Juliette Cara Kennedy

This is the transcript of a conversation between the participants of theWoodin, W. Hugh Arctic Set Theory Workshop VI and Hugh Woodin. The conversation took place on February 24, 2023 at the Biological Research Station of the University of Helsinki in Kilpisjärvi, Finland, and was moderated by Juliette Kennedy. The transcript was created by Beau Madison Mount and is based on detailed notes taken by him during the evening. Questions were asked by David Asperó (DA), Douglas Blue (DB), Juliette Kennedy (JK), Rahman Mohammadpour (RM), Jeffrey Schatz (JS), Corey Switzer (CS), Jouko Väänänen (JV) and Bartosz Wcisło (BW). Unidentified audience members are denoted AM. * * * indicates recording unintelligible or missing.

We introduce a new inner model [Formula: see text] arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively PFA, the regular uncountable cardinals of [Formula: see text] are measurable in the inner model [Formula: see text] and [Formula: see text] satisfies CH. Moreover, assuming a proper class of Woodin cardinals, the theory of [Formula: see text] is (set) forcing absolute. We introduce an auxiliary concept that we call Club Determinacy, which simplifies the construction of [Formula: see text] greatly but may have also independent interest. Based on Club Determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model [Formula: see text].

Assume $\langle \aleph_0, \aleph_1 \rangle \rightarrow \langle \lambda, \lambda^+ \rangle$ . Assume M is a model of a first order theory T of cardinality at most λ+ in a language L(T) of cardinality $\leq \lambda$ . Let N be a model with the same language. Let Δ be a set of first order formulas in L(T) and let D be a regular filter on λ. Then M is $\Delta-embeddable$ into the reduced power $N^\lambda/D$ , provided that every $\Delta-existential$ formula true in M is true also in N. We obtain the following corollary: for M as above and D a regular ultrafilter over $\lambda, M^\lambda/D$ is $\lambda^{++}-universal$ . Our second result is as follows: For $i < \mu$ let Mi and Ni be elementarily equivalent models of a language which has cardinality $\leq \lambda$ . Suppose D is a regular filter on λ and $\langle \aleph_0, \aleph_1 \rangle \rightarrow \langle \lambda, \lambda^+ \rangle$ holds. We show that then the second player has a winning strategy in the $Ehrenfeucht-Fra\ddot{i}ss\acute{e}$ game of length λ+ on $\prod_i M_i/D$ and $\prod_i N_i/D$ . This yields the following corollary: Assume GCH and λ regular (or just $\langle \aleph_0, \aleph_1 \rangle \rightarrow \langle \lambda, \lambda^+ \rangle$ and 2λ = λ+). For L, Mi and Ni be as above, if D is a regular filter on λ, then $\prod_i M_i/D \cong \prod_i N_i/D$