# Consistency formula is strictly stronger in PA than PA-consistency

## TL;DR

This paper shows that the standard consistency statement for PA is strictly stronger than the series of its finite approximations, clarifying their logical relationship within PA.

## Contribution

It proves that Con(PA) is not equivalent to the series ConS(PA) in PA, establishing a strict strength difference between them.

## Key findings

- Con(PA) is not equivalent to ConS(PA) in PA.
- Con(PA) is strictly stronger than ConS(PA) in PA.
- Unprovability of Con(PA) does not imply unprovability of PA's consistency.

## Abstract

In this note, we show that, despite the widespread assumption, the consistency formula for Peano Arithmetic PA, Con(PA), "for all x, x is not a code of a derivation of (0=1)," is not equivalent in PA to the consistency of PA. Specifically, we demonstrate that "PA is consistent" is provably in PA equivalent to the series ConS(PA) of arithmetical sentences "n is not a code of a derivation of (0=1)" for n=0,1,2,.... Since Con(PA) is strictly stronger in PA than ConS(PA), the unprovability of Con(PA) in PA does not yield the unprovability of PA-consistency.