An independence of the MIN principle from the PHP principle

TL;DR

This paper proves that in bounded arithmetic, the minimization principle is independent of the pigeonhole principle when extended with certain formulas, highlighting limitations in provability.

Contribution

It demonstrates that the bounded arithmetic theory with the pigeonhole principle does not prove the minimization principle, showing an independence result.

Findings

MIN principle is independent of PHP in bounded arithmetic

Adding PHP for all Δ^b_1(⊳) formulas does not prove MIN(⊳)

The result clarifies limitations of bounded arithmetic theories

Abstract

The minimization principle $\textsf{MIN}(\triangleleft)$ studied in bounded arithmetic says that a strict linear ordering $\triangleleft$ on any finite interval $[0,\dots,n)$ has the minimal element. We shall prove that bounded arithmetic theory $\textsf{T}^1_2(\triangleleft)$ augmented by instances of the pigeonhole principle for all $\Delta^b_1(\triangleleft)$ formulas does not prove $\textsf{MIN}(\triangleleft)$.