# On the Constructive Truth and Falsity in Peano Arithmetic

## TL;DR

This paper characterizes constructive truth and falsity in Friedman's fragment of Peano Arithmetic, introduces n-constructive falsity, and discusses the independence of certain sentences from constructive provability.

## Contribution

It provides a complete classification of constructive truth and falsity in Friedman's fragment, extending to n-constructive falsity and analyzing independent sentences.

## Key findings

- Complete description of constructive truth and falsity in Friedman's fragment
- Introduction of n-constructive falsity for positive natural numbers
- Identification of classically true but constructively unprovable sentences

## Abstract

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 %unprovable in Peano Arithmetic neither constructively true nor n-constructive false for any n.