Paradox (1) considers the set, R, of all sets that are not members of themselves.   On its surface, it seems like R belongs to itself only if it doesn't belong to itself.  This is where the paradox come from.  Here, a solution is proposed that is similar to Russell's method based on his theory of types (1,2) but is instead based on the definition of why things exist as described in previous work (3).  In that work, it was proposed that a thing exists if it is a grouping defining what is contained within. A corollary is that a thing, such as a set, does not exist until what is contained within is defined. A second corollary is that after a grouping defining what is contained within is present, and the thing exists, if one then alters the definition of what is contained within, the first existent entity is destroyed and a different existent entity is created. Based on this, set R of the Russell Paradox does not even exist until after the list of the elements it contains (e.g. the list of all sets that aren't members of themselves) is defined.  Once this list of elements is completely defined, R then springs into existence. Therefore, because it doesn't exist until after its list of elements is defined, R obviously can't be in this list of elements and, thus, cannot be a member of itself; so, the paradox is resolved.  Additionally, one can't then put R back into its list of elements after the fact because if this were done, it would be a different list of elements, and it would no longer be the original set R, but some new set. This same type of reasoning is then applied to the Godel Incompleteness Theorem, which roughly states that there will always be some statements within a formal system of arithmetic (system P) that are true but that can't be proven to be true. Briefly, this reasoning suggests that arguments such as the Godel sentence and diagonalization arguments confuse references to future, not yet existent statements with a current and existent statement saying that the future statements are unprovable. Current and existent statements are different existent entities than future, not yet existent statements and should not be conflated. In conclusion, a new resolution of the Russell Paradox and some issues with the Godel Incompleteness Theorem are described.
Category: Set Theory and Logic

[124] viXra:1612.0286 [pdf] submitted on 2016-12-17 23:42:11

Thought Experiments About Infinite Sets and Subsets Produce Experimental Artifacts: Relationship to Their Use in Physics

Authors: Roger Granet
Comments: 4 Pages.

Here, the conclusion in set theory that the size of an infinite set is the same as the size of an infinite subset derived from it is questioned.This is done not to try and invalidate any mathematical results because mathematics is an abstract field and does not necessarily have to accurately describe the physical world but in order to prompt the reexamination of the use of this result in physics, which does have to accurately describe the real, physical world and the relationships between its components. The rationale is as follows. First, it is suggested that thought experiments are still experiments and should follow the rules for good experimental technique, which include the need to study a system in a setting as close as possible to the "natural setting" to try and avoid experimental artifacts. Now, starting with the single set of the positive integers, one wants to compare the total number of integers to the total number of even integers within the "natural setting" of the single original set. The traditional experimental processing method extracts the even integers, puts them into a separate subset and pairs off the subset's and set's members one-to-one with a function. After doing this, no elements are left over, and, therefore, the original set and the subset extracted from it are said to be the same size. However, extracting the evens and putting them into a separate subset dramatically alters the original single set system. This is analogous to a biologist extracting the nucleus from a cell, studying the nucleus and remaining parts of the cell in isolation and assuming that the results obtained are the same as in the original intact cell. They often are not. Does extracting the even integers out into a subset alter the results compared to those that would be obtained in the natural single set system? Yes. In the single set system, the positive integers march lockstep and in- phase with the odd integers from one to infinity, meaning that there is a built-in relationship in this system of one positive integer for every two total integers, which means that there are only one-half as many positive integers as total integers. This is a different result than that obtained after the subset extraction method, which means that the result produced by this method is an experimental artifact. This should be unacceptable in a well done experiment even if it is a thought experiment. It is suggested that this artifact may be related to some of the problems associated with infinities in physics.
Category: Set Theory and Logic

[123] viXra:1611.0415 [pdf] submitted on 2016-11-30 16:52:52

Propositional Logic: An Extension Nary Anthropic

Authors: Arthur Shevenyonov
Comments: 6 Pages. new foundations

The proposed extension of propositional logic exhibits a striking similarity to generalized games with a chance player while pointing to accidental applications in quantum computing.
Category: Set Theory and Logic

[122] viXra:1611.0330 [pdf] submitted on 2016-11-24 04:20:28

Ultra Neutrosophic Crisp Sets and Relations

Authors: Hewayda Elghawalby, A. A. Salama
Comments: 8 Pages.

In this paper we present a new neutrosophic crisp family generated from the three components’ neutrosophic crisp sets presented by Salama [4]. The idea behind Salam’s neutrosophic crisp set was to classify the elements of a universe of discourse with respect to an event ”A” into three classes: one class contains those elements that are fully supportive to A, another class contains those elements that totally against A, and a third class for those elements that stand in a distance from being with or against A. Our aim here is to study the elements of the universe of discourse which their existence is beyond the three classes of the neutrosophic crisp set given by Salama. By adding more components we will get a four components’ neutrosophic crisp sets called the Ultra Neutrosophic Crisp Sets. Four types of set’s operations is defined and the properties of the new ultra neutrosophic crisp sets are studied. Moreover, a definition of the relation between two ultra neutrosophic crisp sets is given.
Category: Set Theory and Logic

[121] viXra:1611.0281 [pdf] submitted on 2016-11-19 11:31:44

An Introduction to F-Notation and the Prove of the Cartesian Product of Natural Number is Countably Infinite

Authors: Damodar Rajbhandari
Comments: 3 Pages.

This paper will introduce a new notation named as F-notation. This notation will help us to prove the statement, "The cartesian product of natural numbers is countably infinite".
Category: Set Theory and Logic

[120] viXra:1611.0079 [pdf] submitted on 2016-11-06 06:48:50

Sets, Formulas and Electors

Authors: Max Null, Sergey Belov
Comments: 4 Pages.

This article is a mathematical experiment with the sets and the formulas. We consider new elements which are called the electors. The elector has the properties of the sets and the formulas.
Category: Set Theory and Logic

[119] viXra:1608.0395 [pdf] submitted on 2016-08-29 10:19:42

The Topology on a Complete Semilattice

Authors: Max Null, Sergey Belov
Comments: 12 Pages.

We define the topology atop(χ) on a complete upper semilattice χ = (M, ≤). The limit points are determined by the formula Lim(D,X) = sup{a ∈ M| {x ∈ X| a ≤ x} ∈ D}, where X ⊆ M is an arbitrary set, D is an arbitrary non-principal ultrafilter on X. We investigate Lim(D,X) and topology atop(χ) properties. In particular, we prove the compactness of the topology atop(χ).
Category: Set Theory and Logic

[118] viXra:1608.0057 [pdf] submitted on 2016-08-05 07:48:37

Curry's Non-Paradox and Its False Definition

Authors: Adrian Chira
Comments: 4 Pages.

Curry's paradox is generally considered to be one of the hardest paradoxes to solve. However, it is shown here that the solution is however trivial and the paradox turns out to be no paradox at all. Reviewing the starting point of the paradox, it is concluded that it amounts to a false definition or assertion and therefore it is to be expected, as opposed to being paradoxical, to arrive to a false conclusion. Despite that fact that verifying the truth value of the first statement of the paradox is trivial, mathematicians and logicians have failed to do so and merely assumed that it is true. Taking this into consideration that it is false, the paradox is however dismissed. This conclusion puts to rest an important paradox that preoccupies logicians and points out the importance of verifying one's assumptions.
Category: Set Theory and Logic

[117] viXra:1607.0421 [pdf] submitted on 2016-07-22 08:40:32

The Theory of Ultralogics, the Modified Robinson Approach, and GID

Authors: Robert A. Herrmann
Comments: 8 Pages.

The basic mathematical aspects of the GGU and GID models are discussed. As an illustration, the modified Robinson approach is used to give a more direct prediction as to the composition of ultra-propertons. Relative to logic-systems, the refined developmental paradigm is applied to the General Intelligence Design (GID) model and the basic GID statement are given.
Category: Set Theory and Logic

[116] viXra:1607.0153 [pdf] submitted on 2016-07-13 04:47:18

A Phenomenon in Gödel’s Incompleteness Theorems

Authors: S.Kalimuthu
Comments: 06 Pages. NA

According to James R. Meyer, In mathematics, a theorem is intended to be a term for a very precise and definite concept - a theorem is a statement that is proved, using rigorous mathematical reasoning, to follow according to a set of logical rules, from a set of initial statements. These initial statements are usually called axioms, and these are statements that are accepted without being proven. The set of logical rules which determine how one statement can follow from another are usually called the rules of inference . And basically, Gödel's incompleteness theorem is any statement that says that for every formal mathematical system, there are sentences that cannot be proved to be true or false in that system.
Category: Set Theory and Logic

[115] viXra:1607.0124 [pdf] submitted on 2016-07-11 02:37:20

Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset

Authors: Florentin Smarandache
Comments: 170 Pages.

Neutrosophic Over-/Under-/Off-Set and -Logic were defined for the first time by Smarandache in 1995 and published in 2007. They are totally different from other sets/logics/probabilities. He extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is > 1}, Neutrosophic Underset {when some neutrosophic component is < 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and other neutrosophic component < 0}. This is no surprise with respect to the classical fuzzy set/logic, intuitionistic fuzzy set/logic, or classical/imprecise probability, where the values are not allowed outside the interval [0, 1], since our real-world has numerous examples and applications of over-/under-/off-neutrosophic components.
Category: Set Theory and Logic

[114] viXra:1606.0160 [pdf] submitted on 2016-06-15 08:59:21

The Theory of Ultralogics - Part I of III

Authors: Robert A. Herrmann
Comments: 52 Pages.

This part contains the Contents and Chapters 1, 2, 3 and 4, which include Alphabets, Words, Deduction, The Nonstandard Structure, as well as Adjective, Propositional, Predicate Reasoning, Reasoning from the Prefect and Order.
Category: Set Theory and Logic

[113] viXra:1606.0159 [pdf] submitted on 2016-06-15 08:59:39

The Theory of Ultralogics - Part II of III

Authors: Robert A. Herrmann
Comments: 48 Pages.

This part contains Chapters 5, 6, 7, 8, 9, which include Consequence Operators (Operations) Perception, An Alternate Approach, Developmental Paradigms, Ultrawords, A Neutron Altering Process, The Extended Structure and General Paradigms.
Category: Set Theory and Logic

[112] viXra:1606.0158 [pdf] submitted on 2016-06-15 09:00:00

The Theory of Ultralogics - Part III of III

Authors: Robert A. Herrmann
Comments: 38 Pages.

This part contains Chapters 10, 11, Symbols and the Index, which include Laws and Rules, Propertons (subparticles) and the MA-model.
Category: Set Theory and Logic

[111] viXra:1606.0005 [pdf] submitted on 2016-06-01 09:04:24

Improvements to The Theory of Ultralogics - 1

Authors: Robert A. Herrmann
Comments: 7 Pages.

The major purpose of this article is to establish Theorem 9.3.1 for the ESG, with the modified Robinson approach, and to make other improvements in Section 9 of The Theory of Ultralogics.
Category: Set Theory and Logic

[110] viXra:1605.0231 [pdf] submitted on 2016-05-22 20:23:04

A Totally Ordered Set with Cardinality Strictly Between Natural and Real Numbers

Authors: Philip Druck
Comments: 27 Pages.

A totally ordered set is identified with cardinality strictly between natural (N) and real (R) numbers. This set, denoted DS, is essentially an experimental finding, identified in unrelated patented research on nonuniform data sampling and self-stabilizing computer arithmetic. Its theoretical validation here will provide concrete proof that the Continuum Hypothesis (CH) is false. Note that this is distinct from determining whether CH can or cannot be proven from current axioms of set theory, which is settled. Also note that the Generalized Continuum Hypothesis is not addressed. First, Cantor diagonalization is applied isomorphically to prove that DS has strictly more than Cardinality(N) points. Then three (3) distinct proofs are provided to show that DS contains strictly fewer than Cardinality(R) elements. Each proof relies on a distinct property of primes. It is surmised that the considerable research efforts to-date on CH missed this result due to over-generalization, by considering all Alephi sets, i=0.., ∞. Those efforts thereby missed the impact of primes specifically on Aleph0/Aleph1 sets.
Category: Set Theory and Logic

[109] viXra:1605.0052 [pdf] submitted on 2016-05-04 05:05:38

Adjustable and Mean Potentiality Approach on Decision Making

Authors: J. Martina Jency, I. Arockiarani
Comments: 9 Pages.

In this paper, we design a model based on adjustable and mean potentiality approach to single valued neutrosophic level soft sets. Further, we introduce the notion of weighted single valued neutrosophic soft set and investigate its application in decision making.
Category: Set Theory and Logic

[108] viXra:1605.0051 [pdf] submitted on 2016-05-04 05:07:03

A New Method to Construct Entropy of Interval-Valued Neutrosophic Set

Authors: Chunfang Liu, YueSheng Luo
Comments: 4 Pages.

Interval-valued neutrosophic set (INS) is a generalization of fuzzy set (FS) that is designed for some practical situations in which each element has different truth membership function, indeterminacy membership function and falsity membership function and permits the membership degrees to be expressed by interval values.
Category: Set Theory and Logic

[107] viXra:1605.0050 [pdf] submitted on 2016-05-04 05:08:47

An Extended Grey Relational Analysis Based Multiple Attribute Decision Making in Interval Neutrosophic Uncertain Linguistic Setting

Authors: Partha Pratim Dey, Surapati Pramanik, Bibhas C. Giri
Comments: 10 Pages.

This paper investigates an extended grey relational analysis method for multiple attribute decision making problems under interval neutrosophic uncertain linguistic environment. Interval neutrosophic uncertain linguistic variables are hybridization of uncertain linguistic variables and interval neutrosophic sets and they can easily express the imprecise, indeterminate and inconsistent information which normally exist in real life situations.
Category: Set Theory and Logic

[106] viXra:1605.0049 [pdf] submitted on 2016-05-04 05:09:58

Degree of Dependence and Independence of the (Sub)Components of Fuzzy Set and Neutrosophic Set

Authors: Florentin Smarandache
Comments: 3 Pages.

We have introduced for the first time the degree of dependence (and consequently the degree of independence) between the components of the fuzzy set, and also between the components of the neutrosophic set in our 2006 book’s fifth edition. Now we extend it for the first time to the refined neutrosophic set considering the degree of dependence or independence of subcomponets.
Category: Set Theory and Logic

[105] viXra:1605.0048 [pdf] submitted on 2016-05-04 05:11:04

Isolated Single Valued Neutrosophic Graphs

Authors: Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache
Comments: 5 Pages.

Many results have been obtained on isolated graphs and complete graphs. In this paper, a necessary and sufficient condition will be proved for a single valued neutrosophic graph to be an isolated single valued neutrosophic graph.
Category: Set Theory and Logic

[104] viXra:1605.0047 [pdf] submitted on 2016-05-04 05:12:42

Mapping Causes and Implications of India’s Skewed Sex Ratio and Poverty Problem Using Fuzzy & Neutrosophic Relational Maps

Authors: Gaurav, Megha Kumar, Kanika Bhutani, Swati Aggarwal
Comments: 12 Pages.

This paper employs a new soft computing based methodology for identifying and analyzing the relationships among the causes and implications of the two challenging problems in India: unbalanced sex ratio and poverty.
Category: Set Theory and Logic

[103] viXra:1605.0046 [pdf] submitted on 2016-05-04 05:13:56

Neutrosophic Logic Approach for Evalua

Authors: Nouran Radwan, M. Badr Senousy, Alaa El Din M. Riad
Comments: 5 Pages.

This paper reviews some of the multivalued logic models which are fuzzy set, intuitionistic fuzzy set, and suggests a new approach which is neutrosophic set for handling uncertainty in expert systems to derive decisions. The paper highlights, compares and clarifies the differences of these models in terms of the application area of problem solving.
Category: Set Theory and Logic

[102] viXra:1605.0045 [pdf] submitted on 2016-05-04 05:15:14

Neutrosophic Set Approach for Characterizations of Left Almost Semigroups

Authors: Madad Khan, Florentin Smarandache, Sania Afzal
Comments: 16 Pages.

In this paper we have defined neutrosophic ideals, neutrosophic interior ideals, netrosophic quasi-ideals and neutrosophic bi-ideals (neutrosophic generalized bi-ideals) and proved some results related to them.
Category: Set Theory and Logic

[101] viXra:1605.0044 [pdf] submitted on 2016-05-04 05:16:41

Neutrosophic Soft Graphs

Authors: Nasir Shah, Asim Hussain
Comments: 14 Pages.

The aim of this paper is to propose a new type of graph called neutrosophic soft graphs. We have established a link between graphs and neutrosophic soft sets. Basic operations of neutrosophic soft graphs such as union, intersection and complement are defined here. The concept of strong neutrosophic soft graphs is also discussed in this paper.
Category: Set Theory and Logic

[100] viXra:1605.0043 [pdf] submitted on 2016-05-04 05:18:07

Neutrosophic Soft Multi-Attribute Decision Making Based on Grey Relational Projection Method

Authors: Partha Pratim Dey, Surapati Pramanik, Bibhas C. Giri
Comments: 9 Pages.

The present paper proposes neutrosophic soft multi-attribute decision making based on grey relational projection method. Neutrosophic soft sets is a combination of neutrosophic sets and soft sets and it is a new mathematical apparatus to deal with realistic problems in the fields of medical sciences, economics, engineering, etc.
Category: Set Theory and Logic

[99] viXra:1605.0040 [pdf] submitted on 2016-05-04 05:22:55

The Novel Attempt for Finding Minimum Solution in Fuzzy Neutrosophic Relational Geometric Programming (FNRGP) with (max, min) Composition

Authors: Huda E. Khalid
Comments: 5 Pages.

This article sheds light on the possibility of finding the minimum solution set of neutrosophic relational geometric programming with (max, min) composition. This work examines the privacy enjoyed by both neutrosophic logic and geometric programming, and how it affects the minimum solutions. It is the first attempt to solve this type of problems.
Category: Set Theory and Logic

[98] viXra:1605.0020 [pdf] submitted on 2016-05-03 01:19:33

Neutrosophic Sets and Systems, Vol. 11, 2016

Authors: Florentin Smarandache - Editor-in-Chief
Comments: 113 Pages.

“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Category: Set Theory and Logic

[97] viXra:1604.0392 [pdf] submitted on 2016-04-30 17:18:06

The Multitude Behind the Buddhabrot

Authors: István Aggott Hönsch
Comments: 19 Pages.

A terminological framework is proposed for the mathematical examination and analysis of the Mandelbrot set's correlative ectocopial set. The Apeiropolis and anthropobrot multisets are defined and explained to be the mathematical entities underlying the well-known Buddhabrot visualization. The definitions are presented as tools conducive to finding novel approaches and generating discoveries that might otherwise be missed via a primarily programmatic approach. The anthropobrot multisets are introduced as a new, infinite repository of unique pareidolic figures as richly diverse as the Julia sets.
Category: Set Theory and Logic

[96] viXra:1604.0118 [pdf] submitted on 2016-04-06 06:46:52

A Theory of the Comprehensive Endosemasiopasigraphic Algebraico-Predicate Organon and Its Conformal Catlogographic Interpretations: a General Analytical Solution of Trial Decision Problems for First-Order Predicate Calculus

Authors: Yakov A. Iosilevskii
Comments: 1134 Pages.

In contrast to Church, who proved in 1936, based on papers by Gödel, that a dual decision problem for the conventional axiomatic first-order predicate calculus is unsolvable, I have solved a trial decision problem algebraically (and hence analytically, not tabularily) for a properly designed axiomatic first-order algebraico-predicate calculus, called briefly the trial logic (TL), and have successfully applied the pertinent algebraic decision procedures to all conceivable logical relations of academic or practical interest, including the 19 categorical syllogisms. The structure of the TL is a synthesis of the structure of a conventional axiomatic first-order predicate calculus (briefly CAPC) and of the structure of an abstract integral domain. Accordingly, the TL contains as its autonomous parts the so-called Predicate-Free Relational Trial Logic (PFRTL), which is parallel to a conventional axiomatic sentential calculus (CASC), and the so-called Binder-Free Predicate Trial Logic (BFPTL), which is parallel to the predicate-free part of a pure CAPC. This treatise, presenting some of my findings, is alternatively called “the Theory of Trial Logic” (“the TTL”) or “the Trial Logic Theory” (“the TLT”). The treatise reopens the entire topic of symbolic logic that is called “decision problem” and that Church actually closed by the fact of synecdochically calling the specific dual decision problem, the insolvability of which he had proved, by the generic name “decision problem”, without the qualifier “dual”. Any additional axiom that is incompatible with the algebraic decision method of the trial logic and that is therefore detrimental for that method is regarded as one belonging to either to another logistic system or to mathematics.
Category: Set Theory and Logic

[95] viXra:1602.0198 [pdf] submitted on 2016-02-16 21:38:55

Nidus Idearum. De Neutrosophia

Authors: Florentin Smarandache
Comments: 107 Pages.

Welcome into my scientific lab! My lab[oratory] is a virtual facility with non-controlled conditions in which I mostly perform scientific chats. I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings "ground", "opinion", "expectation"), combining the welly of both science and informal (via internet) talks. In this book, one may find new and old questions and ideas, some of them already put at work, others dead or waiting, referring to various fields of research (e.g. from neutrosophic algebraic structures to Zhang's degree of intersection, or from Heisenberg uncertainty principle to neutrosophic statistics) – email messages to research colleagues, or replies, notes about authors, articles or books, so on. Feel free to budge in the lab or use the scilogs as open source for your own ideas.
Category: Set Theory and Logic

[94] viXra:1601.0193 [pdf] submitted on 2016-01-17 18:49:05

Axiomatics of Mathematics

Authors: Nikolaj Roerich
Comments: 3 Pages.

We show that Riemann Hypothesis is actually an Axiom. Prooving it would mean knowing how to build the Universe. That is the future that people evolution will lead to. To build the Universee one needs to know the details about the corresponding Hilbert Space since Universe = Hilbert Space, and Riemann Hypothesis solution is equivalent to the knowing the linear operator in that Hilbert Space that is called "L" which has the eigenvalues equal to the zeros of Riemann Function.
Category: Set Theory and Logic

[93] viXra:1601.0023 [pdf] submitted on 2016-01-04 06:29:47

Kolmogorov Complexity, Matthew Effect, Godel Theorem and Chaos

Authors: Janis Belov
Comments: 2 Pages.

We solve P vs NP Millenium problem.
Category: Set Theory and Logic

[92] viXra:1512.0357 [pdf] submitted on 2015-12-17 20:25:15

Об одном методе решения оптиальных задач (Method Solution of Optimal Problems)

Authors: Bolonkin A.A.
Comments: 8 Pages.

Предлагается принципиально новый метод оптимизации. В отличие от классической постановки задачи: а) Дан функционал – найти его минималь. Рассматриваются также задачи: б) найти более «узкое» подмножество, содержащее абсолютную минималь; в) найти подмножество решений лучших, чем данное; г) найти оценки снизу данного функционала. В настоящее время большинство исследователей, работающих в области оптимизации заняты решение задачи в классической постановке – отысканием точной минимали. Инженера же, как правило, в реальных задачах интересует подмножество квази-оптимальных решений, выбирая из которого, он заранее уверен, что получит значение функционала не хуже заданной величины (задача в) и оценка снизу, показывающая насколько он далек от точного оптимального решения (задача г). Кроме того у него есть много дополнительных соображений, которые нельзя учесть в математической модели или которые бы ее сильно усложнили. Постановка задачи в форме «в» дает ему определенную свободу выбора. This method, called the “Method of Deformation of Functional (Extreme)”, solves for a total minimum and finds a solution set near the optimum. Solutions found by this method can be exact or approximate. Most other methods solve only for a unique local minimum. The ability to create a set of solutions rather than a unique solution has important practical ramifications in many designs, economic and scientific problems because a unique solution usually is difficult to realize in practice. This method has the additional virtue of a simple proof, one that is useful for studying other methods of optimization, since most other methods can be delivered from the Method of Deformation.
Category: Set Theory and Logic

[91] viXra:1511.0160 [pdf] submitted on 2015-11-18 12:28:07

Take it to Proof: Lie Algebra Symmetry-Checks and Factorization for Goedel’s Theorems

Authors: Alex Patterson
Comments: 7 Pages.

Will be look at (data) type inference for the four major arithmetic types to search for symmetry-checks and factorization in the Lie algebra, using the multiplicative decomposition by such searches in the Lie Algebra to Poincare Group, Poincare Group important only for the theory check.
Category: Set Theory and Logic

[90] viXra:1510.0041 [pdf] submitted on 2015-10-05 06:44:34

Neutrosophic Sets and Systems, Book Series, Vol. 9, 2015

Authors: editors Florentin Smarandache, Mumtaz Ali
Comments: Pages. 98

This volume is a collection of fourteen papers, written by different authors and co-authors (listed in the order of the papers): F. Yuhua, K. Mandal, K. Basu, S. Pramanik, K. Mondal, S. Alkhazaleh, J. Nescolarde-Selva, J. L. Usó-Doménech, A. Betancourt-Vázquez, K. Pérez-Teruel, M. Leyva-Vázquez, A. Aydoğdu, I. Arockiarani, C. A. C. Sweety, F. Smarandache, L. Zhengda, S. Kar, S. Mukherjee, P. Das, and T. K. Kumar.
Category: Set Theory and Logic

[89] viXra:1508.0309 [pdf] submitted on 2015-08-30 19:26:33

Formal Language, Intuition, Kleene Plus and Zorn's Lemma

Authors: Minseong Kim
Comments: 2 Pages.

In computer science, a character set $\Sigma$ is often defined. Then, Kleene plus and Kleene star for formal language are defined. Then, $\Sigma^{+} = \Sigma^{*}\Sigma$ is proved, which means every string (set) in $\Sigma^{+}$ can be represented as a concatenation of a set in $\Sigma^{*}$ and a set in $\Sigma$. However, if one forms a set that cannot be defined by a formula but what people would believe as existing, then while the proof itself does not break down, it may be possible that state of matter is inconsistent. This paper explores this possibility.
Category: Set Theory and Logic

[88] viXra:1508.0299 [pdf] submitted on 2015-08-29 08:36:40

A Short Remark

Authors: Samuel Amok
Comments: 1 Page.

In this paper, I answer to a question that has been raised in http://www.les-mathematiques.net/phorum/read.php?16,1137927,1137947#msg-1137947
Category: Set Theory and Logic

[87] viXra:1508.0284 [pdf] submitted on 2015-08-27 01:08:39

Mappings on Neutrosophic Soft Classes

Authors: Shawkat Alkhazaleh, Emad Marei
Comments: 112 Pages.

In 1995 Smarandache introduced the concept of neutrosophic set which is a mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. In 2013 Maji introduced the concept of neutrosophic soft set theory as a general mathematical tool for dealing with uncertainty.
Category: Set Theory and Logic

[86] viXra:1508.0161 [pdf] submitted on 2015-08-20 09:19:58

A Theoretical Critique of Theory

Authors: Alex Patterson
Comments: 12 Pages. Special thanks to Michael J. Burns

This paper uses its own peculiar lettering system for each paragraph. This paper proposes an overall solution to Godel’s incompleteness theorem and the Gödel sentence. Both are handled as one, by using Gödel numbers as the exemplary objects of incompleteness. New terms and tools are introduced for quantification that creates a more synthetic (logical, reasonable, coherent) intervention and inter-weaving into these now classical problems of the assumptions in the Gödel material and literature. Asymptotes are used within vertical and horizontal graphs to justify a future that need not be seen as a future in the sense of grammatical future-tense, but as a potential part such systems themselves that we deal with respect to incompleteness. The thesis is that we can approach incompleteness by using theoretical reasoning and available tools that are allowed in theoretical reasoning to critique the very theory of incompleteness itself. That is the essential Abstract Thesis. It will be seen that a real attempt is attempted.
Category: Set Theory and Logic

[85] viXra:1508.0089 [pdf] submitted on 2015-08-11 16:25:07

A Probabilistic Proof of the Existence of Extraterrestrial Life

Authors: Peiman Ghasemi
Comments: 6 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.
Category: Set Theory and Logic

Replacements of recent Submissions

[159] viXra:1811.0148 [pdf] replaced on 2018-11-14 07:22:11

Refutation of the Alleged Łukasiewicz Nightmare in Ł4 Logic: (◇p&◇q)→◇(p&q)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

In Prover9 the alleged Łukasiewicz nightmare of (◇p&◇q)→◇(p&q) is not tautologous. However, in Prover 9 the nightmare recast in one variable as (◇p&◇~p)→◇(p&~p) is tautologous. In Meth8/VŁ4, both propositions are tautologous. This speaks to Meth8/VŁ4, based on the corrected modern Square of Opposition as an exact bivalent system, opposed to Prover9, based on the uncorrected modern Square of Opposition as an inexact probabilistic vector space.
Category: Set Theory and Logic

[158] viXra:1811.0018 [pdf] replaced on 2018-11-09 17:10:10

Elementary Set Theory Can Be Used to Prove Fermat's Last Theorem (FLT)

Authors: Phil A. Bloom
Comments: 3 Pages.

An open problem is proving FLT simply for each integral $n>2$. Our proof of FLT is based on our algebraic identity, denoted, {for convenience}, as $r^n+s^n=t^n$. For $n\geq1$ we relate $r,s,t>0$, each a different function of variables comprising $r^n+s^n=t^n$, with $x,y,z>0$ for which $x^n+y^n=z^n$ holds. We infer as true by \emph{direct argument} (not BWOC), for any given $n>2$, that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}$. In addition, we show, for $n>2$, that $\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}=\varnothing$. Thus, for $n\in\mathbb{Z},n>2$, it is true that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\varnothing$.
Category: Set Theory and Logic

[157] viXra:1811.0018 [pdf] replaced on 2018-11-06 14:00:16

Elementary Set Theory Can Be Used to Prove Fermat's Last Theorem (FLT)

Authors: Phil A. Bloom
Comments: 3 Pages.

An open problem is proving FLT simply for each integral $n>2$. Our proof of FLT is based on our algebraic identity, denoted, {for convenience}, as $r^n+s^n=t^n$. For $n\geq1$ we relate $r,s,t>0$, each a different function of variables comprising $r^n+s^n=t^n$, with $x,y,z>0$ for which $x^n+y^n=z^n$ holds. We infer as true by \emph{direct argument} (not BWOC), for any given $n>2$, that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}$. In addition, we show, for $n>2$, that $\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}=\varnothing$. Thus, for $n\in\mathbb{Z},n>2$, it is true that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\varnothing$.
Category: Set Theory and Logic

[156] viXra:1811.0018 [pdf] replaced on 2018-11-05 17:07:33

Elementary Set Theory Can Be Used to Prove Fermat's Last Theorem (FLT)

Authors: Phil A. Bloom
Comments: 3 Pages.

An open problem is proving FLT simply for each integral $n>2$. Our proof of FLT is based on our algebraic identity, denoted, {for convenience}, as $r^n+s^n=t^n$. For $n\geq1$ we relate $r,s,t>0$, each a different function of variables comprising $r^n+s^n=t^n$, with $x,y,z>0$ for which $x^n+y^n=z^n$ holds. We infer as true by \emph{direct argument} (not BWOC), for any given $n>2$, that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}$. In addition, we show, for $n>2$, that $\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}=\varnothing$. Thus, for $n\in\mathbb{Z},n>2$, it is true that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\varnothing$.
Category: Set Theory and Logic

[155] viXra:1810.0047 [pdf] replaced on 2018-11-01 17:24:49

Visualizing the Histograms of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 10 Pages.

The length, displacement, and magnitude distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[154] viXra:1810.0047 [pdf] replaced on 2018-10-31 16:01:54

Visualizing the Histograms of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 10 Pages.

The length, displacement, and magnitude distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[153] viXra:1810.0047 [pdf] replaced on 2018-10-30 14:28:43

Visualizing the Histograms of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 10 Pages.

The length, displacement, and magnitude distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[152] viXra:1810.0047 [pdf] replaced on 2018-10-29 19:02:53

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 10 Pages.

The length, displacement, and magnitude distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[151] viXra:1810.0047 [pdf] replaced on 2018-10-28 21:49:14

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The length, displacement, and magnitude distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[150] viXra:1810.0047 [pdf] replaced on 2018-10-27 14:15:54

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The length, displacement, and magnitude distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[149] viXra:1810.0047 [pdf] replaced on 2018-10-26 09:05:41

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The length, displacement, and magnitude distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[148] viXra:1810.0047 [pdf] replaced on 2018-10-25 14:30:47

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The length, displacement, and magnitude distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[147] viXra:1810.0047 [pdf] replaced on 2018-10-24 09:25:55

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[146] viXra:1810.0047 [pdf] replaced on 2018-10-22 14:39:46

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[145] viXra:1810.0047 [pdf] replaced on 2018-10-21 19:48:03

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[144] viXra:1810.0047 [pdf] replaced on 2018-10-15 17:59:59

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[143] viXra:1810.0047 [pdf] replaced on 2018-10-14 15:52:53

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 6 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[142] viXra:1810.0047 [pdf] replaced on 2018-10-13 19:52:33

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 5 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[141] viXra:1810.0047 [pdf] replaced on 2018-10-12 13:17:32

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 6 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[140] viXra:1810.0047 [pdf] replaced on 2018-10-10 20:56:32

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 5 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[139] viXra:1810.0047 [pdf] replaced on 2018-10-08 19:39:30

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 5 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[138] viXra:1810.0047 [pdf] replaced on 2018-10-07 20:42:42

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 5 Pages.

The length and displacement distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[137] viXra:1810.0047 [pdf] replaced on 2018-10-05 11:11:19

Visualizing the Distributions of the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 4 Pages.

The distributions of the escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[136] viXra:1807.0418 [pdf] replaced on 2018-09-24 22:11:51

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 8 Pages.

The escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[135] viXra:1807.0418 [pdf] replaced on 2018-09-20 15:47:34

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 8 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL 1.x. C++ source code is provided.
Category: Set Theory and Logic

[134] viXra:1807.0418 [pdf] replaced on 2018-09-18 12:32:44

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 8 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[133] viXra:1807.0418 [pdf] replaced on 2018-09-17 17:04:49

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[132] viXra:1807.0418 [pdf] replaced on 2018-08-01 13:29:49

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 4 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[131] viXra:1807.0418 [pdf] replaced on 2018-07-30 09:50:33

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[130] viXra:1807.0418 [pdf] replaced on 2018-07-26 16:18:11

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[129] viXra:1806.0030 [pdf] replaced on 2018-06-27 22:04:44

The Function f(x) = C and the Continuum Hypothesis

Authors: Ron Ragusa
Comments: 14 Pages. email: ron.ragusa@gmail.com

Part 1 examines whether or not an analysis of the behavior of the function f(x) = C, where C is any constant, on the interval (a, b) where a and b are real numbers and a < b, will provide a method of proving the truth or falsity of the Continuum Hypothesis (CH). The argument will be presented in three theorems and one corollary. The first theorem proves, by construction, the countability of the domain d of f(x) = C on the interval (a, b) where a and b are real numbers. The second theorem proves, by substitution, that the set of natural numbers ℕ has the same cardinality as the subset S of real numbers on the given interval. The corollary extends the proof of theorem 2 to show that ℕ and ℝ are of the same cardinality. The third theorem proves, by logical inference, that the CH is true. Part 2 is a demonstration of how the set of natural numbers ℕ can be put into a one to one correspondence with the power set of natural numbers, P(ℕ). From this I will derive the bijective function f : ℕ → P(ℕ). Lastly, I’ll propose a conjecture asserting that f(x) = C can be employed to construct a one to one correspondence between the natural numbers and any infinite set that can be cast as the domain of the function. Appendix A extends the methodology for creating a bijection between infinite sets to the function f(x) = x2 using random real numbers from the domain of the function as input to f(x) = x2 in order to show how the constructed array would appear in practical application.
Category: Set Theory and Logic

[128] viXra:1806.0030 [pdf] replaced on 2018-06-04 08:33:05

The Function f(x) = C and the Continuum Hypothesis

Authors: Ron Ragusa
Comments: 5 Pages.

This paper examines whether or not an analysis of the behavior of the continuous function f(x) = C, where C is any constant, on the interval (a, b) where a and b are real numbers and a < b, will provide a method of proving the truth or falsity of the CH. The argument will be presented in three theorems and one corollary. The first theorem proves, by construction, the countability of the domain d of f(x) = C on the interval (a, b) where a and b are real numbers. The second theorem proves, by substitution, that the set of natural numbers N has the same cardinality as the subset of real numbers S on the given interval. The corollary extends the proof of theorem 2 to show that N and R are of the same cardinality. The third theorem proves, by logical inference, that the CH is true.
Category: Set Theory and Logic

[127] viXra:1805.0302 [pdf] replaced on 2018-05-16 23:04:06

Refutation of Gettier Problem of Justified True/false Belief © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

We find justified true belief is not a theorem. We find justified false belief is also not a theorem. This means the Gettier problem as the superset of the justified belief arguments is refuted as a problem and resolved as a non-problem.
Category: Set Theory and Logic

[126] viXra:1805.0138 [pdf] replaced on 2018-05-09 04:48:16

Refutation of the Paradox of Moses Maimonides © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Does God know or does He not know that a certain individual will be good or bad? If thou sayest 'He knows', then it necessarily follows that the man is compelled to act as God knew beforehand he would to act, otherwise God's knowledge would be imperfect. The question is tautologous. The additional sentence with the first is not tautologous and not contradictory. Therefore the paradox of Maimonides is refuted as a paradox.
Category: Set Theory and Logic

[125] viXra:1805.0124 [pdf] replaced on 2018-05-06 14:30:48

Refutation of the Prisoner Paradox© Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The three states of the prisoner paradox are not contradictory. This means the prisoner paradox is not a paradox.
Category: Set Theory and Logic

[124] viXra:1804.0174 [pdf] replaced on 2018-05-04 01:07:50

The Universe Equation

Authors: Seamus McCelt
Comments: 2 Pages.

If you claim there are particles: there would actually have to be particles.
And that would mean there are about 18 different microscopic things that work flawlessly together -- just like clockwork to make even just one basic atom "gear" set.
If you have larger sized atoms: it would be like throwing more and more gear sets into the clockwork -- but that is ok because no matter what you throw in -- it will still work just fine.
How can an infinity of 18 different things (infinity times 18 different things) just happen to be here, know how work together as a group and also successfully work together as a group(s)?
How is that possible? It isn't...

Stuff cannot be made from what they call "particles."

If there are particles; this is equation of the universe:
Universe = Infinity × {a,b,c,d,f,g,h,j,k,l,m,o,p,q,t,w,x,y,z}
Category: Set Theory and Logic

[123] viXra:1803.0318 [pdf] replaced on 2018-03-21 06:51:30

Refutation of Abductive Reasoning © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Abductive logic of C.S. Peirce is refuted as not tautologous.
Category: Set Theory and Logic

[122] viXra:1803.0180 [pdf] replaced on 2018-03-13 17:40:00

Refutation of the Euathlus Paradox: Neither Pay © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved.

Regardless of who wins the lawsuit of Portagoras, Euathlus does not pay. Hence the Euathlus paradox is refuted and resolved by default in favor of Euathlus.
Category: Set Theory and Logic

[121] viXra:1803.0094 [pdf] replaced on 2018-03-07 17:10:19

Refutation of Cantor's Original Continuum Hypothesis Via Injection and Binary Trees © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

This is the briefest known such refutation of Cantor's continuum conjecture.
Category: Set Theory and Logic

[120] viXra:1712.0403 [pdf] replaced on 2017-12-23 02:39:19

There is no Standard Model of ZFC

Authors: Jaykov Foukzon
Comments: 14 Pages. Journal of Global Research in Mathematical Archives (JGRMA)Vol 5, No 1 (2018): January-2018

Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st),(ii) let k be an inaccessible cardinal then ~Con(ZFC+∃k),[10],11].
Category: Set Theory and Logic

[119] viXra:1712.0403 [pdf] replaced on 2017-12-15 05:52:44

There is no Standard Model of ZFC

Authors: Jaykov Foukzon
Comments: 13 Pages.

Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st),(ii) let k be an inaccessible cardinal then ~Con(ZFC+∃k),[10],[11].
Category: Set Theory and Logic

[118] viXra:1712.0139 [pdf] replaced on 2018-04-03 05:31:20

A Proof of the Falsity of the Axiom of Choice.

Authors: Johan Noldus
Comments: 1 Page.

We show that the axiom of choice is false.
Category: Set Theory and Logic

[117] viXra:1712.0139 [pdf] replaced on 2017-12-22 12:51:03

A Proof of the Falsity of the Axiom of Choice.

Authors: Johan Noldus
Comments: 1 Page.

We show that the axiom of choice is false.
Category: Set Theory and Logic

[116] viXra:1711.0425 [pdf] replaced on 2017-11-26 08:56:58

Refutation of Realizability Semantics for QML © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2017 by Colin James III All rights reserved.

We found that the following are not tautologous: Epistemic Church's Thesis; EZF induction schema; and Scedrov's modal foundation. We did not test subsequent axioms. What follows is that Flagg's construction, Goodman's intensional set theory, and epistemic logic are suspicious.
Category: Set Theory and Logic

[115] viXra:1708.0156 [pdf] replaced on 2018-03-14 06:03:39

Some Critical Notes on the Cantor Diagonal Argument

Authors: Philip Molyneux
Comments: 10 Pages.

This paper critically examines the Cantor Diagonal Argument (CDA) that is used in set theory to draw a distinction between the cardinality of the natural numbers and that of the real numbers. In the absence of a verified English translation of the original 1891 Cantor paper from which it is said to be derived, the CDA is discussed here using a consensus from the forms found in a range of published sources (from "popular" to "professional"). Some general comments are made on these sources. The discussion then focusses on the CDA as applied to the correspondence between the set of the natural numbers, and the set of real numbers in the open range (0,1) in their expansion from decimal digits (0.123… etc.). Four points critical of the CDA are raised: (1) The conventional presentation of the CDA forms a putative new real number (X) from the "diagonal" of the chosen list of real numbers and which is therefore not on this initial list; however, it omits to consider that it may indeed be on the later part of the list, which is never exhausted however far the "diagonal" list is extended. (2) This aspect, combined with the fact that X is still composed of decimal digits, that is, it is a real number as defined, indicates that it must be on the later part of the list, that is, it is not a "new" number at all. (3) The conventional application of the CDA apparently leads to one putative "new" real number (X); however, the logical extension of this in its "exhaustive" application, that is, by using all possible different methods of alteration of the decimal digits on the "diagonal", and by reordering the list in all possible ways, leads to a list of putative "new" real numbers that become orders of magnitude longer than the chosen "diagonal" list. (4) The CDA is apparently considered to be a method that is applicable generally; however, testing this applicability with the natural numbers themselves leads to this contradiction. Following on from this, it is found that it indeed is possible to set up a one-to-one correspondence between the natural numbers and the real numbers in (0,1), that is, ! ⇔ "; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension of the natural number is intended to be a mirror image of the left hand extension of the real number. This may be extended to the general case of real numbers - that is, not limited to the range (0,1) - by intercalation of the digit sequence of its decimal fraction part into the sequence of the natural number part, giving the one-to-one-correspondence: … A3 a3 A2 a2 A1 a1 ⇔ ... A3 A2 A1. a1 a2 a3 … Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998 critical review of "hopeless papers" dealing with the CDA; this form is also examined from the same viewpoints, and to the same conclusions. Finally, some comments are made on the concept of "infinity", pointing out that to consider this as an entity is a category error, since it simply represents an absence, that is, the absence of a termination to a process.
Category: Set Theory and Logic

[114] viXra:1708.0156 [pdf] replaced on 2017-08-24 10:23:39

Some Critical Notes on the Cantor Diagonal Argument

Authors: Philip Molyneux
Comments: 10 Pages.

This paper critically examines the Cantor Diagonal Argument (CDA) that is used in set theory to draw a distinction between the cardinality of the natural numbers and that of the real numbers. In the absence of a verified English translation of the original 1891 Cantor paper from which it is said to be derived, the CDA is discussed here using a consensus from the forms found in a range of published sources (from "popular" to "professional"). Some general comments are made on these sources. The discussion then focusses on the CDA as applied to the correspondence between the set of the natural numbers, and the set of real numbers in the open range (0,1) in their expansion from decimal digits (0.123… etc.). Four points critical of the CDA are raised: (1) The conventional presentation of the CDA forms a putative new real number (X) from the "diagonal" of the chosen list of real numbers and which is therefore not on this initial list; however, it omits to consider that it may indeed be on the later part of the list, which is never exhausted however far the "diagonal" list is extended. (2) This aspect, combined with the fact that X is still composed of decimal digits, that is, it is a real number as defined, indicates that it must be on the later part of the list, that is, it is not a "new" number at all. (3) The conventional application of the CDA leads to one putative "new" real number (X); however, the logical extension of this in its "exhaustive" application, that is, by using all possible different methods of alteration of the decimal digits on the "diagonal", and by reordering the list in all possible ways, leads to a list of putative "new" real numbers that become orders of magnitude longer than the chosen "diagonal" list. (4) The CDA is apparently considered to be a method that is applicable generally; however, testing this applicability with the natural numbers themselves leads to a contradiction. Following on from this, it is found that it indeed is possible to set up a one-to-one correspondence between the natural numbers and the real numbers in (0,1), that is, ℕ ⇔ ℝ; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension of the natural number is intended to be a mirror image of the left hand extension of the real number. It is also shown how this may be extended to real numbers outside the range (0,1). Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998 critical review of "hopeless papers" dealing with the CDA; this form is also examined from the same viewpoints, and to the same conclusions. Finally, some comments are made on the concept of "infinity", pointing out that to consider this as an entity is a category error, since it simply represents an absence, that is, the absence of a termination to a process.
Category: Set Theory and Logic

[113] viXra:1708.0156 [pdf] replaced on 2017-08-22 04:28:33

Some Critical Notes on the Cantor Diagonal Argument

Authors: Philip Molyneux
Comments: 10 Pages.

This paper critically examines the Cantor Diagonal Argument (CDA) that is used in set theory to draw a distinction between the cardinality of the natural numbers and that of the real numbers. In the absence of a verified English translation of the original 1891 Cantor paper from which it is said to be derived, the CDA is discussed here using a consensus from the forms found in a range of published sources (from "popular" to "professional"). Some general comments are made on these sources. The discussion then focusses on the CDA as applied to the correspondence between the set of the natural numbers, and the set of real numbers in the open range (0,1) in their expansion from decimal digits (0.123… etc.). Four points critical of the CDA are raised: (1) The conventional presentation of the CDA forms a putative new real number (X) from the "diagonal" of the chosen list of real numbers and which is therefore not on this initial list; however, it omits to consider that it may indeed be on the later part of the list, which is never exhausted however far the "diagonal" list is extended. (2) This aspect, combined with the fact that X is still composed of decimal digits, that is, it is a real number as defined, indicates that it must be on the later part of the list, that is, it is not a "new" number at all. (3) The conventional application of the CDA leads to one putative "new" real number (X); however, the logical extension of this in its "exhaustive" application, that is, by using all possible different methods of alteration of the decimal digits on the "diagonal", and by reordering the list in all possible ways, leads to a list of putative "new" real numbers that become orders of magnitude longer than the chosen "diagonal" list. (4) The CDA is apparently considered to be a method that is applicable generally; however, testing this applicability with the natural numbers themselves leads to a contradiction. Following on from this, it is found that it indeed is possible to set up a one-to-one correspondence between the natural numbers and the real numbers in (0,1), that is, N ⇔ R; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension of the natural number is intended to be a mirror image of the left hand extension of the real number. It is also shown how this may be extended to real numbers outside the range (0,1). Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998 critical review of "hopeless papers" dealing with the CDA; this form is also examined from the same viewpoints, and to the same conclusions. Finally, some comments are made on the concept of "infinity", pointing out that to consider this as an entity is a category error, since it simply represents an absence, that is, the absence of a termination to a process.
Category: Set Theory and Logic

[112] viXra:1705.0173 [pdf] replaced on 2017-05-12 13:18:20

Dialectical Logic – Negation Of Classical Logic

Authors: Ilija Barukčić
Comments: 24 pages. Copyright © 2017 by Ilija Barukčić, Jever, Germany. Published by:

The division of zero by zero turns out to be a long lasting and not ending puzzle in mathematics and physics. An end of this long discussion is not in sight. In particular zero divided by zero is treated as indeterminate thus that a result cannot be found out. It is the purpose of this publication to solve the problem of the division of zero by zero while relying on the general validity of classical logic. According to classical logic, zero divided by zero is one.
Category: Set Theory and Logic

[111] viXra:1704.0115 [pdf] replaced on 2017-05-16 21:36:12

The Simple Infinite Set

Authors: Ken Seton
Comments: 9 Pages. Enjoy

Many have suggested that the infinite set has a fundamental problem. The usual complaint rails against the actually infinite which (say critics of various finitist persuasions) unjustifiably goes beyond the finite. Here we identify the exact opposite. The problem of the infinite set defined to have an identity (content) that is specified and restricted to be forever finite . Set theory is taken at its word. The existence of the infinite set and the representation of irrational reals as infinite sets of terms is accepted. In this context, it is shown that the standard definition of the infinite countable set is inconsistent with the existence of its own classic convergents of construction. If the set is infinite then it must be quite unlike that which set theory asserts it to be. Set theory found itself in some trouble over a century ago trusting an unrestricted anthropic comprehension. But serious doubt is cast on the validity of infinite sets which have been defined by a comprehension which overly-restricts their content.
Category: Set Theory and Logic

[110] viXra:1704.0115 [pdf] replaced on 2017-05-11 01:29:26

The Simple Infinite Set

Authors: Ken Seton
Comments: 9 Pages. Enjoy

Many have suggested that the infinite set has a fundamental problem. The usual complaint rails against the actually infinite which (say critics of various finitist persuasions) unjustifiably goes beyond the finite. Here we identify the exact opposite. The problem of the infinite set defined to have an identity (content) that is specified and restricted to be forever finite . Set theory is taken at its word. The existence of the infinite set and the representation of irrational reals as infinite sets of terms is accepted. In this context, it is shown that the standard definition of the infinite countable set is inconsistent with the existence of its own classic convergents of construction. If the set is infinite then it must be quite unlike that which set theory asserts it to be. Set theory found itself in some trouble over a century ago trusting an unrestricted anthropic comprehension. But serious doubt is cast on the validity of infinite sets which have been defined by a comprehension which overly-restricts their content.
Category: Set Theory and Logic

[109] viXra:1703.0032 [pdf] replaced on 2017-03-06 04:56:10

The Union is not the Limit.

Authors: Wolfgang Mückenheim
Comments: 5 Pages.

Contrary to the assumptions of transfinite set theory, limit and union of infinite sequences of sets differ.
Category: Set Theory and Logic

[108] viXra:1702.0293 [pdf] replaced on 2017-02-26 04:27:32

Not Enumerating All Positive Rational Numbers

Authors: W. Mückenheim
Comments: 2 Pages.

It is shown that the enumeration of rational numbers cannot be complete.
Category: Set Theory and Logic

[107] viXra:1702.0293 [pdf] replaced on 2017-02-24 05:08:54

Not Enumerating All Positive Rational Numbers

Authors: W. Mückenheim
Comments: 2 Pages.

It is shown that the enumeration of rational numbers cannot be complete.
Category: Set Theory and Logic

[106] viXra:1701.0563 [pdf] replaced on 2017-01-31 11:38:56

بیانیه ی یگانه و انجمن

Authors: امیر دلجو
Comments: 12 Pages.

این سند یک منشور است برای بیان ماهیت ریاضیات و نظریه ی اعداد یا حساب. در اینجا من الگویی را تعریف کرده ام که نشان می دهد، خودآگاهی یک وجود واحده بسیط و همه جاحاضر بوده و هستی به مثابه یک کلیّت، یک تجلّی گرافیکی ست که در جهت افشای این خودآگاهی عارض شده و من هرمس وار، شهود خود بر یگانگی و آفرینش را برای انتقال به انسان های کنجکاو ساده سازی و تحریر کرده ام. از آنجا که تبیین روش شناختی این منشور مستلزم هزاران صفحه است، در اینجا تنها عبارات و معادلات منتج شده ی نهایی اعلان می شود.
Category: Set Theory and Logic

[105] viXra:1608.0395 [pdf] replaced on 2016-11-03 03:06:10

The Topology on a Complete Semilattice

Authors: Max Null, Sergey Belov
Comments: 23 Pages.

We define the topology atop(χ) on a complete upper semilattice χ = (M, ≤). The limit points are determined by the formula lim (X) = sup{a ∈ M | {x ∈ X| a ≤ x} ∈ D}, D where X ⊆ M is an arbitrary set, D is an arbitrary non-principal ultrafilter on X. We investigate lim (X) and topology atop(χ) properties. In particular, D we prove the compactness of the topology atop(χ).
Category: Set Theory and Logic

[104] viXra:1608.0395 [pdf] replaced on 2016-09-14 08:54:47

The Topology on a Complete Semilattice

Authors: Max Null, Sergey Belov
Comments: 17 Pages.

We define the topology atop(χ) on a complete upper semilattice χ = (M, ≤). The limit points are determined by the formula Lim(D,X) = sup{a ∈ M| {x ∈ X| a ≤ x} ∈ D}, where X ⊆ M is an arbitrary set, D is an arbitrary non-principal ultrafilter on X. We investigate Lim(D,X) and topology atop(χ) properties. In particular, we prove the compactness of the topology atop(χ).
Category: Set Theory and Logic

[103] viXra:1608.0358 [pdf] replaced on 2017-10-27 07:50:32

Natural Non-Godel Definitions of Incompleteness

Authors: Vatolin Dm.
Comments: 7 Pages. Russian

Here are definitions of «completeness» and «incompleteness» for mathematical theories. These definitions are different from those that gave Godel. Сontradictions of the Godel's arguments have been eliminated. Found are theo-rems that put everything in its place.
Category: Set Theory and Logic

[102] viXra:1608.0358 [pdf] replaced on 2017-06-04 02:59:26

Natural Non-Godel Definitions of Incompleteness

Authors: Vatolin Dm.
Comments: 7 Pages. Russian

Here are definitions of «completeness» and «incompleteness» for mathematical theories. These definitions are different from those that gave Godel. Сontradictions of the Godel's arguments have been eliminated. Found are theo-rems that put everything in its place.
Category: Set Theory and Logic

[101] viXra:1608.0358 [pdf] replaced on 2017-03-23 21:58:40

Natural Non-Godel Definitions of Incompleteness

Authors: Vatolin Dm.
Comments: 7 Pages. Rassian

Here are definitions of «completeness» and «incompleteness» for mathematical theories. These definitions are different from those that gave Godel. Сontradictions of the Godel's arguments have been eliminated. Found are theo-rems that put everything in its place.
Category: Set Theory and Logic

[100] viXra:1608.0057 [pdf] replaced on 2016-09-07 19:32:34

Curry's Non-Paradox and Its False Definition

Authors: Adrian Chira
Comments: 7 Pages.

Curry's paradox is generally considered to be one of the hardest paradoxes to solve. It is shown here that the paradox can be arrived in fewer steps and also for a different term of the original biconditional. Further, using different approaches, it is also shown that the conclusion of the paradox must always be false and this is not paradoxical but it is expected to be so. One of the approaches points out that the starting biconditional of the paradox amounts to a false definition or assertion which consequently leads to a false conclusion. Therefore, the solution is trivial and the paradox turns out to be no paradox at all. Despite that fact that verifying the truth value of the first biconditional of the paradox is trivial, mathematicians and logicians have failed to do so and merely assumed that it is true. Taking this into consideration that it is false, the paradox is however dismissed. This conclusion puts to rest an important paradox that preoccupies logicians and points out the importance of verifying one's assumptions.
Category: Set Theory and Logic

[99] viXra:1607.0421 [pdf] replaced on 2016-12-07 08:10:32

The Theory of Ultralogics, the Modified Robinson Approach, and GID

Authors: Robert A. Herrmann
Comments: 8 Pages.

The basic mathematical aspects of the GGU and GID models are discussed. As an illustration, the modified Robinson approach is used to give a more direct prediction as to the composition of ultra-propertons. Relative to logic-systems, the refined developmental paradigm is applied to the General Intelligent Design (GID) model and the basic GID statements are given.
Category: Set Theory and Logic

[98] viXra:1607.0421 [pdf] replaced on 2016-08-11 09:56:27

The Theory of Ultralogics, the Modified Robinson Approach, and GID

Authors: Robert A. Herrmann
Comments: 9 Pages.

The basic mathematical aspects of the GGU and GID models are discussed. As an illustration, the modified Robinson approach is used to give a more direct prediction as to the composition of ultra-propertons. Relative to logic-systems, the refined developmental paradigm is applied to the General Intelligence Design (GID) model and basic GID statements are given.
Category: Set Theory and Logic

[97] viXra:1606.0160 [pdf] replaced on 2018-08-12 08:48:39

The Theory of Ultralogics - Part I of III

Authors: Robert A. Herrmann
Comments: 52 Pages.

This part contains the Contents and Chapters 1, 2, 3 and 4, which include Alphabets, Words, Deduction, The Nonstandard Structure, as well as Adjective, Propositional, Predicate Reasoning, Reasoning from the Prefect and Order.
Category: Set Theory and Logic

[96] viXra:1606.0160 [pdf] replaced on 2018-08-08 10:18:20

The Theory of Ultralogics - Part I of III

Authors: Robert A. Herrmann
Comments: 53 Pages.

This part contains the Contents and Chapters 1, 2, 3 and 4, which include Alphabets, Words, Deduction, The Nonstandard Structure, as well as Adjective, Propositional, Predicate Reasoning, Reasoning from the Prefect and Order.
Category: Set Theory and Logic

[95] viXra:1606.0159 [pdf] replaced on 2018-08-17 06:06:55

The Theory of Ultralogics - Part II of III

Authors: Robert A. Herrmann
Comments: 48 Pages.

This part contains Chapters 5, 6, 7, 8, 9, which include Consequence Operators (Operations) Perception, An Alternate Approach, Developmental Paradigms, Ultrawords, A Neutron Altering Process, The Extended Structure and General Paradigms.
Category: Set Theory and Logic

[94] viXra:1606.0159 [pdf] replaced on 2018-08-13 07:55:54

The Theory of Ultralogics - Part II of III

Authors: Robert A. Herrmann
Comments: 48 Pages.

This part contains Chapters 5, 6, 7, 8, 9, which include Consequence Operators (Operations) Perception, An Alternate Approach, Developmental Paradigms, Ultrawords, A Neutron Altering Process, The Extended Structure and General Paradigms
Category: Set Theory and Logic

[93] viXra:1606.0159 [pdf] replaced on 2018-08-12 16:53:12

The Theory of Ultralogics - Part II of III

Authors: Robert A. Herrmann
Comments: 48 Pages.

This part contains Chapters 5, 6, 7, 8, 9, which include Consequence Operators (Operations) Perception, An Alternate Approach, Developmental Paradigms, Ultrawords, A Neutron Altering Process, The Extended Structure and General Paradigms.
Category: Set Theory and Logic

[92] viXra:1606.0159 [pdf] replaced on 2016-12-01 08:07:04

The Theory of Ultralogics - Part II of III

Authors: Robert A. Herrmann
Comments: 48 Pages.

This part contains Chapters 5, 6, 7, 8, 9, which include Consequence Operators (Operations), Perception, An Alternate Approach, Developmental Paradigms, Ultrawords, A Neutron Altering Process, The Extended Structure and General Paradigms.
Category: Set Theory and Logic

[91] viXra:1606.0158 [pdf] replaced on 2018-08-13 08:04:13

The Theory of Ultralogics - Part III of III

Authors: Robert A. Herrmann
Comments: 38 Pages.

This part contains Chapters 10, 11, Symbols and the Index, which include Laws and Rules, Propertons (subparticles) and the MA-model.
Category: Set Theory and Logic

[90] viXra:1606.0158 [pdf] replaced on 2018-08-12 08:58:04

The Theory of Ultralogics - Part III of III

Authors: Robert A. Herrmann
Comments: 38 Pages.

This part contains Chapters 10, 11, Symbols and the Index, which include Laws and Rules, Propertons (subparticles) and the MA-model.
Category: Set Theory and Logic

[89] viXra:1606.0005 [pdf] replaced on 2018-08-10 05:22:51

Improvments to The Theory of Ultralogics - I

Authors: Robert A. Herrmann
Comments: 8 Pages.

The major purpose for this article is to reestablish Theorem 9.3.1, with the modified Robinson approach, and make other improvements in Section 9 of The Theory of Ultralogics (Herrmann, (1978-93, 1999). Additionally, what constitutes a saturated enlargement is now fixed as of this date.
Category: Set Theory and Logic

[88] viXra:1606.0005 [pdf] replaced on 2016-12-16 09:20:24

Improvements to The Theory of Ultralogics - I.

Authors: Robert A. Herrmann
Comments: 8 Pages.

The major purpose for this article is to reestablish Theorem 9.3.1, for the EGS, with the modified Robinson approach and make other improvements in Section 9 of The Theory of Ultralogics (Herrmann, (1978-93, 1999)). Additionally, what constitutes a saturated enlargement is now fixed as of this date.
Category: Set Theory and Logic

[87] viXra:1606.0005 [pdf] replaced on 2016-11-18 08:38:56

Improvements to The Theory of Ultralogics - I

Authors: Robert A. Herrmann
Comments: 7 Pages.

The major purpose for this article is to reestablish Theorem 9.3.1, for the EGS, with the modified Robinson approach and make other improvements in Section 9 of The Theory of Ultralogics (Herrmann, (1978-93)). Additionally, in this version, certain notational conventions are discussed.
Category: Set Theory and Logic

[86] viXra:1606.0005 [pdf] replaced on 2016-08-09 09:58:46

Improvements to The Theory of Ultralogics - I

Authors: Robert A. Herrmann
Comments: 7 Pages.

The major purpose for this article is to reestablish Theorem 9.3.1 for the EGS, with the modified Robinson approach, and make other improvements in Section 9 of The Theory of Ultralogics. Further, an important improvement is made in the (2013) article on Nonstandard Ultra-logic-systems.
Category: Set Theory and Logic

[85] viXra:1606.0005 [pdf] replaced on 2016-06-02 09:45:38

Improvements to The Theory of Ultralogics - I

Authors: Robert A. Herrmann
Comments: 7 Pages.

The major purpose for this article is to reestablish Theorem 9.3.1 for the EGS, with the modified Robinson approach, and make other improvements in Section 9 of The Theory of Ultralogics.
Category: Set Theory and Logic

[84] viXra:1604.0104 [pdf] replaced on 2016-04-10 18:42:42

There Are Infinitely Many Theorems as Difficult to Prove as FERMAT’S Last Theorem: a Characterization of Such Theorems

Authors: Allen D Allen
Comments: Abstract contains 200 words, ms runs 6 pages

By proving that his “last theorem” (FLT) is true for the integral exponent n = 3, Fermat took the first step in a standard method of proving there exists no greatest lower bound on n for which FLT is true, thus proving the theorem. Unfortunately, there are two reasons why the standard method of proof is not available for FLT. First, transitive inequality lies at the heart of that method. Secondly, FLT admits to a change from > to < rendering their transitive natures unavailable. A related, self evident symmetry illustrates another problem that would have plagued Fermat and centuries of successors. FLT asserts such a narrow proposition, it is difficult to find an antecedent while easy to find a non equivalent consequence. For example, if FLT asserted that the exponent n is even, then FLT would be equivalent to the proposition that Fermat’s equation has two solutions, one for positive bases and one for their negative counterparts. This could be addressed with conservative transformations. The example provided by FLT motivates the use of an early paper by the author to prove a theorem on theorems. The theorem on theorems demonstrates there are infinitely many theorems as difficult to prove as FLT.
Category: Set Theory and Logic

[83] viXra:1603.0226 [pdf] replaced on 2016-03-17 02:40:06

Penta and Hexa Valued Representation of Neutrosophic Information

Authors: Vasile Pătraşcu
Comments: 12 Pages.

Starting from the primary representation of neutrosophic information, namely the degree of truth, degree of indeterminacy and degree of falsity, we define a nuanced representation in a penta valued fuzzy space, described by the index of truth, index of falsity, index of ignorance, index of contradiction and index of hesitance. Also, it was constructed an associated penta valued logic and then using this logic, it was defined for the proposed penta valued structure the following operators: union, intersection, negation, complement and dual. Then, the penta valued representation is extended to a hexa valued one, adding the sixth component, namely the index of ambiguity.
Category: Set Theory and Logic

[82] viXra:1603.0226 [pdf] replaced on 2016-03-17 02:40:06

Penta and Hexa Valued Representation of Neutrosophic Information

Authors: Vasile Pătraşcu
Comments: 12 Pages.

Starting from the primary representation of neutrosophic information, namely the degree of truth, degree of indeterminacy and degree of falsity, we define a nuanced representation in a penta valued fuzzy space, described by the index of truth, index of falsity, index of ignorance, index of contradiction and index of hesitance. Also, it was constructed an associated penta valued logic and then using this logic, it was defined for the proposed penta valued structure the following operators: union, intersection, negation, complement and dual. Then, the penta valued representation is extended to a hexa valued one, adding the sixth component, namely the index of ambiguity.
Category: Set Theory and Logic

[81] viXra:1508.0089 [pdf] replaced on 2016-02-13 08:42:50

A Probabilistic Proof of the Existence of Etraterrestrial Life

Authors: Peiman Ghasemi
Comments: 7 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. The first section includes an algebraic probabilistic proof when the event of life is not highly biased and the second section includes a proof by contradiction that describes the event fundamentally. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.
Category: Set Theory and Logic

[80] viXra:1508.0089 [pdf] replaced on 2016-02-13 03:43:30

A Probabilistic Proof of the Existence of Extraterrestrial Life

Authors: Peiman Ghasemi
Comments: 7 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. The first section includes an algebraic probabilistic proof when the event of life is not highly biased and the second section includes a proof by contradiction that describes the event fundamentally. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.
Category: Set Theory and Logic

[79] viXra:1508.0089 [pdf] replaced on 2015-10-13 11:16:10

A Probabilistic Proof of the Existence of Extraterrestrial Life

Authors: Peiman Ghasemi
Comments: 6 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.
Category: Set Theory and Logic