Abstract

A function f from ω1 to the ordinals is called a canonical function for an ordinal α if f represents α in any generic ultrapower induced by forcing with math formula. We introduce here a method for coding sets of ordinals using canonical functions from ω1 to ω1. Combining this approach with arguments from, we show, assuming the Continuum Hypothesis, that for each cardinal κ there is a forcing construction preserving cardinalities and cofinalities forcing that every subset of κ is an element of the inner model math formula.