**April 6**

Zachiri McKenzie,
Zhejiang University

**Topless powerset preserving end-extensions and rank-extensions of countable models of set theory**

This talk will report on ongoing work that is being done in collaboration with Ali Enayat (University of Gothenburg).

For models of set theory $\mathcal{N}$ and $\mathcal{M}$, $\mathcal{N}$ is a powerset preserving end-extension of $\mathcal{M}$ if $\mathcal{N}$ is an end-extension of $\mathcal{M}$ and $\mathcal{N}$ contains no new subsets of sets in $\mathcal{M}$. A model of Kripke-Platek Set Theory, $\mathcal{N}$, is a rank-extension of a model of Kripke-Platek Set Theory, $\mathcal{M}$, if $\mathcal{N}$ is an end-extension of $\mathcal{M}$ and all of the new sets in $\mathcal{N}$ have rank that exceeds the rank of all of the sets in $\mathcal{M}$. A powerset preserving end-extension (rank-extension) $\mathcal{N}$ of $\mathcal{M}$ is topless if $\mathcal{M} \neq \mathcal{N}$ and there is no set in $\mathcal{N} \backslash \mathcal{M}$ containing only sets from $\mathcal{M}$. If $\mathcal{M}= \langle M, E^\mathcal{M} \rangle$ is a model of set theory, then the admissible cover of $\mathcal{M}$, $\mathbb{C}\mathrm{ov}_\mathcal{M}$, is defined to be the smallest admissible structure with $\mathcal{M}$ forming its urelements and whose language contains a unary function function symbol, $F$, that sends each $m \in M$ to the set $\{x \in M \mid x E^\mathcal{M} m\}$. Barwise has shown that if $\mathcal{M}$ is a model of Kripke-Platek Set Theory, then $\mathbb{C}\mathrm{ov}_{\mathcal{M}}$ exists and its minimality facilitates compactness arguments for infinitary languages coded in $\mathbb{C}\mathrm{ov}_\mathcal{M}$. We extend Barwise's analysis by showing that if $\mathcal{M}$ satisfies enough set theory then the expansion of $\mathbb{C}\mathrm{ov}_\mathcal{M}$ obtained by adding the powerset function remains admissible. This allows us to build powerset preserving end-extensions and rank-extensions of countable models of certain subsystems of $\mathrm{ZFC}$ satisfying any given recursive subtheory of the model being extended. In particular, we show that

- Every countable model of $\mathrm{KP}^\mathcal{P}$ has a topless rank-extension that satisfies $\mathrm{KP}^\mathcal{P}$.
- Every countable $\omega$-standard model of $\mathrm{MOST}+\Pi_1\textrm{-collection}$ has a topless powerset preserving end-extension that satisfies $\mathrm{MOST}+\Pi_1\textrm{-collection}$.