On a Hierarchy of Reflection Principles in Peano Arithmetic

TL;DR

This paper explores the hierarchy of reflection principles in Peano Arithmetic, showing their equivalence to specific provability conditions and establishing a non-collapsing order of their deductive strength.

Contribution

It characterizes all reflection principles in PA as equivalent to either basic provability or iterated provability, forming a strict hierarchy.

Findings

Reflection principles are equivalent to specific provability conditions.

The hierarchy of reflection principles is non-collapsing.

A strict order of deductive strength is established among these principles.

Abstract

We study reflection principles of Peano Arithmetic PA which are based on both proof and provability. Any such reflection principle in PA is equivalent to either $\Box P\!\rightarrow\! P$ ($\Box P$ stands for `$P$ is provable') or $\Box^k u\!\!:\!\!P\!\rightarrow\! P$ for some $k\geq 0$ ($t:P$ states `$t$ is a proof of $P$'). Reflection principles constitute a non-collapsing hierarchy with respect to their deductive strength $$u\!\!:\!\!P\!\rightarrow\! P\ \ \prec\ \ \Box u\!\!:\!\!P\!\rightarrow\! P\ \ \prec\ \ \Box^2 u\!\!:\!\!P\!\rightarrow\! P \ \ \prec\ \ldots\ \prec\ \ \Box P\!\rightarrow\! P.$$