Coding into K by reasonable forcing
Ralf Schindler
Abstract
Assuming that there is no inner model with a strong cardinal, the following is shown: any subset of ω1 can be made Δ13 (in the codes) by a reasonable set-forcing; there is a reasonable set-generic extension with a Δ13 well-ordering of its reals; 2-step Σ14 absoluteness fails w.r.t. set-sized reasonable forcings.
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