On partial disjunction properties of theories containing Peano arithmetic
type: Item
instance of: scholarly article
title: On partial disjunction properties of theories containing Peano arithmetic
zbMATH Open document ID: 1446.03102
author: Taishi Kurahashi
published in: Archive for Mathematical Logic
publication date: 2019-01-14T00:00:00Z
review text: A theory is said to have the disjunction property (DP) when for any two sentences, the theory proves at least one of them if it proves their disjunction. This property holds for many interesting intuitionistic theories, and so it is considered to be a constructive property. Another such property is the (numerical) existence property (EP) which holds for a theory \(T\) when for any formula \(\varphi(x)\), if \(T\vdash\exists x\varphi(x)\), then \(T\vdash\varphi(\bar{n})\) for some \(n\in\mathbb{N}\); here \(\bar{n}\) is the term representing the number \(n\) in the language of \(T\). By the incompleteness theorem of Gödel (and Rosser) no consistent and sufficiently strong arithmetical theory can have DP or EP over the classical logic. However, those theories may enjoy these properties for a (restricted) class of sentences. Let \(\Gamma\) be a class of sentences; such as \(\Sigma_n\) or \(\Pi_n\), for natural \(n\). The \(\Gamma\)-DP or \(\Gamma\)-EP are the disjunction and existence properties when the considered sentences are chosen from \(\Gamma\). In this paper, the author thoroughly studies these properties for theories containing Peano's arithmetic PA. One of the oldest results in this field is a theorem of \textit{D. Guaspari} [Trans. Am. Math. Soc. 254, 47--68 (1979; Zbl 0417.03030)] which states that a \(\Sigma_1\)-definable consistent extension of PA has \(\Sigma_1\)-DP if and only if it is \(\Sigma_1\)-sound. Let us note that PA (and its every extension) is \(\Sigma_1\)-complete. This is generalized in the paper as follows: For \(n\geq 1\), a \(\Sigma_n\)-definable and \(\Sigma_n\)-complete theory has \(\Sigma_n\)-DP if and only if it is \(\Sigma_n\)-sound if and only if it has \(\Sigma_n\)-EP. The paper has very many other interesting results, with technical proofs, that cannot be listed all in this short review. One of the nice characterizations of the \(\Sigma_n\)-DP for a theory \(T\) in the paper is the following: for any models \(M_1,M_2\vDash T\) the…
reviewed by: Saeed Salehi
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03F40
zbMATH DE Number: 7001427
zbMATH Keywords: partial disjunction properties
zbMATH Keywords: partial existence properties
zbMATH Keywords: formal arithmetic
zbMATH Keywords: Gödel's incompleteness theorems
MaRDI profile type: Publication
full work available at URL: s00153-018-0618-3
OpenAlex ID: W2786661286
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DOI: 10.1007/S00153-018-0618-3
description: scientific article; zbMATH DE number 7001427