On the Constructive Truth and Falsity in Peano Arithmetic

05/25/2019

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by   Hirohiko Kushida, et al.

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Recently, Artemov [4] offered the notion of constructive consistency for Peano Arithmetic and generalized it to constructive truth and falsity 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 where n is any positive natural number. We also establish similar classification results for constructive truth and n-constructive falsity of Friedman's formulas. Then, we discuss `extremely' independent sentences in the sense that they are classically true but neither constructively true nor n-constructive false for any n.