Set Theory and Logic

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Recent submissions

Any replacements are listed further down

[70] viXra:1409.0056 [pdf] submitted on 2014-09-08 16:49:27

An Origin of Transitivity and Other Useful Relation’s Properties

Authors: Misha Mikhaylov
Comments: 11 Pages.

There was an attempt to formalize an appearance of main relation properties in contrast to the usual one. There were paid an attention only for natural relations. Surely, such relations as “better than” are not observed here since the notion of “good” itself is still not defined.
Category: Set Theory and Logic

[69] viXra:1409.0041 [pdf] submitted on 2014-09-06 09:58:42

The Axiomatic Definition of Infinitesimals

Authors: Ihsan Raja Muda Nasution
Comments: 1 Page.

In this paper, we redefine the infinitesimals by using axiomatic method.
Category: Set Theory and Logic

[68] viXra:1409.0006 [pdf] submitted on 2014-09-02 01:36:05

What Mathematics Is The Most Fundamental?

Authors: Felix M. Lev
Comments: 5 Pages.

Standard mathematics involves such notions as infinitely small/large, continuity and standard division. However, some of these notions are treated differently in traditional and constructive versions. This mathematics is usually treated as fundamental while finite mathematics is treated as inferior. Standard mathematics has foundational problems (as follows, for example, from G\"{o}del's incompleteness theorems) but people usually believe that this is less important than the fact that it describes many experimental data with high accuracy. We argue that the situation is the opposite: standard mathematics is only a special case of finite one in the formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity. Therefore foundational problems in standard mathematics are not fundamental.
Category: Set Theory and Logic

[67] viXra:1408.0211 [pdf] submitted on 2014-08-29 10:11:03

On the Infinitesimals

Authors: Ihsan Raja Muda Nasution
Comments: 1 Page.

Our result is the explicit form of the infinitesimals.
Category: Set Theory and Logic

[66] viXra:1408.0039 [pdf] submitted on 2014-08-07 22:42:05

Simplifying Compound Proposition and Logical Equation

Authors: Oh Jung Uk
Comments: 8 Pages.

If B(P) is the truth value of proposition P then connectives could be translated to arithmetic operation in the congruent expression of 2 as described below. B(~p)≡1+B(p)( mod 2 ),B(p∧q)≡B(p)B(q)( mod 2 ) B(p∨q)≡B(p)+B(q)+B(p)B(q)( mod 2 ) B(p⟶q)≡1+B(p)+B(p)B(q)( mod 2 ) B(p⟷q)≡1+B(p)+B(q)(mod 2) By using this, logical laws could be proved, compound proposition could be simplified, and logical equation could be solved.
Category: Set Theory and Logic

[65] viXra:1407.0033 [pdf] submitted on 2014-07-04 06:53:06

Building a Foolproof Navigation System: Fuzzy Logic Emulating the Brain

Authors: Subhajit Ganguly
Comments: 19 Pages.

Our aim is to help build a machine that can reduce the possibility of mishaps in navigation to zero. For that devise a new system of numbers, in which the real numbers are represented on the y-axis and complex numbers on the x-axis. Inside such a system, we incorporate the equivalent Ideal Fuzzy Logic that can be used by the machine to predict and avoid mishaps.
Category: Set Theory and Logic

[64] viXra:1406.0115 [pdf] submitted on 2014-06-18 12:02:18

The Sohraab-Hyder or SH Set Theory

Authors: Mustafa A. Khan
Comments: 9 Pages.

This paper introduces a new type of set called the Sohraab-Hyder set or the SH set which is quite different from the well known and studied set of the Set Theory. This new set theory is logically consistent and leads to some fascinating results such as, (1) the necessary existence of an ultimate SH set that created itself in pre-eternity and will continue to exist to post-eternity, (2) the ultimate SH set is self sufficient, (2) the ultimate SH set must give rise to infinite number of other SH sets which represent logically consistent algebras and geometries, (3) the ultimate SH set must give rise to the existence of infinite universes, including our own, (4) the ultimate SH set must give rise to mathematical cells, organs and organisms which undergo must undergo evolution, (5) some of these mathematical cells, organs and organisms are same as the organic cells, organs and organisms which leads to the conclusion that the evolution of organisms is a mathematical fact and finally, (6) it is shown that every living and non-living entity in the universe, with the universe included of course, is a matrix of 0's and 1's.
Category: Set Theory and Logic

[63] viXra:1406.0100 [pdf] submitted on 2014-06-16 05:54:18

An Alteration in the Foundations of the Theory of Ultralogics

Authors: Robert A. Herrmann
Comments: 4 Pages.

In 1963, nonstandard analysis was applied to the semantics of formal languages. In 1978, nonstandard analysis was first applied to the syntax of formal and informal languages via the notion of the consequence operator (operation) and later the logic-system. In order to incorporate both of these approaches into the Theory of Ultralogics, a foundational change is necessary.
Category: Set Theory and Logic

[62] viXra:1405.0360 [pdf] submitted on 2014-05-30 12:59:54

The Logic of Complexity. When One and One do not Add up to Two

Authors: Ricardo Alvira
Comments: 42 Pages.

Presentation based on book expected to be published in 1-2 months: "A Unified Theory of Complexity". ISBN: 978-1499335859. 250 pages aprox.
Category: Set Theory and Logic

[61] viXra:1405.0227 [pdf] submitted on 2014-05-14 03:52:18

Jedyne na ŚWIECIE ROZWIĄZANIE Najtrudniejszej Zagadki ŚWIATA

Authors: Leszek W. Guła
Comments: 2 Pages. no

Rozwiązanie najtrudniejszej zagadki świata
Category: Set Theory and Logic

[60] viXra:1405.0225 [pdf] submitted on 2014-05-13 07:06:04

On a Counterexample to Cantor's Diagonal Argument

Authors: Jailton C. Ferreira
Comments: 3 Pages.

A counterexample to Cantor's Diagonal Argument is revisited.
Category: Set Theory and Logic

[59] viXra:1405.0202 [pdf] submitted on 2014-05-10 01:05:38

Godel's Incompleteness Theorem And Logic

Authors: Bertrand Wong
Comments: 3 Pages.

This article raises some important points about logic, e.g., mathematical logic.
Category: Set Theory and Logic

[58] viXra:1404.0454 [pdf] submitted on 2014-04-24 08:43:58

A 'logical' Approach to Urban Sustainability Indicator Design

Authors: Ricardo Alvira
Comments: 14 Pages. To appear as “Un acercamiento lógico al diseño de indicadores de sostenibilidad urbana” en Urbanismo y Sostenibilidad en La Ciudad. Edited by BREEAM ES and ITG [currently in press]

The present text proposes a methodology for designing urban sustainability indicators based on set theory / fuzzy logic. The aim is to provide a sound basis that increases ‘understandability’ and ‘shareability’ of sustainability indicators. While the analysis focuses on urban sustainability indicators, its conclusions can be applied to many different type of indicators, not only referred to sustainability but to many different issues.
Category: Set Theory and Logic

[57] viXra:1402.0109 [pdf] submitted on 2014-02-16 14:19:12

Antisets as Multisets

Authors: Heitor Baldo
Comments: 3 Pages.

This paper is a complement of my previous paper [3] which I introduce the concept of Antisets as multisets of negative spaces. Here I pretend to explain more clearly the structure of antisets; to be seen as multisets consisting of peculiar mathematical entities called negative spaces or "holes".
Category: Set Theory and Logic

[56] viXra:1401.0149 [pdf] submitted on 2014-01-22 05:43:50

What Gödel's Theorem Really Proves

Authors: Antonio Leon
Comments: 14 Pages.

It is proved in this paper the undecidable formula involved in Gödel's first incompleteness theorem would be inconsistent if the formal system where it is defined were complete. So, before proving the formula is undecidable it is necessary to assume the system is not complete in order to ensure the formula is not inconsistent. Consequently, Gödel proof does not prove the formal system is incomplete but that, once assumed it is incomplete, it is possible to define an undecidable formula within the system. This conclusion makes Gödel's incompleteness theorems devoid of substance.
Category: Set Theory and Logic

[55] viXra:1310.0242 [pdf] submitted on 2013-10-28 02:44:24

Infinity at Stake

Authors: Antonio Leon
Comments: 190 Pages.

Selected set theory and supertask arguments on the formal consistency of the actual infinity hypothesis subsumed by the Axiom of Infinity.
Category: Set Theory and Logic

[54] viXra:1310.0221 [pdf] submitted on 2013-10-24 16:37:16

The Characteristic Function of a Neutrosophic Set

Authors: A. A. Salama, Mohamed eisa, Florentin Smarandache
Comments: 1 Page.

The purpose of this paper is to introduce and study the characteristic function of a neutrosophic set. After given the fundamental definitions of neutrosophic set operations generated by, we obtain several properties, and discussed the relationship between neutrosophic sets generated by Ng and others. Finally, we introduce the neutrosophic topological spaces generated by Ng . Possible application to GIS topology rules are touched upon.
Category: Set Theory and Logic

[53] viXra:1310.0055 [pdf] submitted on 2013-10-08 11:54:01

Sets and Modern Non-Linear Graph Theory

Authors: F. U. Yu, M. R. Catra
Comments: 12 Pages.

Assume there exists a pseudo-elliptic monoid. We wish to extend the results of [39] to meromorphic, positive, contra-everywhere open planes. We show that E = ;. It has long been known that u < i [39]. In [39], it is shown that sF;p 6= V00.
Category: Set Theory and Logic

[52] viXra:1309.0028 [pdf] submitted on 2013-09-05 20:36:23

N-Valued Refined Neutrosophic Logic and Its Applications to Physics

Authors: Florentin Smarandache
Comments: 9 Pages.

In this paper we present a short history of logics: from particular cases of 2-symbol or numerical valued logic to the general case of n-symbol or numerical valued logic. We show generalizations of 2-valued Boolean logic to fuzzy logic, also from the Kleene’s and Lukasiewicz’ 3-symbol valued logics or Belnap’s 4-symbol valued logic to the most general n-symbol or numerical valued refined neutrosophic logic. Two classes of neutrosophic norm (n-norm) and neutrosophic conorm (n-conorm) are defined. Examples of applications of neutrosophic logic to physics are listed in the last section. Similar generalizations can be done for n-Valued Refined Neutrosophic Set, and respectively n- Valued Refined Neutrosopjhic Probability.
Category: Set Theory and Logic

[51] viXra:1309.0013 [pdf] submitted on 2013-09-04 07:32:54

General Logic-Systems that Determine Significant Collections of Consequence Operators

Authors: Robert A. Herrmann
Comments: 19 Pages.

Relative to universal logic, it is demonstrated how useful it is to utilize general logic-systems to investigate finite consequence operators (operations). Among many other examples relative to the lattice of finite consequence operators, a general characterization for the lattice-theoretic supremum for a nonempty collection of finite consequence operators is given. Further, it is shown that for any denumerable language there is rather simple collection of finite consequence operators and for a propositional language, three simple modifications to the finitary rules of inference that demonstrate that the lattice of finite consequence operators is not meet-complete.
Category: Set Theory and Logic

[50] viXra:1307.0124 [pdf] submitted on 2013-07-23 18:54:31

Proceedings of the First International Conference on Smarandache Multispace & Multistructure

Authors: editor Linfan Mao
Comments: 135 Pages.

The First International Conference on Smarandache Multispace and Multistructure was organized by Prof. Linfan Mao, and it was held in the Beijing University of Civil Engineering and Architecture of P. R. China on June 28-30, 2013. There were 46 researchers from China, India, Iran, Nigeria, and USA that have taken part in this conference with 14 papers on Smarandache multispace and geometry, birings, neutrosophy,neutrosophic groups, regular maps and topological graphs with applications to non-solvable equation systems. Definition. In any domain of knowledge, a Smarandache multispace (or S-multispace) with its multistructure is a finite or infinite (countable or uncountable) union of many spaces that have various structures. The spaces may overlap. The notions of multispace (also spelt multi-space) and multi- structure (also spelt multi-structure) were introduced by Smarandache in 1969 under his idea of hybrid science: combining different fields into a unifying field, which is closer to our real life world since we live in a heterogeneous space. Today, this idea is widely accepted by the world of sciences. S-multispace is a qualitative notion, since it is too large and includes both metric and non-metric spaces. It is believed that the smarandache multispace with its multistructure is the best candidate for 21st century Theory of Everything in any domain. It unifies many knowledge fields. Applications. A such multispace can be used for example in physics for the Unified Field Theory that tries to unite the gravitational, electromagnetic, weak and strong interactions. Or in the parallel quantum computing and in the mu-bit theory, in multi-entangled states or particles and up to multi-entangles objects. We also mention: the algebraic multispaces (multi-groups, multi-rings, multi-vector spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold, etc.), geometric multispaces (combinations of Euclidean and Non-Euclidean geometries into one space as in Smarandache geometries), theoretical physics, including the relativity theory, the M-theory and the cosmology, then multi-space models for p-branes and cosmology, etc. - The multispace and multistructure were first used in the Smarandache geometries (1969), which are combinations of different geometric spaces such that at least one geometric axiom behaves differently in each such space. - In paradoxism (1980), which is a vanguard in literature, arts, and science, based on finding common things to opposite ideas [i.e. combination of contradictory fields]. - In neutrosophy (1995), which is a generalization of dialectics in philosophy, and takes into con- sideration not only an entity < A > and its opposite < antiA > as dialectics does, but also the neutralities < neutA > in between. Neutrosophy combines all these three < A >,< antiA > and < neutA > together. Neutrosophy is a metaphilosophy. - Then in neutrosophic logic (1995), neutrosophic set (1995), and neutrosophic probability (1995), which have, behind the classical values of truth and falsehood, a third component called indeterminacy (or neutrality, which is neither true nor false, or is both true and false simultaneously - again a combination of opposites: true and false in indeterminacy). - Also used in Smarandache algebraic structures (1998), where some algebraic structures are in- cluded in other algebraic structures.
Category: Set Theory and Logic

[49] viXra:1306.0209 [pdf] submitted on 2013-06-25 11:06:36

A 15-Page Collection of Notes on my Mathematical Articles

Authors: Andrew Nassif
Comments: 15 Pages. May need formatting, but not sure because they are notes.

A 15 page collection of my notes on mathematical articles I have created.
Category: Set Theory and Logic

[48] viXra:1303.0084 [pdf] submitted on 2013-03-11 13:38:33

A Solution to The Unexpected Examination Paradox

Authors: Mosayyeb Fakhreslam
Comments: 2 Pages.

In this paper we offer a solution to the unexpected examination paradox by introducing the unexpected examinationless day paradox.
Category: Set Theory and Logic

[47] viXra:1302.0137 [pdf] submitted on 2013-02-20 13:01:57

Restructuring Logic

Authors: Joachim Derichs
Comments: 44 Pages.

The outline of a programme for restructuring mathematical logic. We explain what we mean by ‘restructuring’ and carry out exemplary parts of the programme.
Category: Set Theory and Logic

[46] viXra:1302.0048 [pdf] submitted on 2013-02-08 12:33:56

Inconsistent Countable Set

Authors: Jaykov Foukzon
Comments: 13 Pages.

In this article we derived an importent example of the inconsistent countable set. Main result is: ~con(ZFC+E(\omega-model of ZFC)).
Category: Set Theory and Logic

[45] viXra:1212.0127 [pdf] submitted on 2012-12-20 14:29:35

Ultra-Universal Models

Authors: Colin Naturman, Henry Rose
Comments: 10 Pages.

The concept of ultra-universal algebras in varieties is generalized to models of first order theories. Characterizations of theories which have ulta-universal models are found and general examples of ultra-universal models are investigated. In particular we show that a theory has an ultra-universal model iff it is consistent and its class of models satisfies the joint embedding property.
Category: Set Theory and Logic

[44] viXra:1212.0088 [pdf] submitted on 2012-12-13 06:50:57

Difficulties of the Set of Natural Numbers

Authors: Qiu Kui Zhang
Comments: 8 Pages.

In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.
Category: Set Theory and Logic

[43] viXra:1209.0070 [pdf] submitted on 2012-09-20 17:26:27

Interpreting Sergeyev's Numerical Methodology Within a Hyperreal Number System

Authors: Nader Vakil
Comments: 4 Pages.

In this paper we show the consistency of the essential part of Sergeyev's numerical methodology (\cite{Yarov 1}, \cite{Yarov 2}) by constructing a model of it within the framework of an ultrapower of the ordinary real number system.
Category: Set Theory and Logic

[42] viXra:1207.0064 [pdf] submitted on 2012-07-17 02:27:13

Zorn's Lemma

Authors: Pierre-Yves Gaillard
Comments: 1 Page.

We give a short proof of Zorn's Lemma.
Category: Set Theory and Logic

[41] viXra:1207.0039 [pdf] submitted on 2012-07-11 06:01:20

Universes

Authors: Pierre-Yves Gaillard
Comments: 2 Pages.

The book "Categories and Sheaves" by Kashiwara and Schapira starts with a few statements which are not proved, a reference being given instead. We spell out the proofs in a short and self-contained way.
Category: Set Theory and Logic

[40] viXra:1207.0009 [pdf] submitted on 2012-07-03 22:07:09

Definition à la Bourbaki of the Basic Notions of Category Theory

Authors: Pierre-Yves Gaillard
Comments: 2 Pages.

We give definitions in the spirit of Bourbaki's Set Theory for the basic notions of category theory. The goal is to avoid using either Grothendieck's universes axiom, or ``classes'' (or ``collections'') of sets which are not sets.
Category: Set Theory and Logic

[39] viXra:1206.0106 [pdf] submitted on 2012-07-01 00:39:37

A Question About the Book "Categories and Sheaves"

Authors: Pierre-Yves Gaillard
Comments: 6 Pages.

This is the beginning of an attempt at rewriting the book "Categories and Sheaves" by Kashiwara and Schapira without using Grothendieck's universes axiom.
Category: Set Theory and Logic

[38] viXra:1206.0030 [pdf] submitted on 2012-06-09 09:13:06

On Cantor’s Actual Infinity

Authors: Andrew Banks
Comments: 8 Pages.

This paper will demonstrate a diagonal argument by listing all non-empty finite ordinals in a table according to their ε order using their subset representation, meaning {0,1,2…n-1} is listed for the ordinal n. Next, the axiom of choice is applied to all of these ordinals and selects the maximal element. This selection process forms a diagonal which satisfies the axiom of infinity, hence, the diagonal is a limit ordinal. However, it will also be shown for the nth choice made by the choice function, the diagonal is the successor ordinal number n = {0,1,2…n-1} and this is true for all n. So, at the n+1 choice, the diagonal is the ordinal n+1 and so on. Therefore, based on all the actions of the choice function, it is provable from ZFC on one hand that this diagonal cannot ever be anything other than a successor ordinal and on the other hand, the diagonal is a limit ordinal.
Category: Set Theory and Logic

[37] viXra:1204.0030 [pdf] submitted on 2012-04-08 14:42:29

Solution for Hadwiger Conjecture

Authors: Wilber Valgusbitkevyt
Comments: 2 Pages.

Reverse Modus Ponens followed by set theory using lines followed by considering the maximum number of colours that can be used using graph homomorphism.
Category: Set Theory and Logic

[36] viXra:1204.0012 [pdf] submitted on 2012-04-03 20:00:22

Solution to Ringel–Kotzig Conjecture

Authors: Wilber Valgusbitkevyt
Comments: 3 Pages. I may upload another draft for more descriptive and elaborative explanations.

If there is something I am not explaining very elaborately or descriptively, let me know. I have had a math professor who showed me her inconsistent equations saying "these equations have no solution" although my equations are consistent and homogeneous which always have a solution. Also, I have had another math professor who told me it is wrong to assign specific values to variables although I was assigning factor variables to composite variables not to mention how there is an underlying condition how these numbers are positive integers excluding zero. In this paper, I am creating a new theorem called Victoria Hayanisel Theorem dedicated to Princess Eugenie of York to describe the state of numbers, circles, and lines. Followed by the theorem, I am using the set theory and Fermat's Infinite Descent Method (if my method is different, I will name it) to show how the conjecture is true.
Category: Set Theory and Logic

[35] viXra:1204.0011 [pdf] submitted on 2012-04-03 20:03:25

Solution to Collatz Conjecture

Authors: Wilber Valgusbitkevyt
Comments: 2 Pages.

There is a pattern for the arbitary sequence which can be divided into four groups to be formalized as a recursive formula which shows the conjecture is true.
Category: Set Theory and Logic

[34] viXra:1204.0009 [pdf] submitted on 2012-04-03 20:08:46

Solution for Erdős–Faber–Lovász Conjecture

Authors: Wilber Valgusbitkevyt
Comments: 2 Pages.

Translate the graph into sets by using the shared vertexes as indexes. Then consider several different cases to see how the conjecture is true in each case.
Category: Set Theory and Logic

[33] viXra:1203.0101 [pdf] submitted on 2012-03-28 18:07:22

On Cantor’s Completed Infinity

Authors: Andrew Banks
Comments: 9 Pages.

The debate between process infinity and Cantor’s eigentlich Unendliche “completed infinity” has occurred since before Greek times. Prior to Cantor, the prevailing view of infinity was that it is a process that continues on forever and there is only one type of infinity. Cantor, on the other hand, produced the current foundations of mathematics with his hierarchy of completed infinite objects. In particular, the completed infinite set ω contains all natural numbers and none are missing from the set. This paper will demonstrate, however, a specific method under ZFC of assembling all finite ordinals into the completed set ω such that ω ε ω is a necessary condition of that formation. Then, from ω ε ω, it will be shown ZFC is inconsistent.
Category: Set Theory and Logic

[32] viXra:1202.0070 [pdf] submitted on 2012-02-21 04:37:25

Théorie Mathématique Platoniste-Théorie Aléatoire Des Nombres

Authors: Thierry Delort
Comments: 120 Pages.

This document, written in French, contains 2 parts: In the 1st part, Théorie mathématique Platoniste (Platonic mathematical theory) we expose a complete Platonic theory, covering all the fiels of logic and foundation of mathematics, including a complete set theory. In the 2nd part, Théorie aléatoire des nombres (random theory of number)we expose a theory of random in mathematics, that can be considered as as a branch of logic as well as a branch of number theory. In particular we show that it gives a theoretical justification of the Goldbach conjecture (weak and strong) as well of the twin prime Conjecture.
Category: Set Theory and Logic

[31] viXra:1110.0055 [pdf] submitted on 18 Oct 2011

Analytic Gauge Functions, Invariances, and Modular Curves

Authors: Thomas Evans
Comments: 19 pages

It is the underlying purpose of the author throughout this and subsequent related papers to consider the examination of conjectures such as the Birch-Swinnerton-Dyer conjecture, the Riemann Hypotheses, as well as a number of other misunderstood or unacknowledged phenomena. It is the author's hope that through such considerations, both autonomous and presented herein, that it may become evident that the introduction of fundamental, new practices is a necessity to any advancement in the directions of the aforementioned. This represents the first in a series of eight (8) papers regarding these materials. Throughout the remaining 7 the author presents, to a much greater degree of rigor, the basic theory of analytic gauge functions, associated phenomenology, and there from a solution to the (two) above conjectures. This paper facilitates an introduction to the theory of analytic gauges. In the first section the author presents a re-examination of the concepts of geometries of connections. Very briefly introduced are the basic concepts of analytic numbers, analytic fields, analytic gauge functions, etc.
Category: Set Theory and Logic

[30] viXra:1108.0025 [pdf] submitted on 19 Aug 2011

Logical Ramblings

Authors: Thomas Evans
Comments: 11 pages

I present extensions to logic theory whose utilitarian application contains itself in the form of a developmental, logical framework determinant of all being, and then derive several applications thereof to areas of general quantum theory and pure mathematics, providing solutions to 2 longstanding relevant problems: P vs NP and the Riemann Hypothesis.
Category: Set Theory and Logic

[29] viXra:1108.0011 [pdf] submitted on 4 Aug 2011

Neutrosophic Logics on Non-Archimedean Structures

Authors: Andrew Schumann
Comments: 23 pages

We present a general way that allows to construct systematically analytic calculi for a large family of non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes' axiom. These logics are built as different extensions of standard many-valued logics (namely, Lukasiewicz's, Gödel's, Product, and Post's logics). The informal sense of Archimedes' axiom is that anything can be measured by a ruler. Also logical multiple-validity without Archimedes' axiom consists in that the set of truth values is infinite and it is not well-founded and well-ordered. We consider two cases of non-Archimedean multi-valued logics: the first with many-validity in the interval [0; 1] of hypernumbers and the second with many-validity in the ring Zp of p-adic integers. On the base of non-Archimedean valued logics, we construct non-Archimedean valued interval neutrosophic logics by which we can describe neutrality phenomena.
Category: Set Theory and Logic

[28] viXra:1107.0045 [pdf] submitted on 23 Jul 2011

A Different Approach to Logic

Authors: Mauro Avon
Comments: 158 pages

This paper outlines an approach to mathematical logic which is different from the standard one. We list the most relevant features of the system. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. The set-builder notation is enclosed as an expression-building pattern. In our system we can easily express second-order and all-order conditions (the set to which a quantifier refers is explicitly written in the expression). The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved, as well as the fact that our system is not affected by the most known types of paradox. The paper provides both the theoretical material and two fully documented examples of deduction. The author has built the whole system with the idea to provide a faithful model of human mathematical deductive process. He believes this objective has been achieved but obviously the reader is free to examine the system and get his own opinion about it.
Category: Set Theory and Logic

[27] viXra:1010.0052 [pdf] submitted on 20 Mar 2010

Neutrosophic Degree of a Paradoxicity

Authors: Florentin Smarandache
Comments: 3 pages

A paradox is called a statement <P> which is true and false in the same time. Therefore, if we suppose that statement <P> is true, it results that <P> is false; and reciprocally, if we suppose that <P> is false, it results that <P> is true.
Category: Set Theory and Logic

[26] viXra:1010.0002 [pdf] submitted on 1 Oct 2010

The Gëdel Thesis 'about a Incompleteness of Formal Arithmetics' is not Proved and This Hypothesis is not Similar to the Truth

Authors: Dm. Vatolin
Comments: 11 pages, Russian.

In this paper the question is examined that a incompleteness of enough advanced theories of arithmetics does not follow from the Gëdel statements.
Category: Set Theory and Logic

[25] viXra:1008.0091 [pdf] submitted on 31 Aug 2010

A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics

Authors: Florentin Smarandache
Comments: 157 pages

It was a surprise for me when in 1995 I received a manuscript from the mathematician, experimental writer and innovative painter Florentin Smarandache, especially because the treated subject was of philosophy - revealing paradoxes - and logics. He had generalized the fuzzy logic, and introduced two new concepts: a) "neutrosophy" - study of neutralities as an extension of dialectics; b) and its derivative "neutrosophic", such as "neutrosophic logic", "neutrosophic set", "neutrosophic probability", and "neutrosophic statistics" and thus opening new ways of research in four fields: philosophy, logics, set theory, and probability/statistics.
Category: Set Theory and Logic

[24] viXra:1005.0059 [pdf] submitted on 14 May 2010

Geometrical Axioms Refuting the Continuum-Hypothesis

Authors: Dm. Vatolin
Comments: 15 pages, Russian.

This article formulates three geometrical axioms from which it follows that the continuum power is greater then any well-ordered set power.
Category: Set Theory and Logic

[23] viXra:1005.0006 [pdf] submitted on 10 Mar 2010

Neutrality and Many-Valued Logics

Authors: Andrew Schumann, Florentin Smarandache
Comments: 121 pages

This book written by A. Schumann & F. Smarandache is devoted to advances of non-Archimedean multiple-validity idea and its applications to logical reasoning. Leibnitz was the first who proposed Archimedes' axiom to be rejected. He postulated infinitesimals (infinitely small numbers) of the unit interval [0, 1] which are larger than zero, but smaller than each positive real number. Robinson applied this idea into modern mathematics in [117] and developed so-called non-standard analysis. In the framework of non-standard analysis there were obtained many interesting results examined in [37], [38], [74], [117].
Category: Set Theory and Logic

[22] viXra:1004.0112 [pdf] submitted on 21 Apr 2010

Some Characterisations of α-Cut in Intuitionistic Fuzzy Set Theory

Authors: V. Veeramani, Roque Batulan
Comments: 7 pages

This paper contains the Basic Definitions of an Intuitionstic Fuzzy Set theory and operations on it. Mainly we discussed the basic concepts of α - cut with examples and Characterisations.
Category: Set Theory and Logic

[21] viXra:1004.0096 [pdf] submitted on 19 Apr 2010

A Note on Neutrosophy

Authors: Cheng-Gui Huang
Comments: 1 pages.

I claim that Neutrosophy, by Professor Florentin Smarandache, is a deep thought in human culture. That gives advantage to break the mechanical understanding of human culture. For example, according to the mechanical theory: existence and non-existence could not be simultaneously. Actually existence and non-existence are simultaneously. Everyone knows that human life is like a way in the empty space of a bird flying. Everyone can not see himself a second ago, everyone can not see himself for the time being and everyone can not see himself a second future. Everyone could not know what is the existence of self. Everyone is also difficult to say the non-existence of self. So the existence and non-existence of self are simultaneously. And the existence and nonexistence of everything are simultaneously, where, the law of excluded middle does not apply. These basic facts express the depth of Smarandache's Neutrosophy. He has a lot of friends in ancient and in nowadays, in the West and in the East.
Category: Set Theory and Logic

[20] viXra:1004.0092 [pdf] submitted on 19 Apr 2010

Toward Excitation and Inhibition in Neutrosophic Logic

Authors: Feng Liu
Comments: 8 pages.

Logic should have been defined as the unity of contradiction between logic director and logic implementation. Chinese Daoism asserts that everything is defined in the unity of opposites, namely yin and yang, accordingly yang conducts change and yin brings it up (I-Ching, also known as Book of Changes). In this way logic is redefined in an indeterminate style to facilitate "both A and Anti-A" etc. in neutrosophics of logic. The unity of opposites is also described as neutrality in neutrosophy. An intermediate multi-referential model of excitation and inhibition is developed to derive a multiagent architecture of logic, based on Chinese yin-yang philosophy. This methodology of excitation/inhibition suggests a rhymed way of logic, leading to a dynamic methodology of weight strategy that links logic with neural network approach. It also confirms the crucial role of indeterminacy in logic as a fatal criticism to classical mathematics and current basis of science.
Category: Set Theory and Logic

[19] viXra:1004.0091 [pdf] submitted on 19 Apr 2010

Truth and Absolute Truth in Neutrosophic Logic

Authors: Feng Liu
Comments: 12 pages.

As a philosophical analysis of some fatal paradoxes, the paper distinguishes the conceptual difference between representation of truth and source of truth, and leads to the conclusion that in order to acquire the genuine source of truth, independently of specific representations possibly belonging to different worlds, one is necessary to ignore all the ideas, logics, conceptions, philosophies and representable knowledge even himself belonging to those misleading worlds, returning to his infant nature, as a preliminary step for his cultivation of unconstrained wisdom. It also carries out some coordinative crucial issues as natural-doctrine, minded-unwitting, logic-infancy, conception-deconception, determinacy-indeterminacy. The paper tries to verify the role of neutrosophy and neutrosophic logic in religious issues and open a gateway toward the oriental classics, excavating the lost treasure.
Category: Set Theory and Logic

[18] viXra:1004.0065 [pdf] submitted on 10 Apr 2010

S-Denying a Theory

Authors: Florentin Smarandache
Comments: 8 pages

In this paper we introduce the operators of validation and invalidation of a proposition, and we extend the operator of S-denying a proposition, or an axiomatic system, from the geometric space to respectively any theory in any domain of knowledge, and show six examples in geometry, in mathematical analysis, and in topology.
Category: Set Theory and Logic

[17] viXra:1004.0051 [pdf] submitted on 8 Mar 2010

Single Valued Neutrosophic Sets

Authors: Haibin Wang, Florentin Smarandache, Yan-Qing Zhang, Rajshekhar Sunderraman
Comments: 4 pages

Neutrosophic set is a part of neutrosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Neutrosophic set is a powerful general formal framework that has been recently proposed. However, neutrosophic set needs to be specified from a technical point of view. To this effect, we define the settheoretic operators on an instance of neutrosophic set, we call it single valued neutrosophic set (SVNS). We provide various properties of SVNS, which are connected to the operations and relations over SVNS.
Category: Set Theory and Logic

[16] viXra:1004.0026 [pdf] submitted on 3 Apr 2010

Neutrosophic Diagram and Classes of Neutrosophic Paradoxes

Authors: Florentin Smarandache
Comments: 14 pages

These paradoxes are called "neutrosophic" since they are based on indeterminacy (or neutrality, i.e. neither true nor false), which is the third component in neutrosophic logic. We generalize the Venn Diagram to a Neutrosophic Diagram, which deals with vague, inexact, ambiguous, illdefined ideas, statements, notions, entities with unclear borders. We define the neutrosophic truth table and introduce two neutrosophic operators (neuterization and antonymization operators) give many classes of neutrosophic paradoxes.
Category: Set Theory and Logic

[15] viXra:1004.0016 [pdf] submitted on 8 Mar 2010

N-norm and N-conorm in Neutrosophic Logic and Set, and the Neutrosophic Topologies

Authors: Florentin Smarandache
Comments: 11 pages

In this paper we present the N-norms/N-conorms in neutrosophic logic and set as extensions of T-norms/T-conorms in fuzzy logic and set. Also, as an extension of the Intuitionistic Fuzzy Topology we present the Neutrosophic Topologies.
Category: Set Theory and Logic

[14] viXra:1004.0013 [pdf] submitted on 8 Mar 2010

n-ary Fuzzy Logic and Neutrosophic Logic Operators

Authors: Florentin Smarandache, V. Christianto
Comments: 15 pages

We extend Knuth's 16 Boolean binary logic operators to fuzzy logic and neutrosophic logic binary operators. Then we generalize them to n-ary fuzzy logic and neutrosophic logic operators using the smarandache codification of the Venn diagram and a defined vector neutrosophic law. In such way, new operators in neutrosophic logic/set/probability are built.
Category: Set Theory and Logic

[13] viXra:1004.0010 [pdf] submitted on 8 Mar 2010

Strategy on T, I, F Operators.

Authors: Florentin Smarandache
Comments: 6 pages

The paper presents an initial explorations on T, I, F operations based on genetic concept hierarchy and genetic referential hierarchy, as a novel proposal to the indeterminacy issue in neutrosophic logic, in contrast to the T, I, F values inherited from conventional logics in which those values would fail to demonstrate the genetic aspect of a concept and accordingly loose the connection between generality and practicality. Based on the novel definition of logic and on the relativity of T, F concept, it illustrates that T, F are hierarchical operations which inter-consist and inter-complement each other, that "I" relates to a learning behavior profiled by an inspiration from I-ching, and that the neutralization operation, as the means to solve contradictions, will eventually come to the unification of opposites, leading to the fundamental issues in Buddhism and such alike. It also implies that Buddhism and Daoism are not religions.
Category: Set Theory and Logic

[12] viXra:1004.0006 [pdf] submitted on 8 Mar 2010

Toward Dialectic Matter Element of Extenics Model

Authors: Florentin Smarandache
Comments: 10 pages

Based on the authors intensive investigation on the oriental dialectics, the paper presents a novel theoretical frame of matter element in the world leading science, extenics dealing with inconsistency or incompatibility, covering the widest range of application area from informatics, system engineering to management and finance. The dialectic matter-element is defined as the integral of all existing and prospecting ones based on all the infinite possible cognitive models. The novel model serves as the origin of constraint matter elements, the unity of both state description and cognitive action (cognition force with respect to neural science), a latent part of extenics, and possibly as essence of matter element. It explains, in a novel perspective, the origin of a name, and uncovers the source of contradiction and even the impetus of cognition.
Category: Set Theory and Logic

[11] viXra:1003.0269 [pdf] submitted on 8 Mar 2010

The Smarandache's Class of Paradoxes

Authors: C. Le
Comments: 4 pages, edited by C. Le, and translated into German by Bernd Hutschenreuther

The Smarandache's Class of Paradoxes are semantic paradoxes of the form "All is <A>, the <nonA> too!", where <nonA> is what is not <A>. As a particular case, replacing <A> but an attribute (or, in general, by an idea) it is well know the Smarandache semantic paradox: "All is possible, the impossible too!" which is the motto of the Paradoxism movement in arts, letters, and sciences.
Category: Set Theory and Logic

[10] viXra:1003.0224 [pdf] submitted on 7 Mar 2010

Introduction to Neutrosophic Logic

Authors: Charles Ashbacher
Comments: 145 pages

As someone who works heavily in both math and computers, I can truly appreciate the role that logic plays in our modern world. One cannot understand the foundations of mathematics while lacking knowledge of the basics of logic and how proofs are constructed. Two of the first classes I took as a graduate student in mathematics were in the foundations of mathematics, and hardly a day goes by where I do not use some topic from those courses.
Category: Set Theory and Logic

[9] viXra:1003.0171 [pdf] submitted on 6 Mar 2010

Deducibility Theorems in Boolean Logic

Authors: Florentin Smarandache
Comments: 4 pages

In this paper we give two theorems from the Propositional Calculus of the Boolean Logic with their consequences and applications and we prove them axiomatically.
Category: Set Theory and Logic

[8] viXra:1003.0167 [pdf] submitted on 6 Mar 2010

A Class of Recursive Sets

Authors: Florentin Smarandache
Comments: 7 pages

In this article one builds a class of recursive sets, one establishes properties of these sets and one proposes applications. This article widens some results of [1].
Category: Set Theory and Logic

[7] viXra:1003.0119 [pdf] submitted on 6 Mar 2010

Funny Problems!

Authors: Florentin Smarandache
Comments: 6 pages

Thirty original and collected problems, puzzles, and paradoxes in mathematics and physics are explained in this paper, taught by the author to the elementary and high school teachers at the University of New Mexico - Gallup in 1997-8 and afterwards. They have a more educational interest because make the students think different!
Category: Set Theory and Logic

[6] viXra:1003.0117 [pdf] submitted on 6 Mar 2010

Linguistic Paradoxes and Tautologies

Authors: Florentin Smarandache
Comments: 12 pages

Classes of linguistic paradoxes are introduced with examples and explanations. They are part of the author's work on the Paradoxist Philosophy based on mathematical logic. The general cases exposed below are modeled on the English language structure in a rigid way. In order to find nice particular examples of such paradoxes one grammatically adjusts the sentences.
Category: Set Theory and Logic

[5] viXra:1003.0065 [pdf] submitted on 6 Mar 2010

Logic: a Misleading Concept. a Contradiction Study Toward Agent's Logic Ontology

Authors: Feng Liu, Florentin Smarandache
Comments: 10 pages

The paper presents a fresh new comprehensive ideology on Neutrosophic Logic based on contradiction study in a broad sense: general critics on conventional logic by examining the essence of logic, fresh insights on logic definition based on Chinese philosophical survey, and a novel and genetic logic model as the elementary cell against Von Neumann oriented ones based on this novel definition. As for the logic definition, the paper illustrates that logic is rather a tradeoff between different factors than truth and false abstraction. It is stressed that the kernel of any intelligent system is exactly a contradiction model. The paper aims to solve the chaos of logic and exhibit the potential power of neutrosophy: a new branch of scientific philosophy.
Category: Set Theory and Logic

[4] viXra:1003.0062 [pdf] submitted on 6 Mar 2010

Intentionally and Unintentionally. On Both, A and Non-A, in Neutrosophy

Authors: Feng Liu, Florentin Smarandache
Comments: 7 pages

The paper presents a fresh new start on the neutrality of neutrosophy in that "both A and Non-A" as an alternative to describe Neuter-A in that we conceptualize things in both intentional and unintentional background. This unity of opposites constitutes both objective world and subjective world. The whole induction of such argument is based on the intensive study on Buddhism and Daoism including I-ching. In addition, a framework of contradiction oriented learning philosophy inspired from the Later Trigrams of King Wen in I-ching is meanwhile presented. It is shown that although A and Non-A are logically inconsistent, but they are philosophically consistent in the sense that Non-A can be the unintentionally instead of negation that leads to confusion. It is also shown that Buddhism and Daoism play an important role in neutrosophy, and should be extended in the way of neutrosophy to all sciences according to the original intention of neutrosophy.
Category: Set Theory and Logic

[3] viXra:0912.0017 [pdf] submitted on 8 Dec 2009

Resolving Russell's Paradox Within Cantor's Intuitive Set Theory

Authors: Feng Xu
Comments: 6 pages, first published in 2006 in Hadronic Journal, volume 29, page 227

The set of all the subsets of a set is its power set, and the cardinality of the power set is always larger than the set and its subsets. Based on the definition and the inequality in cardinality, a set cannot include its power set as element, and a power set cannot include itself as element. "Russell's set" is a putative set of all the sets that don't include themselves as element. It can be shown, however, that "Russell's set" can never take in all such sets. This is because its own power set, which (like any power set) is a set that doesn't include itself (thus qualifies as an element for "Russell's set"), cannot (although should) be taken in due to the cardinality inequality. Thus "Russell's set" can never be formed. Without it, Russell's paradox, which forced the modification of Cantor's intuitive set theory into a more restricted axiomatic theory, can never be formulated. The reported approach to resolve Russell's paradox is fundamentally different from the conventional approaches. It may restore the self-consistency of Cantor's original set theory, make the Axiom of Regularity unnecessary, and expand the coverage of set to assemblies that include themselves as element.
Category: Set Theory and Logic

[2] viXra:0910.0041 [pdf] submitted on 21 Oct 2009

Experimental Proof for Gödel Theorem on Time

Authors: Amrit S. Sorli
Comments: 2 pages

In 1949, Gödel postulated a theorem that stated: "In any universe described by the theory of relativity, time cannot exist". Gödel idea was that forth coordinate of space-time is not time. Fourth coordinate is spatial too. In this article will be shown that on the base of elementary perception and experimental data Gödel theorem is right. With eyes one observes universe is in a continuous change. A change n gets transformed into a change n+1, the change n+1 into a change n+2 and so on. Clocks measure a frequency, velocity and numerical order of change. Experimental date confirms that changes and clocks do not run time; they run in space only. Time is not a part of space. Fourth coordinate of space-time is spatial too. Space itself is timeless. Physical time that is clocks run is man created physical reality. Fundamental arena of the universe is timeless space. In the timeless space into which massive bodies and elementary particles move there is no past and no future. Past and future belong to the inner neuronal space-time that is a result of neuronal activity of the brain.
Category: Set Theory and Logic

[1] viXra:0909.0039 [pdf] submitted on 16 Sep 2009

Formalistics of Generalization

Authors: Victor Porton
Comments: 2 Pages.

In the framework of ZF formally considered generalizations, such as whole numbers generalizing natural number, rational numbers generalizing whole numbers, real numbers generalizing rational numbers, complex numbers generalizing real numbers, etc. The formal consideration of this may be especially useful for computer proof assistants.
Category: Set Theory and Logic

Replacements of recent Submissions

[54] viXra:1409.0056 [pdf] replaced on 2014-09-14 04:03:45

An Origin of Transitivity and Other Useful Relation’s Properties

Authors: Misha Mikhaylov
Comments: 11 Pages.

There was an attempt to formalize an appearance of main relation properties in contrast to the usual one. There was paid an attention only for natural relations. In fact, such relations as “better than” are not observed here since the notion of “good” is still not defined.
Category: Set Theory and Logic

[53] viXra:1408.0211 [pdf] replaced on 2014-08-29 22:58:49

On the Infinitesimals

Authors: Ihsan Raja Muda Nasution
Comments: 2 Pages.

Our result is the explicit form of the infinitesimals.
Category: Set Theory and Logic

[52] viXra:1405.0360 [pdf] replaced on 2014-07-23 14:52:39

The Logic of Complexity. When One and One do not Add up to Two

Authors: Ricardo Alvira
Comments: 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 278 pages aprox. The book is currently available in spanish as 'Una Teoría Unificada de la Complejidad'
Category: Set Theory and Logic

[51] viXra:1405.0360 [pdf] replaced on 2014-06-30 05:41:35

The Logic of Complexity. When One and One do not Add up to Two

Authors: Ricardo Alvira
Comments: 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 270 pages aprox. The book is currently available in spanish as 'Una Teoria Unificada de la Complejidad'
Category: Set Theory and Logic

[50] viXra:1405.0360 [pdf] replaced on 2014-06-23 06:15:21

The Logic of Complexity. When One and One do not Add up to Two

Authors: Ricardo Alvira
Comments: 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 270 pages aprox. The book is currently available in spanish as 'Una Teoria Unificada de la Complejidad'
Category: Set Theory and Logic

[49] viXra:1405.0360 [pdf] replaced on 2014-05-30 17:14:51

The Logic of Complexity. When One and One do not Add up to Two

Authors: Ricardo Alvira
Comments: 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 250 pages aprox. The book is expected to be published soon.
Category: Set Theory and Logic

[48] viXra:1405.0202 [pdf] replaced on 2014-05-13 09:26:34

Godel's Incompleteness Theorem And Mathematics

Authors: Bertrand Wong
Comments: 3 Pages.

This article raises some important points about logic, e.g., mathematical logic.
Category: Set Theory and Logic

[47] viXra:1404.0454 [pdf] replaced on 2014-06-16 09:48:58

A 'logical' Approach to Urban Sustainability Indicator Design

Authors: Ricardo Alvira
Comments: 14 Pages.

The present text proposes a methodology for designing urban sustainability indicators based on set theory / fuzzy logic. The aim is to provide a sound basis that increases ‘understandability’ and ‘shareability’ of sustainability indicators. While the analysis focuses on urban sustainability indicators, its conclusions can be applied to many different type of indicators, not only referred to sustainability but to many different issues.
Category: Set Theory and Logic

[46] viXra:1310.0242 [pdf] replaced on 2014-08-30 05:09:07

Infinity at Stake

Authors: Antonio Leon
Comments: 198 Pages. Third edition

Selected set theory and supertask arguments on the formal consistency of the actual infinity hypothesis subsumed by the Axiom of Infinity.
Category: Set Theory and Logic

[45] viXra:1310.0242 [pdf] replaced on 2014-04-03 00:18:26

Infinity at Stake

Authors: Antonio Leon
Comments: 190 Pages.

Selected set theory and supertask arguments on the formal consistency of the actual infinity hypothesis subsumed by the Axiom of Infinity.
Category: Set Theory and Logic

[44] viXra:1310.0242 [pdf] replaced on 2014-01-21 13:17:58

Infinity at Stake

Authors: Antonio Leon
Comments: 190 Pages.

Selected set theory and supertask arguments on the formal consistency of the actual infinity hypothesis subsumed by the Axiom of Infinity.
Category: Set Theory and Logic

[43] viXra:1310.0242 [pdf] replaced on 2014-01-17 03:54:29

Infinity at Stake

Authors: Antonio Leon
Comments: 190 Pages.

Selected set theory and supertask arguments on the formal consistency of the actual infinity hypothesis subsumed by the Axiom of Infinity.
Category: Set Theory and Logic

[42] viXra:1310.0242 [pdf] replaced on 2014-01-08 13:15:23

Infinity at Stake

Authors: Antonio Leon
Comments: 192 Pages.

Selected set theory and supertask arguments on the formal consistency of the actual infinity hypothesis subsumed by the Axiom of Infinity.
Category: Set Theory and Logic

[41] viXra:1310.0075 [pdf] replaced on 2014-04-18 11:11:16

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis by categorical logic, proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that the theorems of the theory of cardinals are theorems of the theory of initial ordinals, and conversely, the theorems of the theory of initial ordinals are theorems of the theory of cardinals, and so, since isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic, cardinals and initial ordinals are isomorphic structures, we use the definition of a theory, the definition of an isomorphism of structures in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms, so as to apply both the theorem on the comparablity of ordinals to the theory of cardinals and the fundamental theorem of cardinal arithmetic to the theory of ordinals.
Category: Set Theory and Logic

[40] viXra:1310.0075 [pdf] replaced on 2014-03-18 20:34:36

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis by categorical logic, proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that the theorems of the theory of cardinals are theorems of the theory of initial ordinals, and conversely, the theorems of the theory of initial ordinals are theorems of the theory of cardinals, and so, since isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic, cardinals and initial ordinals are isomorphic structures, we use the definition of a theory, the definition of an isomorphism of structures in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms, so as to apply both the theorem on the comparablity of ordinals to the theory of cardinals and the fundamental theorem of cardinal arithmetic to the theory of ordinals.
Category: Set Theory and Logic

[39] viXra:1310.0075 [pdf] replaced on 2014-03-18 13:30:15

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis by categorical logic, proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that the theorems of the theory of cardinals are theorems of the theory of initial ordinals, and conversely, the theorems of the theory of initial ordinals are theorems of the theory of cardinals, and so, since isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic, cardinals and initial ordinals are isomorphic structures, we use the definition of a theory, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic and the axioms of the theory of categories which include the Gödel-Bernays-von Neumann axioms, so as to apply both the theorem on the comparablity of ordinals to the theory of cardinals and the fundamental theorem of cardinal arithmetic to the theory of ordinals.
Category: Set Theory and Logic

[38] viXra:1310.0075 [pdf] replaced on 2014-01-01 19:40:12

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the definition of a theory, the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[37] viXra:1310.0075 [pdf] replaced on 2013-12-27 16:55:44

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the definition of a theory, the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[36] viXra:1310.0075 [pdf] replaced on 2013-12-27 11:39:22

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the definition of a theory, the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[35] viXra:1310.0075 [pdf] replaced on 2013-12-24 17:36:51

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the definition of a theory, the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[34] viXra:1310.0075 [pdf] replaced on 2013-12-23 21:00:03

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the definition of a theory, the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[33] viXra:1310.0075 [pdf] replaced on 2013-12-22 17:33:39

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a theory, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[32] viXra:1310.0075 [pdf] replaced on 2013-12-21 18:29:42

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the definition of a theory, the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[31] viXra:1310.0075 [pdf] replaced on 2013-12-19 21:43:23

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a theory, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[30] viXra:1310.0075 [pdf] replaced on 2013-12-18 18:06:04

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic, by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[29] viXra:1310.0075 [pdf] replaced on 2013-12-12 18:31:32

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic, by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the definition of an isomorphism of theories, in its equivalent form, the definition of an isomorphism of categories, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[28] viXra:1310.0075 [pdf] replaced on 2013-12-10 12:36:03

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic, by proving that the theory of initial ordinals and the theory of cardinals are equivalent. We use the definition of an isomorphism of theories in mathematical logic, in its equivalent form, the definition of an isomorphism of categories from the theory of categories, and also, we use the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets, and so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[27] viXra:1310.0075 [pdf] replaced on 2013-12-09 19:21:45

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 3 Pages.

In this paper we prove the continuum hypothesis, by proving that the theory of initial ordinals and the theory of cardinals are equivalent. We use the definition of an isomorphism of theories in mathematical logic, in its equivalent form, the definition of an isomorphism of categories from the theory of categories, and also, we use the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets, and so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.
Category: Set Theory and Logic

[26] viXra:1310.0075 [pdf] replaced on 2013-12-04 11:07:24

The Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 2 Pages.

In this paper we give a categorical proof of the continuum hypothesis.
Category: Set Theory and Logic

[25] viXra:1310.0006 [pdf] replaced on 2014-06-29 13:08:43

The Intractability Computational Complexity Class

Authors: Daniel Cordero Grau
Comments: 2 Pages.

In this paper we prove the fundamental theorem of computational complexity, in particular the deterministic polynomially time bounded computational complexity class P and the nondeterministic polynomially time bounded computational complexity class NP are distinct, and in general the intractability computational complexity class exists for every computational complexity class, proving the intractability limit of quantum computation and any other future form of computation.
Category: Set Theory and Logic

[24] viXra:1302.0048 [pdf] replaced on 2013-02-20 04:58:47

Inconsistent Countable Set

Authors: Jaykov Foukzon
Comments: 15 Pages. Advances in Pure Mathematics.DOI: 10.4236/apm.2013.33053

In this article we derived an importent example of the inconsistent countable set. Main result is: ~con(ZFC+E(\omega-model of ZFC)).
Category: Set Theory and Logic

[23] viXra:1302.0048 [pdf] replaced on 2013-02-16 11:11:07

Inconsistent Countable Set

Authors: Jaykov Foukzon
Comments: 15 Pages.

In this article we derived an importent example of the inconsistent countable set. Main result is: ~con(ZFC+E(\omega-model of ZFC)).
Category: Set Theory and Logic

[22] viXra:1212.0088 [pdf] replaced on 2013-06-04 05:22:15

Difficulties of the Set of Natural Numbers

Authors: Qiu Kui Zhang
Comments: 10 Pages.

In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.
Category: Set Theory and Logic

[21] viXra:1212.0088 [pdf] replaced on 2013-02-17 04:14:16

Difficulties of the Set of Natural Numbers

Authors: Qiu Kui Zhang
Comments: 10 Pages.

In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.
Category: Set Theory and Logic

[20] viXra:1212.0088 [pdf] replaced on 2012-12-28 03:13:00

Difficulties of the Set of Natural Numbers

Authors: Qiu Kui Zhang
Comments: 9 Pages.

In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.
Category: Set Theory and Logic

[19] viXra:1212.0088 [pdf] replaced on 2012-12-22 07:01:34

Dificulties of the Set of Natural Numbers

Authors: Qiu Kui Zhang
Comments: 9 Pages.

In this article some dificulties are deduced from the set of natural numbers. The demonstrated dificulties suggest that if the set of natural numbers exists it would confict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.
Category: Set Theory and Logic

[18] viXra:1209.0103 [pdf] replaced on 2014-06-07 09:17:38

Fernan: the Value of the Intuition

Authors: Fernando Sanchez-Escribano
Comments: Spanish original(11 pages) and English translation (11 pages)

Hereby it is presented a new set theory (generalized as the so-called system theory, fully respectful with the dictates of intuition (including the existence and numerability of the universal set, of which all beings are elements) and capable to overcome all the logical difficulties that forced logicians in last century to accept never desired axioms for avoiding contradictions. The somewhat philosophical nature of the essay, imposed by the need to distinghish between concepts before not sufficiently clear, if not confused, and now denoted with distinct names (some neologisms), such as entema, concept, reflexo, set, system, aente, uente…, should not prevent the appreciation of its mathematical significance. 
Category: Set Theory and Logic

[17] viXra:1209.0103 [pdf] replaced on 2014-06-07 09:05:43

Fernan: the Value of the Intuition

Authors: Fernando Sanchez-Escribano
Comments: Spanish original (11 pages) and English translation (11 pages)

Hereby it is presented a new set theory (generalized as the so-called system theory, fully respectful with the dictates of intuition (including the existence and numerability of the universal set, of which all beings are elements) and capable to overcome all the logical difficulties that forced logicians in last century to accept never desired axioms for avoiding contradictions. The somewhat philosophical nature of the essay, imposed by the need to distinghish between concepts before not sufficiently clear, if not confused, and now denoted with distinct names (some neologisms), such as entema, concept, reflexo, set, system, aente, uente…, should not prevent the appreciation of its mathematical significance. 
Category: Set Theory and Logic

[16] viXra:1209.0103 [pdf] replaced on 2013-05-29 12:49:35

The Value of the Intuition (On an Intuitive Set Theory Without Antinomies)

Authors: Fernando Sánchez-Escribano
Comments: Pages.

Hereby it is presented a new set theory (generalized as the so-called system theory, fully respectful with the dictates of intuition (including the existence and numerability of the universal set, of which all beings are elements) and capable to overcome all the logical difficulties that forced logicians in last century to accept never desired axioms for avoiding contradictions. The somewhat philosophical nature of the essay, imposed by the need to distinghish between concepts before not sufficiently clear, if not confused, and now denoted with distinct names (some neologisms), such as entema, concept, reflexo, set, system, aente, uente…, should not prevent the appreciation of its mathematical significance. 
Category: Set Theory and Logic

[15] viXra:1209.0103 [pdf] replaced on 2013-02-09 03:54:15

Fernán: el Valor de la Intuición (Sobre Una Teoría Intuitiva de Conjuntos Sin Antinomias)

Authors: Fernando Sánchez-Escribano
Comments: 11 pages for Spanish original; 11 pages for English translation.

A new set theory (generalized as the so-called system theory) is presented, that respects all the dictates of intuition (including the existence of a universal set, the one including all beings, and its numerability) and is able not only to overcome all the logical difficulties that forced logicians in last century to accept axioms never desired, in order to avoid contradictions, but also to make obvious the mistakes that caused these.
Category: Set Theory and Logic

[14] viXra:1207.0009 [pdf] replaced on 2012-07-09 00:35:18

Definition à la Bourbaki of the Basic Notions of Category Theory

Authors: Pierre-Yves Gaillard
Comments: 3 Pages.

We unsuccessfully try to give definitions in the spirit of Bourbaki's set theory for the basic notions of category theory. The goal is to avoid using either Grothendieck's universes axiom, or "classes" (or "collections") of sets which are not sets. We explain why our attempt fails.
Category: Set Theory and Logic

[13] viXra:1207.0009 [pdf] replaced on 2012-07-07 01:06:47

Definition à la Bourbaki of the Basic Notions of Category Theory

Authors: Pierre-Yves Gaillard
Comments: 2 Pages.

We give definitions in the spirit of Bourbaki's Set Theory for the basic notions of category theory. The goal is to avoid using either Grothendieck's universes axiom, or ``classes'' (or ``collections'') of sets which are not sets.
Category: Set Theory and Logic

[12] viXra:1204.0012 [pdf] replaced on 2012-04-04 14:58:56

Solution for Ringel-Kotzig Conjecture

Authors: Wilber Valgusbitkevyt
Comments: 5 Pages. Further elaboration on Victoria Hayanisel Theorems were explained.

In this paper, I am creating three new theorems called Victoria Hayanisel Theorem dedicated to Princess Eugenie of York to describe the state of numbers, circles, and lines. Followed by the theorem, I am using the set theory and Fermat's Infinite Descent Method (if my method is different, I will name it) to show how the conjecture is true.
Category: Set Theory and Logic

[11] viXra:1109.0016 [pdf] replaced on 2014-01-02 21:11:02

The N-th Root Algorithm

Authors: Daniel Cordero Grau
Comments: 2 Pages.

In this paper we give the N-th root algorithm in completions of fraction semfields of normed euclidean semialgebras for every natural N > 1. The algorithm starts with a nonzero element of arbitrary length n in terms of its p-adic expansion for a nonunit p of nonzero degree of a normed euclidean semialgebra, thereafter, for a nonzero natural m = O(n), writes O(m) elements of length O(1) to go through O(m) steps in each of which compares, calculates and writes O(1) elements of length O(m^(N-1)), and so, in time O(n^N).
Category: Set Theory and Logic

[10] viXra:1107.0045 [pdf] replaced on 2013-12-08 11:38:28

A Different Approach to Logic

Authors: Mauro Avon
Comments: 201 Pages.

The paper is about an approach to logic that differs from the standard first-order logic and other known approaches. It should be a new approach the author has created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. We list the most relevant features of the system. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. The set-builder notation is enclosed as an expression-building pattern. In our system we can easily express second-order and all-order conditions (the set to which a quantifier refers is explicitly written in the expression). The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external `structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved, as well as the fact that our system is not affected by the most known types of paradox. The paper provides both the theoretical material and two fully documented examples of deduction. The author believes his aims have been achieved, obviously the reader is free to examine the system and get his own opinion about it.
Category: Set Theory and Logic

[9] viXra:1107.0045 [pdf] replaced on 2011-12-11 13:41:47

A Different Approach to Logic

Authors: Mauro Avon
Comments: 159 Pages.

The paper is about an approach to logic that differs from the standard first-order logic and other known approaches. It should be a new approach the author has created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. We list the most relevant features of the system. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. The set-builder notation is enclosed as an expression-building pattern. In our system we can easily express second-order and all-order conditions (the set to which a quantifier refers is explicitly written in the expression). The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved, as well as the fact that our system is not affected by the most known types of paradox. The paper provides both the theoretical material and two fully documented examples of deduction. The author believes his aims have been achieved but obviously the reader is free to examine the system and get his own opinion about it.
Category: Set Theory and Logic

[8] viXra:1107.0045 [pdf] replaced on 8 Sep 2011

A Different Approach to Logic

Authors: Mauro Avon
Comments: 159 pages

The paper is about an approach to logic that differs from the standard first-order logic and other known approaches. It should be a new approach the author has created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. We list the most relevant features of the system. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. The set-builder notation is enclosed as an expression-building pattern. In our system we can easily express second-order and all-order conditions (the set to which a quantifier refers is explicitly written in the expression). The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved, as well as the fact that our system is not affected by the most known types of paradox. The paper provides both the theoretical material and two fully documented examples of deduction. The author believes his aims have been achieved but obviously the reader is free to examine the system and get his own opinion about it.
Category: Set Theory and Logic

[7] viXra:1107.0045 [pdf] replaced on 12 Aug 2011

A Different Approach to Logic

Authors: Mauro Avon
Comments: 159 pages

This paper outlines an approach to mathematical logic which is different from the standard one. We list the most relevant features of the system. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. The set-builder notation is enclosed as an expression-building pattern. In our system we can easily express second-order and all-order conditions (the set to which a quantifier refers is explicitly written in the expression). The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved, as well as the fact that our system is not affected by the most known types of paradox. The paper provides both the theoretical material and two fully documented examples of deduction. The author has built the whole system with the idea to provide a faithful model of human mathematical deductive process. He believes this objective has been achieved but obviously the reader is free to examine the system and get his own opinion about it.
Category: Set Theory and Logic

[6] viXra:1010.0002 [pdf] replaced on 19 Apr 2011

The Gëdel Thesis 'about a Incompleteness of Formal Arithmetics' is not Proved and This Hypothesis is not Similar to the Truth

Authors: Dm. Vatolin
Comments: 9 pages, in Russian.

In this paper the question is examined that an incompleteness of advanced enough theories of arithmetic does not follow from the Gëdel statements.
Category: Set Theory and Logic

[5] viXra:1010.0002 [pdf] replaced on 6 Apr 2011

The Gëdel Thesis 'about a Incompleteness of Formal Arithmetics' is not Proved and This Hypothesis is not Similar to the Truth

Authors: Dm. Vatolin
Comments: 9 pages, Russian.

In this paper the question is examined that a incompleteness of enough advanced theories of arithmetics does not follow from the Gëdel statements.
Category: Set Theory and Logic

[4] viXra:1010.0002 [pdf] replaced on 31 Oct 2010

The Gëdel Thesis 'about a Incompleteness of Formal Arithmetics' is not Proved and This Hypothesis is not Similar to the Truth

Authors: Dm. Vatolin
Comments: 11 pages, Russian.

In this paper the question is examined that a incompleteness of enough advanced theories of arithmetics does not follow from the Gëdel statements.
Category: Set Theory and Logic

[3] viXra:1005.0059 [pdf] replaced on 22 Nov 2010

Geometrical Axioms Refuting the Continuum-Hypothesis

Authors: Dm. Vatolin
Comments: 14 pages, Russian.

This article formulates three geometrical axioms from which it follows that the continuum power is greater then any well-ordered set power.
Category: Set Theory and Logic

[2] viXra:1004.0026 [pdf] replaced on 22 Apr 2010

Neutrosophic Diagram and Classes of Neutrosophic Paradoxes

Authors: Florentin Smarandache
Comments: 14 pages

These paradoxes are called "neutrosophic" since they are based on indeterminacy (or neutrality, i.e. neither true nor false), which is the third component in neutrosophic logic. We generalize the Venn Diagram to a Neutrosophic Diagram, which deals with vague, inexact, ambiguous, illdefined ideas, statements, notions, entities with unclear borders. We define the neutrosophic truth table and introduce two neutrosophic operators (neuterization and antonymization operators) give many classes of neutrosophic paradoxes.
Category: Set Theory and Logic

[1] viXra:1002.0003 [pdf] replaced on 15 Mar 2010

Shifting Assignments Between Infinite Sets

Authors: Willi Penker
Comments: 3 pages

To shift assignments between infinite sets is to create a disturbance within the assignment itself that cannot be removed. An assignment carrying such a disturbance cannot be regarded as static.
Category: Set Theory and Logic