Set Theory and Logic

1907 Submissions

[23] viXra:1907.0628 [pdf] submitted on 2019-07-31 07:45:14

Refutation of Completeness for Inclusion and Equivalence of Universality

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate a formula for inclusion and equivalence of universality. It is not tautologous, refuting the conjecture of completeness, and forming a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[22] viXra:1907.0613 [pdf] submitted on 2019-07-30 06:28:12

Refutation of Turing's Halting Problem as Logically Unsolvable

Authors: Colin James III
Comments: Pages.

We confirm the halting conjecture as tautologous and hence refute the halting problem as unsolvable. What follows is that first order logic is decidable. This proof was made possible by the universal logic VL4.
Category: Set Theory and Logic

[21] viXra:1907.0583 [pdf] submitted on 2019-07-29 15:57:59

Refutation of Probabilistic Approximate Logic (Palo) and Logical Imagination Engine

Authors: Colin James III
Comments: 1 Page. .. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

A key property of probabilistic approximate logic (PALO) as one form of inference (of many) is evaluated as not tautologous. This refutes its semantics of the logical imagination engine and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[20] viXra:1907.0582 [pdf] submitted on 2019-07-29 17:17:55

Refutation of Leibniz' Identity of Indiscernibles and Leo-III Theorem Prover

Authors: Colin James III
Comments: Pages. Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com; email: info@cec-services . com

Leibniz' identity of indiscernibles as ∀x∀y[∀F(Fx <-> Fy) -> x=y] is not tautologous. Consequently the Leo-III theorem prover for higher-order paramodulated extensional logic is also refuted. These form a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[19] viXra:1907.0543 [pdf] submitted on 2019-07-27 20:40:04

Refutation of the Algebra of Binary Relations as Basis of Free Kleene Algebras with Domain

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The definitions for composition of relations and set-theoretic union are not tautologous. This refutes the algebra of binary relations on which is based the free Kleene algebras with domain, to form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[18] viXra:1907.0503 [pdf] submitted on 2019-07-25 06:32:58

Refutation of a Formula for Systems of Boolean Polynomials to Parameterized Complexity

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Three formulas defining Boolean polynomial arithmetic are not tautologous, to refute the conjecture. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[17] viXra:1907.0495 [pdf] submitted on 2019-07-25 11:33:12

An Anomaly in the Set of Algebraic Real Numbers

Authors: James Edwin Rock
Comments: 1 Page.

Depending on their representation, the ratio of irrational algebraic numbers to rationals in (0, 1) is either 0.0 or infinite.
Category: Set Theory and Logic

[16] viXra:1907.0453 [pdf] submitted on 2019-07-25 05:47:22

Refutation of the Orthomodular Law

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com; email: info@cec-services . com

We evaluate the orthomodular law x≤y implies y=x∨(y∧x′), with ′ as negation, which is not tautologous. This forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[15] viXra:1907.0417 [pdf] submitted on 2019-07-22 23:57:17

Refutation of the Borel Base and Hull

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate in two equations the Borel base and hull as not tautologous and contradictory, refuting the conjectures and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[14] viXra:1907.0382 [pdf] submitted on 2019-07-19 10:42:33

Refutation of Bounded Homomorphisms and Finitely Generated Fiber Products of Lattices

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate two equations for the standard method for constructing subdirect products, which are not tautologous. Hence the conjecture of bounded homomorphisms and finitely generated fiber products of lattices is refuted. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[13] viXra:1907.0366 [pdf] submitted on 2019-07-18 07:16:12

Refutation of Shallow Embedding in Martin-Löf Type Theory

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In Martin-Löf type theory (MLTT), we evaluate shallow embedding as the following conjecture: “if we add the rewrite rule ∀x. f x (not x) = true, the expression f true false will not be rewritten to true, since it does not rigidly match the not x on the left hand side”. The conjecture is not tautologous, hence refuting shallow embedding in MLTT and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[12] viXra:1907.0252 [pdf] submitted on 2019-07-16 05:37:16

Refutation of Translation of Implicit Logic (Il) to Explicit Logic (El)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

For translation of implicit logic to explicit logic, using intuitionistic and epistemic logic, seven equations are evaluated, with none tautologous. Two refute the recursive translation of “the intuitionistic truth definition into a syntactic recipe”; three refute “key features of intuitionistic logic in modal terms”; and one refutes “the recursion law after knowledge update [as] the basic dynamic equation of hard information” for public announcement logic (PAL). These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[11] viXra:1907.0196 [pdf] submitted on 2019-07-11 17:52:23

Refutation of Hilbert's First Epsilon Theorem in Intuitionistic and Intermediate Logics

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

From the universal quantifier shift of (∀xA(x)→B)→∃x(B→A(x)) as not tautologous, the intermediate logic L is refuted, refuting Hilbert’s first epsilon theorem and intuitionistic logic, and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[10] viXra:1907.0175 [pdf] submitted on 2019-07-10 10:02:20

Refutation of Interpretability Logics

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Of the non trivial logics for axioms as evaluated, none is tautologous. Hence the interpretability logic IL is refuted, and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[9] viXra:1907.0142 [pdf] submitted on 2019-07-08 07:01:55

Refutation of the Axiom of Dependent Choices in Mice

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The axiom of dependent choices as ∀a∈X∃b∈XP(a, b) ⇒ ∃f: ω → X∀n P(f(n), f(n+1)) is not tautologous. What follows is that the axiom of determinacy is also not tautologous, hence relegating both axioms to a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[8] viXra:1907.0133 [pdf] submitted on 2019-07-08 18:18:44

Refutation of Isabelle/HOL

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Eight of eleven equations are evaluated as not tautologous. This means the rewrite-engine or simplifier tool is not confirmed, the conjunction is not effectively defined by three rules, and other reasoning steps are not expressed similarly, hence refuting Isabelle/HOL. These anomalies form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[7] viXra:1907.0130 [pdf] submitted on 2019-07-08 20:22:52

Refutation of Analytic Choice Principles: Axioms of Choice and Dependent Choice

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The axiom of Γ choice and axiom of Σ11-dependent choice are not tautologous. Therefore an open problem on the Weihrauch degree of parallelization of the Σ11-choice principle on the integers is not solved by using those axioms. These axioms form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[6] viXra:1907.0118 [pdf] submitted on 2019-07-07 09:58:37

Shorter Refutation of Gödel’s Completeness Theorem

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The completeness theorem rendered as (∀x.R(x,x))→(∀x∃y.R(x,y)) is not tautologous. The application of Isabelle/HOL to prove the same also is not tautologous, to invalidate that tool. These demonstrations form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[5] viXra:1907.0103 [pdf] submitted on 2019-07-06 16:38:32

Refutation of the Axiom of Infinity

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the following. Two definitions of the axiom of infinity are both contradictory. Two definitions to extract natural numbers from the infinite are are not equivalent and not tautologous. Therefore the four definitions for the axiom of infinity are refuted. Therefore these form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[4] viXra:1907.0098 [pdf] submitted on 2019-07-05 08:29:54

Refutation of Open-Universe Causal Reasoning

Authors: Colin James III
Comments: Pages.

The proposition and axiom as tested are not tautologous. This does not “validate an intuitive and familiar set of principles about subjunctive conditionals and the relation of causal influence”. This also does not support “an important class of implicit generative models that can plausibly be treated as genuine causal models” or “enable reasoning beyond the propositional level”. The conjecture of open-universe causal reasoning forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[3] viXra:1907.0053 [pdf] replaced on 2019-07-05 07:09:11

Refutation of the Curry-Howard Correspondence

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In the Curry-Howard correspondence, the identity and composition combinators are not tautologous. In fact, the examples result in equivalent truth table values. Further demonstrated is that the instances of Hilbert, lambda, and sequent fragments are also not tautologous with a recent paper rendered moot. These artifacts form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[2] viXra:1907.0048 [pdf] submitted on 2019-07-02 07:31:03

Refutation of a Weak Set Theory H that Proves Its Own Consistency

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

For theory H, the axioms of extensionality and separation are not tautologous. The theorem to prove any instances of the scheme of ε-induction is also not tautologous. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[1] viXra:1907.0040 [pdf] submitted on 2019-07-02 10:21:41

Refutation of the Fixed-Point Property of Self-Proving for Predicate Modal Logics

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Refutation of the fixed-point property of self-proving for predicate modal logics
Category: Set Theory and Logic