On a Hierarchy of Reflection Principles in Peano Arithmetic

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 $$u!!:!!P!\rightarrow! P\ \ \prec\ \ \Box u!!:!!P!\rightarrow! P\ \ \prec\ \ \Box^2 u!!:!!P!\rightarrow! P \ \ \prec\ \ldots\ \prec\ \ \Box P!\rightarrow! P.$$