Consistency strength of the axiom of full reflection at large cardinals (Q1912778)

Let \(\kappa\) be an uncountable regular cardinal and let \(S\) be a stationary subset of \(\kappa\) consisting of regular cardinals. We say that \(S\) reflects fully at regular cardinals if whenever \(T\) is a stationary subset of \(\kappa\) consisting of regular cardinals, with higher order than \(S\), then \(S \cap \alpha\) is stationary in \(\alpha\) for almost all \(\alpha\) in \(T\). The Axiom of Full Reflection at \(\kappa\) is the statement that every stationary subset of \(\kappa\) reflects fully at regular cardinals. In this paper the authors prove that the Axiom of Full Reflection at a measurable cardinal is equiconsistent with the existence of a measurable cardinal. They show the corresponding result holds for some other large cardinals, including strong cardinals and supercompact cardinals. The proof is by a complicated iterated forcing argument.