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Gödel's incompleteness theorem and the Anti-Mechanist Argument: revisited

Yong Cheng

TL;DR

The paper interrogates whether Gödel’s incompleteness theorems force that the mind cannot be mechanized, arguing that $G1$ alone does not entail mechanizability failure and that precise formal frameworks are needed to assess related claims. Building on Koellner’s approach, it distinguishes relative vs absolute provability and truth via systems like EA and EA$_ ext{T}$, and formalizes Gödel’s Disjunctive Thesis ($GD$) within these frameworks, showing $GD$ is provable in $ ext{EA}_ ext{T}$ while respecting nuances introduced by different notions of truth and provability. It analyzes three mechanistic theses—weak, strong, and super-strong—demonstrating results such as the consistency of $ ext{EA}_ ext{T}+ ext{WMT}$ and the inconsistency of $ ext{EA}_ ext{T}+ ext{SSMT}$, and discussesPenrose-style arguments under type-free truth ($ ext{DTK}$). The discussion on Gödel’s Undemonstrability of Consistency ($G2$) and the definability of natural numbers highlights the intensional nature of these claims, showing that the status of $G2$ depends on base theory, choice of provability predicate, consistency formulation, numbering, and axiom numeration, with natural-number definability likewise being structure-dependent. In sum, the work clarifies that Gödelian limits do not straightforwardly entail anti-mechanist conclusions and provides precise logical tools to navigate the landscape of mind-mechanization debates.

Abstract

This is a paper for a special issue of the journal "Studia Semiotyczne" devoted to Stanislaw Krajewski's paper [30]. This paper gives some supplementary notes to Krajewski's [30] on the Anti-Mechanist Arguments based on Gödel's incompleteness theorem. In Section 3, we give some additional explanations to Section 4-6 in Krajewski's [30] and classify some misunderstandings of Gödel's incompleteness theorem related to Anti-Mechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel's Disjunctive Thesis, Gödel's Undemonstrability of Consistency Thesis and the definability of natural numbers as in Section 7-8 in Krajewski's [30], describing how recent advances bear on these issues.

Gödel's incompleteness theorem and the Anti-Mechanist Argument: revisited

TL;DR

The paper interrogates whether Gödel’s incompleteness theorems force that the mind cannot be mechanized, arguing that alone does not entail mechanizability failure and that precise formal frameworks are needed to assess related claims. Building on Koellner’s approach, it distinguishes relative vs absolute provability and truth via systems like EA and EA, and formalizes Gödel’s Disjunctive Thesis () within these frameworks, showing is provable in while respecting nuances introduced by different notions of truth and provability. It analyzes three mechanistic theses—weak, strong, and super-strong—demonstrating results such as the consistency of and the inconsistency of , and discussesPenrose-style arguments under type-free truth (). The discussion on Gödel’s Undemonstrability of Consistency () and the definability of natural numbers highlights the intensional nature of these claims, showing that the status of depends on base theory, choice of provability predicate, consistency formulation, numbering, and axiom numeration, with natural-number definability likewise being structure-dependent. In sum, the work clarifies that Gödelian limits do not straightforwardly entail anti-mechanist conclusions and provides precise logical tools to navigate the landscape of mind-mechanization debates.

Abstract

This is a paper for a special issue of the journal "Studia Semiotyczne" devoted to Stanislaw Krajewski's paper [30]. This paper gives some supplementary notes to Krajewski's [30] on the Anti-Mechanist Arguments based on Gödel's incompleteness theorem. In Section 3, we give some additional explanations to Section 4-6 in Krajewski's [30] and classify some misunderstandings of Gödel's incompleteness theorem related to Anti-Mechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel's Disjunctive Thesis, Gödel's Undemonstrability of Consistency Thesis and the definability of natural numbers as in Section 7-8 in Krajewski's [30], describing how recent advances bear on these issues.

Paper Structure

This paper contains 5 sections, 11 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notes on Gödel-based Anti-Mechanist Arguments
  4. Gödel's disjunctive thesis
  5. Gödel's Undemonstrability of Consistency Thesis and the definability of natural numbers

Key Result

Theorem 2.1

Let $T_1$ and $T_2$ be two consistent theories such that $T_2$ is interpretable in $T_1$. If $T_2$ is essentially undecidable, then $T_1$ is also essentially undecidable.

Theorems & Definitions (15)

  • Theorem 2.1: undecidable, Theorem 7, p. 22
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1: Gödel's first incompleteness theorem $(\sf G1)$
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 5 more