Relative leftmost path principles and omega-model reflections of transfinite inductions

TL;DR

This paper characterizes the relationship between leftmost path principles and omega-model reflections of transfinite inductions, revealing their equivalences and strength hierarchies within reverse mathematics.

Contribution

It establishes the equivalence between omega-model reflection of transfinite induction and relative leftmost path principles, and compares their strengths across different levels.

Findings

Omega-model reflection of $ ext{Pi}^1_{n+1}$ transfinite induction equals $ ext{Sigma}^0_n$ relative leftmost path principle for $n > 1$

$ ext{Sigma}^0_{n+1} ext{LPP}$ is strictly stronger than $ ext{Sigma}^0_{n} ext{LPP}$

The results clarify the hierarchy of principles in reverse mathematics.

Abstract

In this paper, we give characterizations of Towsner's relative leftmost path principles in terms of omega-model reflections of transfinite inductions. In particular, we show that the omega-model reflection of $\Pi^1_{n+1}$ transfinite induction is equivalent to the $\Sigma^0_n$ relative leftmost path principle over $\mathsf{RCA}_0$ for $n > 1$. As a consequence, we have that $\Sigma^0_{n+1}\mathsf{LPP}$ is strictly stronger than $\Sigma^0_{n}\mathsf{LPP}$.