Coding with canonical functions

by Larson and Shelah. [LrSh:1076]
Math Logic Quarterly, 2017
A function f from omega_{1} to the ordinals is called a canonical function for an ordinal alpha if f represents alpha in any generic ultrapower induced by forcing with {P}(omega_{1})/ mathrm {NS}_{omega_{1}} . We introduce here a method for coding sets of ordinals using canonical functions from omega_{1} to omega_{1} . Combining this approach with arguments from [Sh:f], we show that for each cardinal kappa there is a forcing construction preserving cardinalities and cofinalities forcing that every subset of kappa is in the inner model L({P}(omega_{1})) .

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