Journal of Mathematical Logic

We show that every definable nested family of closed and bounded subsets of a P-minimal field K has nonempty intersection. As an application we answer a question of Darnière and Halupczok showing that P-minimal fields satisfy the “extreme value property”: for every closed and bounded subset $U\subseteq K$ and every interpretable continuous function $f:U\to {\mathrm{\Gamma }}_K$ (where ${\mathrm{\Gamma }}_K$ denotes the value group), f(U) admits a maximal value. Two further corollaries are obtained as a consequence of their work. The first one shows that every interpretable subset of $K\times {\mathrm{\Gamma }}_K^n$ is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every P-minimal field is polynomially bounded. The second one characterizes those P-minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues.

The Ultrapower Axiom is a combinatorial principle concerning the structure of large cardinals that is true in all known canonical inner models of set theory. A longstanding test question for inner model theory is the equiconsistency of strongly compact and supercompact cardinals. In this paper, it is shown that under the Ultrapower Axiom, the least strongly compact cardinal is supercompact. A number of stronger results are established, setting the stage for a complete analysis of strong compactness and supercompactness under UA that will be carried out in the sequel to this paper.

We introduce bounded category forcing axioms for well-behaved classes $\mathrm{\Gamma }$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{{\lambda }_{\mathrm{\Gamma }}^{+}}$ modulo forcing in $\mathrm{\Gamma }$, for some cardinal ${\lambda }_{\mathrm{\Gamma }}$ naturally associated to $\mathrm{\Gamma }$. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation ${\lambda }_{\mathrm{\Gamma }}=\omega$ — to classes $\mathrm{\Gamma }$ with ${\lambda }_{\mathrm{\Gamma }}>\omega$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on $V$. We also show the existence of many classes $\mathrm{\Gamma }$ with ${\lambda }_{\mathrm{\Gamma }}={\omega }_1$ giving rise to pairwise incompatible theories for $H_{{\omega }_2}$.

Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle ${\mathrm{\text{SR}}}^{-}$ introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle ${\mathrm{\text{SR}}}^{-}$ for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak ${\mathrm{\Pi }}_n^1$-indescribability, can be canonically characterized through localized versions of the principle ${\mathrm{\text{SR}}}^{-}$. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak ${\mathrm{\Pi }}_1^1$-indescribability.

We generically construct a model in which the ${\mathrm{\Sigma }}_3^1$-separation property is true, i.e. every pair of disjoint ${\mathrm{\Sigma }}_3^1$-sets can be separated by a ${\mathrm{\Delta }}_3^1$-definable set. This answers an old question from the problem list “Surrealist landscape with figures” by A. Mathias from 1968. We also construct a model in which the (lightface) ${\mathrm{\Sigma }}_3^1$-separation property is true.