Set Theory and Logic

1905 Submissions

[6] viXra:1905.0590 [pdf] submitted on 2019-05-31 03:08:42

Representation of Processive Functions in Robinson Arithmetic ?

Authors: Hannes Hutzelmeyer
Comments: 10 Pages.

In connection with his so-called incompleteness theorem Gödel discovered the beta-function. The beta-function theorem is important for the representation of recursive functions in the concrete calcule ALPHA of Robinson arithmetic. The other features are composition and minimization of primitive recursive functions. Recursive functions are no part of Robinson arithmetic, but they are representable by certain formulae. The author has developed an approach to logics that comprises, but goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and so on. The concrete calcule LAMBDA of a natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. They include e.g. Ackermann's function and similar constructions. All recursive functions (that are obtained by minimization too) can be represented in LAMBDA . As long as there is no proof that all processive functions are minimitive recursive (recursive but not primitive recursive) one has the problem of representing them in concrete calcule ALPHA of Robinson arithmetic. As long as the challenge of such a proof is not met there is the conjecture that there are calculative functions that are not representable in Robinson arithmetic. An abstract calcule alphakappa of Robinson-Crusoe arithmetic shows that there exists an even weaker adequate arithmetic than Robinson's.
Category: Set Theory and Logic

[5] viXra:1905.0358 [pdf] replaced on 2019-09-12 06:06:15

Proposal of Replacing Classical Logic with Free Logic for Reasoning with Non-Referring Names in Ordinary Discourse

Authors: Bornali Paul
Comments: 9 Pages.

Reasoning carried out in ordinary language, can not avoid using non-referring names if occasion arises. Semantics of classical logic does not fit well for dealing with sentences with non-referring names of the language. The principle of bivalence does not allow any third truth-value, it does not allow truth-value gap also. The outcome is an ad hoc stipulation that no names should be referentless. The aim of this paper is to evaluate how far free logic with supervaluational semantics is appropriate for dealing with the problems of non-referring names used in sentences of ordinary language, at the cost of validity of some of the classical logical theses/ principles.
Category: Set Theory and Logic

[4] viXra:1905.0275 [pdf] submitted on 2019-05-17 10:56:36

Programming Primitive Recursive Functions and Beyond

Authors: Hannes Hutzelmeyer
Comments: 17 Pages. The details given here relate to document viXra: 1905.0221

In addition to the publication 'The Snark, a counterexample for Church's thesis?' examples and details are offered in the form of two appendices C6 and C7 that allow for better understanding of the general method and the particular problem related to Church's thesis. The author has developed an approach to logics that comprises, but also goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers etc. The most usual approach to calculative (effectively calculable) functions is done by register machines or similar storage-based computers like the Abacus or Turing machines. Another usual approach to computable functions is to start with primitive recursive functions. However, one has to find a way to put this into a form that does not rely on a pre-knowledge about functions and higher logic. The concrete calcule LAMBDA of decimal pinitive arithmetic allows for such an access. It is based on a machine that is completely different from the storage-based machines: the PINITOR does not use storages but rather many microprocessors, one for each appearance of a command in the code of primitive recursive functior. the codes are decimal numbers, called pinons , where only the characters 0 1 2 8 9 appear. There are four kind of commands only: 0 nullification, 1 succession, 2 straight recursion and 8 composition. The PINITOR is a calculator which means that there is no halting problem. Computers have halting problems, per defintion calculators do not. Appendix C6 gives the programming of the codes of most of the usual primitive function and goes even farther, e.g. it introduces generator technique that allows for the straight-forward calculation of so-called processive function, that are not primitive recursive. The most famous examples of Ackermann and other hyperexponential functions are programmed. Appendix C7 turns to the Boojum-function and the Snark-function that have been introduced as calculative functions in the above publication in connection with Church's thesis. A list is provided that gives the lowest values for these functions and gives some more insight into these functions.
Category: Set Theory and Logic

[3] viXra:1905.0231 [pdf] submitted on 2019-05-15 08:39:24

A Methodology for Urban Sustainability Indicator Design

Authors: Ricardo Alvira
Comments: 19 Pages.

In recent times we have witnessed proliferation of indicators and models for measuring sustainability. This reveals the lack of common and shared scientific paradigm/common framework from which to confront the issue of quantitatively assessing the sustainability of our society. With the aim of moving forward the definition of such common framework, in this article we explain an easy formal methodology for designing urban sustainability indicators based on Fuzzy Logic / Fuzzy Sets Theory. The interest of this methodology is threefold: Firstly, formal procedures enable easier testing, a most fundamental issue forgotten in many current proposals of sustainability indicators. Secondly, a formal procedure is easily understandable and can become a common language allowing shared use of the indicators and facilitating their continuous improvement. And thirdly, fuzzy logic is widely used in computing and artificial intelligence, thus facilitating the progressive automation of our sustainability monitoring models. To help understand the procedure, the design of two indicators is reviewed.
Category: Set Theory and Logic

[2] viXra:1905.0221 [pdf] submitted on 2019-05-16 03:15:48

The Snark, a Counterexample for Church's Thesis ?

Authors: Hannes Hutzelmeyer
Comments: 7 Pages.

In 1936 Alonzo Church put forward his thesis that recursive functions comprise all effectively calculative functions. Whereas recursive functions are precisely defined, effectively calculative functions cannot be defined with a rigor that is requested by mathematicians. There has been a considerable amount of talking about the plausibility of Church's thesis, however, this is not relevant for a strict mathematical analysis. The only way to end the discussion is obtained by a counterexample. The author has developed an approach to logics that comprises, but goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and so on. The concrete calcule LAMBDA of natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. All recursive functions can be represented in LAMBDA too. The unary Snark-function is defined by a diagonalization procedure such that it can be calculated in a finite number of steps. However, this calculative function transcends the compinitive functions and presumably the recursive functions. The defenders of Church's thesis are challenged to show that the Snark-function is recursive. Another challenge asks for an example of a recursive function that cannot be expressed as a compinitive function, i.e. without minimization.
Category: Set Theory and Logic

[1] viXra:1905.0153 [pdf] submitted on 2019-05-10 18:37:20

The Thot-Queen Paradox

Authors: Analytic Birb
Comments: 2 Pages.

The apparent contradiction of the status of women as both thots and queens, as expressed by the sentences ‘if she breathes, she’s a thot’ and ‘all women are queens’ is examined. The problem is formalized using the law of the excluded middle and a counterexample to the assertion is provided. The implications of the lack of a paradox are discussed.
Category: Set Theory and Logic