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The limitless First Incompleteness Theorem

Yong Cheng

TL;DR

The paper investigates the limits of the First Incompleteness Theorem by analyzing two degree structures for recursively enumerable theories with G1: Turing-based degrees $D_T$ and interpretation-based degrees $D_I$. It establishes that the Turing-degree structure is a dense distributive lattice with no minimal elements and that every nonrecursive RE degree can be realized by a theory sharing G1, via a Turing-persistent construction $T_d$. In the interpretation-weakening setting, it shows $D_I$ is countable with many incomparable elements, no finitely axiomatizable theories, and that minimality is precluded by generalized PV-style constructions; it also provides a broad generalization framework proving no minimal elements for a wide class of properties. The results collectively reinforce the view that incompleteness phenomena are ubiquitous and cannot be bounded by simple minimality notions, while offering a unified toolkit for extending G1 results to broader theory families.

Abstract

This work is motivated by the problem of finding the limit of the applicability of the first incompleteness theorem ($\sf G1$). A natural question is: can we find a minimal theory for which $\sf G1$ holds? We examine the Turing degree structure of recursively enumerable (RE) theories for which $\sf G1$ holds and the interpretation degree structure of RE theories weaker than the theory $\mathbf{R}$ with respect to interpretation for which $\sf G1$ holds. We answer all questions that we posed in [2], and prove more results about them. It is known that there are no minimal essentially undecidable theories with respect to interpretation. We generalize this result and give some general characterizations which tell us under what conditions there are no minimal RE theories having some property with respect to interpretation.

The limitless First Incompleteness Theorem

TL;DR

The paper investigates the limits of the First Incompleteness Theorem by analyzing two degree structures for recursively enumerable theories with G1: Turing-based degrees and interpretation-based degrees . It establishes that the Turing-degree structure is a dense distributive lattice with no minimal elements and that every nonrecursive RE degree can be realized by a theory sharing G1, via a Turing-persistent construction . In the interpretation-weakening setting, it shows is countable with many incomparable elements, no finitely axiomatizable theories, and that minimality is precluded by generalized PV-style constructions; it also provides a broad generalization framework proving no minimal elements for a wide class of properties. The results collectively reinforce the view that incompleteness phenomena are ubiquitous and cannot be bounded by simple minimality notions, while offering a unified toolkit for extending G1 results to broader theory families.

Abstract

This work is motivated by the problem of finding the limit of the applicability of the first incompleteness theorem (). A natural question is: can we find a minimal theory for which holds? We examine the Turing degree structure of recursively enumerable (RE) theories for which holds and the interpretation degree structure of RE theories weaker than the theory with respect to interpretation for which holds. We answer all questions that we posed in [2], and prove more results about them. It is known that there are no minimal essentially undecidable theories with respect to interpretation. We generalize this result and give some general characterizations which tell us under what conditions there are no minimal RE theories having some property with respect to interpretation.

Paper Structure

This paper contains 7 sections, 28 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The structure $\langle {\sf D_T}, \leq_{\sf T}\rangle$
  4. The structure $\langle {\sf D_I}, \lhd\rangle$
  5. Some generalizations
  6. Conclusion
  7. Appendix

Key Result

Proposition 1.2

For any consistent RE theory $T$, $\sf G1$ holds for $T$ iff $T$ is essentially incomplete iff $T$ is essentially undecidable.

Theorems & Definitions (65)

  • Definition 1.1: Cheng20
  • Proposition 1.2: undecidable, see also Cheng20
  • Definition 1.3: Cheng20
  • Definition 2.1: Robinson Arithmetic $\mathbf{Q}$, undecidable
  • Definition 2.2: The theory $\mathbf{R}$, undecidable
  • Definition 2.3: Basic recursion theory, Rogers87
  • Definition 2.4: Incompleteness and undecidability, undecidable
  • Definition 2.5: Interpretations
  • Theorem 2.6: Visser, Theorem 6, Visser 14
  • Theorem 2.7: Cobham, Visser16
  • ...and 55 more