Full reflection of stationary sets at regular cardinals

Abstract

A stationary subset S of a regular uncountable cardinal kappa reflects fully at regular cardinals if for every stationary set T subseteq kappa of higher order consisting of regular cardinals there exists an alpha in T such that S cap alpha is a stationary subset of alpha. We prove that the Axiom of Full Reflection which states that every stationary set reflects fully at regular cardinals, together with the existence of n-Mahlo cardinals is equiconsistent with the existence of Pi^1_n-indescribable cardinals. We also state the appropriate generalization for greatly Mahlo cardinals.