The extent of determinacy in omega-small mice

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Abstract

Given sufficient large cardinals, the minimal iterable proper class mouse M with infinitely many Woodin cardinals satisfies "the reals R are wellordered in L(R)". However, determinacy holds essentially as far as possible in $L(R\cap M)$, in that it satisfies "there is an ordinal $\delta$ such that $L_\delta(R)$ models determinacy and there is a wellorder of R in $L_{\delta+1}(R)$". In fact, there is a $\Sigma_1$-elementary embedding from $L_\delta(R\cap M)$ into (the true) L(R). Rudominer and Steel conjectured in 1999 that a similar phenomenon should arise in all $\omega$-small mice which model ZF-minus + "R exists" (these include M and all mice which are below it in the mouse-order and which model ZF-minus + "R exists"). They confirmed the conjecture in certain cases, but the remaining cases have remained open. We report on some further progress toward a positive resolution of the conjecture. This is joint work with John Steel.

Speakers from the University of Münster

Schlutzenberg, Farmer

Junior professorship for mathematical logic (Prof. Schlutzenberg)