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FINDING THE LIMIT OF INCOMPLETENESS I

Part of: Proof theory and constructive mathematics Computability and recursion theory

Published online by Cambridge University Press:  16 April 2021

YONG CHENG
YONG CHENG*
Affiliation:
SCHOOL OF PHILOSOPHY WUHAN UNIVERSITYWUHAN430072, HUBEIPEOPLE’S REPUBLIC OF CHINA E-mail: [email protected]
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Abstract

In this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

Keywords

Gödel’s first incompleteness theoreminterpretationessential undecidabilityRobinson’s R

MSC classification

Primary: 03F40: Gödel numberings and issues of incompleteness
Secondary: 03F30: First-order arithmetic and fragments 03D35: Undecidability and degrees of sets of sentences
Type
Communication
Information
Bulletin of Symbolic Logic , Volume 26 , Issue 3-4 , December 2020 , pp. 268 - 286
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Copyright
© The Association for Symbolic Logic 2021

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