A case for weakening the Church-Turing Thesis

We conclude from Goedel's Theorem VII of his seminal 1931 paper that every recursive function f(x_{1}, x_{2}) is representable in the first-order Peano Arithmetic PA by a formula [F(x_{1}, x_{2}, x_{3})] which is algorithmically verifiable, but not algorithmically computable, if we assume that the negation of a universally quantified formula of the first-order predicate calculus is always indicative of the existence of a counter-example under the standard interpretation of PA. We conclude that the standard postulation of the Church-Turing Thesis does not hold if we define a number-theoretic formula as effectively computable if, and only if, it is algorithmically verifiable; and needs to be replaced by a weaker postulation of the Thesis as an equivalence.Comment: 22pages. an updated version of this manuscript is accessible at http://alixcomsi.com/30_Church_Turing_Thesis_Update.pdf . arXiv admin note: substantial text overlap with arXiv:1003.560

Three Theorems on modular sieves that suggest the Prime Difference is O(Number of primes < (p(n)^1/2))

This 1964 paper developed as an off-shoot to the foundational query: Do we discover the natural numbers (Platonically), or do we construct them linguistically? The paper also assumes computational significance in the light of Agrawal, Kayal and Saxena's August 2000 paper, "PRIMES is in P", since both the TRIM and Compact Number Generating algorithms - each of which generates all the primes - are deterministic algorithms that run in polynomial time and suggest that the Prime Difference, d(n), is O(Number of primes < (p(n)^1/2)).Comment: 6 pages; an HTML version is available at http://alixcomsi.com/Three_Theorems.ht

Some consequences of interpreting the associated logic of the first-order Peano Arithmetic PA finitarily

We show that the classical interpretations of Tarski's inductive definitions actually allow us to define the satisfaction and truth of the quantified formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two essentially different ways: (a) in terms of algorithmic verifiabilty; and (b) in terms of algorithmic computability. We show that the classical Standard interpretation I_PA(N, Standard) of PA essentially defines the satisfaction and truth of the formulas of the first-order Peano Arithmetic PA in terms of algorithmic verifiability. It is accepted that this classical interpretation---in terms of algorithmic verifiabilty---cannot lay claim to be finitary; it does not lead to a finitary justification of the Axiom Schema of Finite Induction of PA from which we may conclude---in an intuitionistically unobjectionable manner---that PA is consistent. We now show that the PA-axioms---including the Axiom Schema of Finite Induction---are, however, algorithmically computable finitarily as satisfied / true under the Standard interpretation I_PA(N, Standard) of PA; and that the PA rules of inference do preserve algorithmically computable satisfiability / truth finitarily under the Standard interpretation I_PA(N, Standard). We conclude that the algorithmically computable PA-formulas can provide a finitary interpretation I_PA(N, Algorithmic) of PA from which we may classically conclude that PA is consistent in an intuitionistically unobjectionable manner. We define this interpretation, and show that if the associated logic is interpreted finitarily then (i) PA is categorical and (ii) Goedel's Theorem VI holds vacuously in PA since PA is consistent but not omega-consistent.Comment: 37pages; significant revision; differentiated the foundational perspective of this paper from the more common perspectives. An updated version of this manuscript is accessible at http://alixcomsi.com/30_Finitary_Logic_PA_Update.pdf . arXiv admin note: substantial text overlap with arXiv:1003.5602, arXiv:1108.4597, arXiv:0902.106

Does resolving PvNP require a paradigm shift?

I shall argue that a resolution of the PvNP problem requires building an iff bridge between the domain of provability and that of computability. The former concerns how a human intelligence decides the truth of number-theoretic relations, and is formalised by the first-order Peano Arithmetic PA following Dededekind's axiomatisation of Peano's Postulates. The latter concerns how a human intelligence computes the values of number-theoretic functions, and is formalised by the operations of a Turing Machine following Turing's analysis of computable functions. I shall show that such a bridge requires objective definitions of both an `algorithmic' interpretation of PA, and an `instantiational' interpretation of PA. I shall show that both interpretations are implicit in the definition of the subjectively defined `standard' interpretation of PA. However the existence of, and distinction between, the two objectively definable interpretations---and the fact that the former is sound whilst the latter is not---is obscured by the extraneous presumption under the `standard' interpretation of PA that Aristotle's particularisation must hold over the structure N of the natural numbers. I shall argue that recognising the falseness of this belief awaits a paradigm shift in our perception of the application of Tarski's analysis (of the concept of truth in the languages of the deductive sciences) to the `standard' interpretation of PA. I shall then show that an arithmetical formula [F] is PA-provable if, and only if, [F] interprets as true under an algorithmic interpretation of PA. I shall finally show how it then follows from Goedel's construction of a formally `undecidable' arithmetical proposition that there is a Halting-type PA formula which---by Tarski's definitions---is algorithmically verifiable as true, but not algorithmically computable as true, under a sound interpretation of PA.Comment: 35 pages; major revision; an update of this investigation is accessible at http://alixcomsi.com/27_Resolving_PvNP_Update.pd

Naive Philosophical Foundations

This soliloquy outlines some naive philosophical arguments underlying the thesis that mathematics ought to be viewed simply as a universal set of languages, some of precise expression, and some of effective communication.Comment: 15 pages; an HTML version is available at http://alixcomsi.com/Naive_Philosophical_Foundations.ht

Is the Halting problem effectively solvable non-algorithmically, and is the Goedel sentence in NP, but not in P?

We consider the thesis that an arithmetical relation, which holds for any, given, assignment of natural numbers to its free variables, is Turing-decidable if, and only if, it is the standard representation of a PA-provable formula. We show that, classically, such a thesis is, both, unverifiable and irrefutable, and, that it implies the Turing Thesis is false; that Goedel's arithmetical predicate R(x), treated as a Boolean function, is in the complexity class NP, but not in P; and that the Halting problem is effectively solvable, albeit not algorithmically.Comment: 12 pages; an HTML version is available at http://alixcomsi.com/Is_the_Halting_problem.ht

Are there parts of our arithmetical competence that no sound formal system can duplicate?

In 1995, David Chalmers opined as implausible that there may be parts of our arithmetical competence that no sound formal system could ever duplicate. We prove that the recursive number-theoretic relation x=Sb(y 19|Z(y)) - which is algorithmically verifiable since Goedel's recursive function Sb(y 19|Z(y)) is Turing-computable - cannot be "duplicated" in any consistent formal system of Arithmetic.Comment: v2; introduced standardised ACI compliant notation for citations; 9 pages; this paper reproduces Meta-theorem 1 and related Meta-lemmas from my earlier paper http://arXiv.org/abs/math.GM/0210078 ; an HTML version is available at http://alixcomsi.com/index01.ht

Reviewing Goedel's and Rosser's meta-reasoning of undecidability

I review the classical conclusions drawn from Goedel's meta-reasoning establishing an undecidable proposition GUS in standard PA. I argue that, for any given set of numerical values of its free variables, every recursive arithmetical relation can be expressed in PA by different, but formally equivalent, propositions. This asymmetry yields alternative Representation and Self-reference meta-Lemmas. I argue that Goedel's meta-reasoning can thus be expressed avoiding any appeal to the truth of propositions in the standard interpretation IA of PA. This now establishes GUS as decidable, and PA as omega-inconsistent. I argue further that Rosser's extension of Goedel's meta-reasoning involves an invalid deduction.Comment: v3: Introduced ACI compliant notation for citations. 30 pages. An HTML version is available on the web at http://alixcomsi.com/index01.ht

The Goebbellian Syndrome

Can we really falsify truth by dictat? A critical note on J. R. Lucas' 1996 remarks concerning non-standard models of first-order Peano Arithmetic.Comment: 4 pages; an HTML version is available at http://alixcomsi.com/The Goebbellian Syndrome.ht

The formal roots of Platonism

We present some arguments for the thesis that a set-theoretic inspired faith, in the ability of intuitive truth to faithfully reflect relationships between elements of a Platonic universe, may be as misplaced as an assumption that such truth cannot be expressed in a constructive, and effectively verifiable, manner.Comment: v2; revised para 2.2(ix); introduced standardised ACI compliant notation for citations; 24 pages; an HTML version is available at http://alixcomsi.com/The_formal_roots_of_Platonism.ht