Combinatorial principles equivalent to weak induction

Abstract

We consider two combinatorial principles, ${\sf{ERT}}$ and ${\sf{ECT}}$. Both are easily proved in ${\sf{RCA}}_0$ plus ${\Sigma^0_2}$ induction. We give two proofs of ${\sf{ERT}}$ in ${\sf{RCA}}_0$, using different methods to eliminate the use of ${\Sigma^0_2}$ induction. Working in the weakened base system ${\sf{RCA}}_0^*$, we prove that ${\sf{ERT}}$ is equivalent to ${\Sigma^0_1}$ induction and ${\sf{ECT}}$ is equivalent to ${\Sigma^0_2}$ induction. We conclude with a Weihrauch analysis of the principles, showing ${\sf{ERT}} {\equiv_{\rm W}} {\sf{LPO}}^* {<_{\rm W}}{{\sf{TC}}_{\mathbb N}}^* {\equiv_{\rm W}} {\sf{ECT}}$.