Easton’s theorem and large cardinals

Abstract

The continuum function α ↦ 2 α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β , cf ( F ( α ) ) > α and α < β → F ( α ) ≤ F ( β ) . The classic example of an Easton function is the continuum function α ↦ 2 α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V [ G ] ; we say that F is realised in V [ G ] . However if we also wish to preserve measurable cardinals, new restrictions must be put on F . We say that κ is F ( κ ) - hypermeasurable iff there is an elementary embedding j : V → M with critical point κ such that H ( F ( κ ) ) V ⊆ M ; j will be called a witnessing embedding. We will show that if GCH holds then for any Easton function F there is a cofinality-preserving generic extension V [ G ] such that if κ , closed under F , is F ( κ ) -hypermeasurable in V and there is a witnessing embedding j such that j ( F ) ( κ ) ≥ F ( κ ) , then κ will remain measurable in V [ G ] .