The internal consistency of Easton’s theorem

An Easton function is a monotone function C from infinite regular cardinals to cardinals such that C ( α ) has cofinality greater than α for each infinite regular cardinal α . Easton showed that assuming GCH, if C is a definable Easton function then in some cofinality-preserving extension, C ( α ) = 2 α for all infinite regular cardinals α . Using “generic modification”, we show that over the ground model L , models witnessing Easton’s theorem can be obtained as inner models of L [ 0 # ] , for Easton functions which are L -definable with parameters at most ω 1 L [ 0 # ] . And using a gap 1 morass, we obtain an inner model of L [ 0 # ] with the same cofinalities as L in which ω 1 L [ 0 # ] is a strong limit cardinal and 2 ω 1 L [ 0 # ] equals ω 2 L [ 0 # ] .