On $\alpha$-largeness and the Paris-Harrington principle in $\mathrm{RCA}_0$ and $\mathrm{RCA}_0^{\displaystyle{*}}$

TL;DR

This paper investigates the use of $oldsymbol{\alpha}$-largeness in relation to the Paris-Harrington principle within weak base systems, extending previous results to unrestricted dimensions and confirming the approach's validity in $oldsymbol{ ext{EFA}}$ without transfinite induction.

Contribution

It extends the treatment of $oldsymbol{\alpha}$-largeness and the Paris-Harrington principle to $oldsymbol{ ext{RCA}_0^{*}}$ for unrestricted dimensions, showing it can be done in $oldsymbol{ ext{EFA}}$ without transfinite induction.

Findings

The proof works in $ ext{RCA}_0^{*}$ for fixed standard dimensions.

The proof can be modified to work for unrestricted dimensions.

The approach is confirmed to be valid in $ ext{EFA}$ without transfinite induction.

Abstract

We examine, within $\mathrm{RCA}_0$, the treatment by Ketonen and Solovay on the use of $\alpha$-largeness for giving an upper bound for the Paris--Harrington principle. This proof works fine in $\mathrm{RCA}_0^{\displaystyle{*}}$ for every fixed standard dimension. We also show how to modify the arguments to work within $\mathrm{RCA}_0^{\displaystyle{*}}$ for unrestricted dimensions. To the author's knowledge, this is the first time that it is confirmed that the treatment can be done within $\mathrm{EFA}$ without some transfinite induction added.