A case for weakening the Church-Turing Thesis

TL;DR

This paper argues that the traditional Church-Turing Thesis should be weakened, based on Gödel's Theorem VII, suggesting that some recursive functions are verifiable but not computable, challenging the standard interpretation.

Contribution

It introduces a revised perspective on the Church-Turing Thesis, proposing a weaker version aligned with algorithmic verifiability rather than computability.

Findings

Recursive functions are verifiable but not necessarily computable.

The standard Church-Turing Thesis does not fully capture all effectively verifiable functions.

A weaker postulation of the Thesis is proposed, aligning with algorithmic verifiability.

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.