On Tao's "finitary" infinite pigeonhole principle

TL;DR

This paper examines different finitizations of Tao's infinite pigeonhole principle, providing counterexamples, analyzing their logical equivalences, and exploring the strength of related principles within reverse mathematics.

Contribution

It introduces an alternative finitization FIPP2, analyzes the faithfulness of various versions, and formalizes Tao's correspondence principle as CUB within reverse mathematics.

Findings

Counterexample to FIPP1 disproves its faithfulness.

FIPP2 is shown to be equivalent to IPP in reverse mathematics.

The strength of the continuous uniform boundedness principle CUB is characterized.

Abstract

In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the "finitary" infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alternative version FIPP2. Tao himself formulated yet another version FIPP3 in a revised version of his essay. We give a counterexample to FIPP1 and discuss for both of the versions FIPP2 and FIPP3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP <-> FIPP2 and IPP <-> FIPP3 in the context of reverse mathematics. In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the "big five" subsystems of second order arithmetic.