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

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 ${\mathrm{RCA}}_0^{\ast }$ for every fixed standard dimension. We also show how to modify the arguments to work within ${\mathrm{RCA}}_0^{\ast }$ 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.

arXiv comment11 pages