A case for weakening the Church-Turing Thesis
Abstract
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.
Keywords
Cite
@article{arxiv.1108.4597,
title = {A case for weakening the Church-Turing Thesis},
author = {Bhupinder Singh Anand},
journal= {arXiv preprint arXiv:1108.4597},
year = {2011}
}
Comments
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:1003.5602