Models of Bounded Arithmetic and variants of Pigeonhole Principle

TL;DR

The paper provides an elementary proof that certain bounded arithmetic theories augmented with the weak pigeonhole principle cannot prove the bijective pigeonhole principle, illustrating a separation in provability within these theories.

Contribution

It introduces a new elementary proof showing the limitations of $T^1_2(R)$ with the weak pigeonhole principle in proving the bijective pigeonhole principle, advancing understanding of bounded arithmetic.

Findings

Existence of a model where $ontoPHP^{n+1}_n(R)$ fails for some nonstandard $n$

$PHP^{m+1}_m$ holds for all $ riangle^b_1(R)$-definable relations and $m \,\leq\, n^{1-\epsilon}$

Supports an open question by M. Ajtai regarding bounded arithmetic and pigeonhole principles

Abstract

We give elementary proof that theory $T^1_2(R)$ augmented by the weak pigeonhole principle for all $\Delta^b_1(R)$-definable relations does not prove the bijective pigeonhole principle for $R$. This can be derived from known more general results but our proof yields a model of $T^1_2(R)$ in which $ontoPHP^{n+1}_n(R)$ fails for some nonstandard element $n$ while $PHP^{m+1}_m$ holds for all $\Delta^b_1(R)$-definable relations and all $m \leq n^{1-\epsilon}$, where $\epsilon > 0$ is a fixed standard rational parameter. This can be seen as a step towards solving an open question posed by M. Ajtai.