We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a single \-sentence \ that codes Kurepa trees to prove the following statements: The spectrum of \ is consistently equal to \ and also consistently equal to \\), where \ is weakly inaccessible.The amalgamation spectrum of \ is consistently equal to \ and \\), where again \ is weakly inaccessible. This is the first example of an \-sentence whose spectrum and amalgamation spectrum are consistently both right-open and (...) right-closed. It also provides a positive answer to a question in Souldatos :533–551, 2014).Consistently, \ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in Baldwin et al. :545–565, 2016) and Baldwin and Souldatos :444–452, 2019) of sentences with maximal models in countably many cardinalities.Consistently, \ and there exists an \-sentence with models in \, but no models in \. This relates to a conjecture by Shelah that if \, then any \-sentence with a model of size \ also has a model of size \. Our result proves that \ can not be replaced by \, even if \. (shrink)

Given an uncountable regular cardinal κ, we study the structural properties of the class of all sets of functions from κ to κ that are definable over the structure 〈H,∈〉 by a Σ1-formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of ZFC together with large cardinal axioms. In this paper, we present other canonical extensions of ZFC that provide a strong structure theory for these classes. These axioms are (...) variations of the Maximality Principle introduced by Stavi and Väänänen and later rediscovered by Hamkins. (shrink)

We prove results that falsify Silver’s dichotomy for Borel equivalence relations on the generalized Baire space under the assumptionV=L.

We study Σ1-definable sets in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1-definable, the set of all stationary subsets of ω1 is not Σ1-definable and the complement of every Σ1-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1-definable well-ordering (...) of H and the existence of a Δ1-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1-definable subsets of κκ, in the case where κ itself has certain large cardinal properties. (shrink)