Abstract
We present a technique for coding sets “into K K ,” where K K is the core model below a strong cardinal. Specifically, we show that if there is no inner model with a strong cardinal then any X ⊂ ω 1 X\subset \omega _1 can be made Δ 3 1 \boldsymbol {\Delta }^1_3 (in the codes) in a reasonable and stationary preserving set generic extension.
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