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On $\Sigma^1_3$ sets in the Sacks model

(2026) Schilhan, Jonathan

We show that in the iterated Sacks model over the constructible universe the Mansfield–Solovay theorem holds for $\Sigma^1_3$ sets. In particular, every $\mathbf{\Sigma}^1_3$ set is Marczewski measurable and the optimal complexity for a Bernstein set is $\Delta^1_4$. Based on a result by Kanovei, we also briefly show how to separate the Mansfield–Solovay theorem at non-trivial levels of the projective hierarchy.

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Isomorphism spectra and computably composite structures

(2026) Lakerdas-Gayle, Joey

Adapting a result of Bazhenov, Kalimullin & Yamaleev (2020), we show that if a Turing degree $\mathbf{d}$ is the degree of categoricity of a computable structure $\mathcal{M}$ and is not the strong degree of categoricity of any computable structure, then $\mathcal{M}$ has a pair of computable copies whose isomorphism spectrum is not finitely generated. Motivated by this result, we introduce a class of computable structures, called computably composite structures, with the property that the isomorphisms between arbitrary computable copies of these structures are exactly the unions of isomorphisms between the computable copies of their components. We use this to show that any computable union of isomorphism spectra is also an isomorphism spectrum. In particular, this gives examples of isomorphism spectra that are not finitely generated.

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On the theory of exponential integer parts

(2026) Jeřábek, Emil

We axiomatize the first-order theories of exponential integer parts of real-closed exponential fields in a language with $2^x$, in a language with a predicate for powers of $2$, and in the basic language of ordered rings. In particular, the last theory extends $\mathsf{IOpen}$ by sentences expressing the existence of winning strategies in a certain game on integers; we show that it is a proper extension of $\mathsf{IOpen}$, and give upper and lower bounds on the required number of rounds needed to win the game.

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Pinto, Jaqueline

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Schilhan, Jonathan

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