Set Theory and Logic

1412 Submissions

[8] viXra:1412.0269 [pdf] submitted on 2014-12-29 20:01:52

Axiomatical System HST^# Asinconsistent Generalization of Hrbacek Set Theory HST.

Authors: Jaykov Foukzon
Comments: 29 Pages.

In this paper paraconsistent first-order logic LP^# with infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth^# is proposed. Axiomatical system HST^#,as inconsistent generalization of Hrbacek set theory HST is considered.
Category: Set Theory and Logic

[7] viXra:1412.0235 [pdf] replaced on 2015-07-28 04:51:49

ZFC is Inconsistent. A Condition by Paul of Venice (1369-1429) Solves Russell's Paradox, Blocks Cantor's Diagonal Argument, and Provides a Challenge to ZFC

Authors: Thomas Colignatus
Comments: 2 Pages. The paper refers to the book FMNAI that supersedes the paper

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) axioms for set theory appear to be inconsistent. They are still too lax on the notion of a well-defined set. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations of set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).
Category: Set Theory and Logic

[6] viXra:1412.0234 [pdf] replaced on 2015-07-28 05:05:15

Contra Cantor Pro Occam - Proper Constructivism with Abstraction

Authors: Thomas Colignatus
Comments: 2 Pages. The paper refers to the book FMNAI that supersedes the paper

> Context • In the philosophy of mathematics there is the distinction between platonism (realism), formalism, and constructivism. There seems to be no distinguishing or decisive experiment to determine which approach is best according to non-trivial and self-evident criteria. As an alternative approach it is suggested here that philosophy finds a sounding board in the didactics of mathematics rather than mathematics itself. Philosophers can go astray when they don’t realise the distinction between mathematics (possibly pure modeling) and the didactics of mathematics (an empirical science). The approach also requires that the didactics of mathematics is cleansed of its current errors. Mathematicians are trained for abstract thought but in class they meet with real world students. Traditional mathematicians resolve their cognitive dissonance by relying on tradition. That tradition however is not targetted at didactic clarity and empirical relevance with respect to psychology. The mathematical curriculum is a mess. Mathematical education requires a (constructivist) re-engineering. Better mathematical concepts will also be crucial in other areas, such as e.g. brain research. > Problem • Aristotle distinguished between potential and actual infinite, Cantor proposed the transfinites, and Occam would want to reject those transfinites if they aren’t really necessary. My book “A Logic of Exceptions” already refuted ‘the’ general proof of Cantor's Conjecture on the power set, so that the latter holds only for finite sets but not for ‘any’ set. There still remains Cantor’s diagonal argument on the real numbers. > Results • There is a bijection by abstraction between N and R. Potential and actual infinity are two faces of the same coin. Potential infinity associates with counting, actual infinity with the continuum, but they would be ‘equally large’. The notion of a limit in R cannot be defined independently from the construction of R itself. Occam’s razor eliminates Cantor’s transfinites. > Constructivist content • Constructive steps S1, ..., S5 are identified while S6 gives non-constructivism (possibly the transfinites). Here S3 gives potential infinity and S4 actual infinity. The latter is taken as ‘proper constructivism with abstraction'. The confusions about S6 derive rather from logic than from infinity.
Category: Set Theory and Logic

[5] viXra:1412.0233 [pdf] submitted on 2014-12-25 05:58:23

A Difficulty in Proof Theory

Authors: Thomas Colignatus
Comments: 10 Pages. Paper of 2007, written in Mathematica

Adding some reasonable properties to the Gödelian system of Peano Arithmetic creates a new system for which Gödel's completeness theorems collapse and the Gödeliar becomes the Liar paradox again. Rejection of those properties is difficult since they are reasonable. Three-valued logic is a better option to deal with the Liar and its variants.
Category: Set Theory and Logic

[4] viXra:1412.0201 [pdf] replaced on 2015-01-27 21:14:44

Proof of the Existence of Transfinite Cardinals Strictly Smaller Than Aleph Zero with an Ensuing Solution to the Twin Prime Conjecture

Authors: Karan Doshi
Comments: 11 Pages.

In this paper the author submits a proof using the Power Set relation for the existence of a transfinite cardinal strictly smaller than Aleph Zero, the cardinality of the Naturals. Further, it can be established taking these arguments to their logical conclusion that even smaller transfinite cardinals exist. In addition, as a lemma using these new found and revolutionary concepts, the author conjectures that some outstanding unresolved problems in number theory can be brought to heel. Specifically, a proof of the twin prime conjecture is given.
Category: Set Theory and Logic

[3] viXra:1412.0155 [pdf] replaced on 2015-01-30 13:43:13

Neutrosophic Theory and its Applications, Collected Papers, Vol. 1

Authors: Florentin Smarandache
Comments: 480 Pages.

Neutrosophic Theory means Neutrosophy applied in many fields in order to solve problems related to indeterminacy. Neutrosophy considers every entity <A> together with its opposite or negation <antiA>, and with their spectrum of neutralities <neutA> in between them (i.e. entities supporting neither nor <antiA>). Where , which of course depends on <A>, can be indeterminacy, neutrality, tie (game), unknown, vagueness, contradiction, ignorance, incompleteness, imprecision, etc. Hence, in one hand, the Neutrosophic Theory is based on the triad <A>, <neutA>, and <antiA>. In the other hand, Neutrosophic Theory studies the indeterminacy in general, labelled as I, with I^n = I for n ≥ 1, and mI + nI = (m+n)I, in neutrosophic structures developed in algebra, geometry, topology etc. This volume contains 45 papers, written by the author alone or in collaboration with the following co-authors: Mumtaz Ali, Said Broumi, Sukanto Bhattacharya, Mamoni Dhar, Irfan Deli, Mincong Deng, Alexandru Gal, Valeri Kroumov, Pabitra Kumar Maji, Maikel Leyva- Vazquez, Feng Liu, Pinaki Majumdar, Munazza Naz, Karina Perez-Teruel, Rıdvan Sahin, A. A. Salama, Muhammad Shabir, Rajshekhar Sunderraman, Luige Vladareanu, Magdalena Vladila, Stefan Vladutescu, Haibin Wang, Hongnian Yu, Yan-Qing Zhang, about discounting of a neutrosophic mass in terms of reliability and respectively the importance of the source, evolution of sets from fuzzy set to neutrosophic set, classes of neutrosophic norm (n-norm) and neutrosophic conorm (n-conorm), applications of neutrosophic logic to physics, connections between extension logic and refined neutrosophic logic, approaches of neutrosophic logic to RABOT real time control, some applications of the neutrosophic logic to robotics, correlation coefficients of interval valued neutrosophic set, cosine similarity between interval valued neutrosophic sets, distance and similarity measures of interval neutrosophic soft sets, generalized interval neutrosophic soft sets and their operations, G-neutrosophic space, neutrosophic orbit, neutrosophic stabilizer, intuitionistic neutrosophic sets, intuitionistic neutrosophic soft sets, neutrosophic multi relation (NMR) defined on neutrosophic multisets, neutrosophic loops and biloops, neutrosophic N-loops and soft neutrosophic N-loops, operations on intuitionistic fuzzy soft sets, fuzzy soft matrix and new operations, such as fuzzy soft complement matrix and trace of fuzzy soft matrix based on reference function related properties, neutrosophic parameterized (NP) soft sets, NP-aggregation operator, and many more.
Category:
Set Theory and Logic

[2] viXra:1412.0130 [pdf] replaced on 2015-02-27 12:54:44

Relevant First-Order Logic LP# and Curry’s Paradox Resolution. Applications to da Costa’s Paraconsistent Set Theories.

Authors: Jaykov Foukzon
Comments: 8 Pages. DOI: 10.11648/j.pamj.s.2015040101.12

In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei's paradox without rejection any contraction postulate is proposed.
Category: Set Theory and Logic

[1] viXra:1412.0006 [pdf] submitted on 2014-12-01 09:42:44

Neutrosophic Information in the Framework of Multi-Valued Representation

Authors: Vasile Patrascu
Comments: 10 Pages.

The paper presents some steps for multi-valued representation of neutrosophic information. These steps are provided in the framework of multi-valued logics using the following logical value: true, false, neutral, unknown and saturated. Also, this approach provides some calculus formulae for the following neutrosophic features: truth, falsity, neutrality, ignorance, under-definedness, over-definedness, saturation and entropy. In addition, it was defined net truth, definedness and neutrosophic score.
Category: Set Theory and Logic