On partial disjunction properties of theories containing Peano arithmetic
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 \(M_1,M_2\vDash T\) there are \(N,M_1',M_2'\vDash T\) such that \(M_1\equiv M_1'\), \(M_2\equiv M_2'\) and \(N\) is \(\Sigma_{n-1}\)-elementarily embeddable in both \(M_1'\) and \(M_2'\). The fact that one cannot necessarily have \(M_1'=M_1,M_2=M_2'\) has been mentioned in the footnote of page 960 giving credit to the referee. One wishes that the author had put the referee's proof in the paper for the sake of completeness, as another observation of the referee has been put as Theorem 8 on page 968. Even with this minor gap, the paper has lots of good characterizations for the mentioned properties, and gives a perspective to its readers on the discussed topics. It has a solid structure, and is strongly recommended to anyone interested in the issues of incompleteness of classical arithmetical theories.
scientific article; zbMATH DE number 7001427
scientific article