Constructive truth and falsity in Peano arithmetic

Abstract

Abstract Artemov (2019, The provability of consistency) offered the notion of constructive truth and falsity of arithmetical sentences in the spirit of Brouwer–Heyting–Kolmogorov semantics and its formalization, the logic of proofs. In this paper, we provide a complete description of constructive truth and falsity for Friedman’s constant fragment of Peano arithmetic. For this purpose, we generalize the constructive falsity to $n$-constructive falsity in Peano arithmetic where $n$ is any positive natural number. Based on this generalization, we also analyse the logical status of well-known Gödelean sentences: consistency assertions for extensions of PA, the local reflection principles, the ‘constructive’ liar sentences and Rosser sentences. Finally, we discuss ‘extremely’ independent sentences in the sense that they are classically true but neither constructively true nor $n$-constructively false for any $n$.