Epistemic Church’s Thesis and Absolute Undecidability

Abstract

On December 26, 1951, Gödel delivered the 25th]. W. Gibbs Lecture at a Meeting of the American Mathematical Association at Brown University. In the lecture, he formulated a disjunctive thesis conceming the limits of mathematical reasoning and the possibility of the existence of mathematical truths that are inaccessible to the human rnind. This thesis, known as Gödel's disjunction, is introduced as a direct consequence of the incompleteness theorems [Gödel 1951, p. 310): Either ... the human mind ( even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems [henceforth, absolutely undecidable mathematical sentences] ... ( where the case that both tenns of the disjunction are true is not excluded, so that there are, strictly speaking, three alternatives).That is, either the output of the human mathematical rnind exceeds the output of a Turing machine ( called the anti-mechanist thesis) or there are true mathematical sentences that are undecidable "not just within some particular axiomatic system, but by any mathematical proof the human mind can conceive." The latter are called absolutely undecidable mathematical sentences, i.e. mathematical sentences that cannot be either absolutely proved or refuted [Gödel 1951].According to Gödel, the fact that the disjunctive thesis above holds is a "mathematically established fact" of great philosophical interest which follows from the incompleteness theorems, and as such, it is "entirely independent from the standpoint taken toward the foundation of mathematics" [ Gödel 1951]. Indeed, most commentators agree that Gödel's arguments for this disjunctive thesis are compelling. Since Gödel's disjunction was first formulated in 1951, much effort has gone into finding equally compelling arguments for or against either of the disjuncts. In particular, attemptsGödtl's Disjunction. First Edition. Leon Horsten & Philip Welch ( eds}.

Human-Effective Computability†

“…For a detailed exposition of Feferman's completeness theorem see[Franzén 2004] and [AntonuttiMarfori 2013].For a discussion of different types of models for ECT see [AntonuttiMarfori and Horsten 2016] and[Carlson 2016].…”