### 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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