The internal consistency of Easton's theorem

Let Card denote the class of infinite cardinals and Reg the class of infinite regular cardinals. The continuum function on regulars is the function κ 7→ 2, defined on Reg. This function C has the following two properties: α ≤ β → C(α) ≤ C(β) and α < cof (C(α)). Easton [2] showed that, assuming GCH, any function F : Reg → Card with these two properties (any “Easton function”) is the continuum function on regulars of a cofinalitypreserving generic extension of the universe. We say that this generic extension “realises” the Easton function F . In particular, the statement “2 = κ for all regular κ” is consistent, as by Easton’s result it can be forced over Gödel’s universe L.