arXiv · 2405.13741
We show that Kim-forking satisfies existence in all NSOP$_1$ theories.
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Byunghan Kim, Joonhee Kim, Hyoyoon Lee. 2025-12-26. Existence in NSOP$_1$ theories. https://arxiv.org/abs/2405.13741
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We study the Borel and analytic subsets of the spaces ${}^κκ$ and ${}^κ2$ endowed with ideal topologies, where $κ$ is a regular uncountable cardinal, thereby addressing some open problems of the literature. We provide a systematic analysis of the Borel hierarchy for an arbitrary ideal topology. In particular, we formulate a sufficient condition ensuring that the hierarchy does not collapse, demonstrate that every Borel set in such a topology is analytic, and establish the existence of a set that is not Borel. We show that, when the underlying ideal contains an unbounded subset, the collection of analytic sets coincides with the full power set of the ambient space. Finally, we prove that the Approximation Lemma holds in the setting of ideal topologies.
Let $M_0$ denote either the field of $p$-adic numbers $\mathbb{Q}_p$ or an $o$-minimal expansion of the real field $\mathbb{R}$. We study the minimal flows and Ellis groups of groups definable over $M_0$ from the viewpoint of definable topological dynamics. The computation of Ellis groups, together with the closely related Newelski conjecture, has so far been available only under restrictive hypotheses: in the $p$-adic setting, the results of \cite{BY-APAL} apply to reductive algebraic groups, while in the $o$-minimal setting the known computations rely on a compact-torsion-free (Iwasawa) decomposition. In this paper we remove both restrictions. We show that every definable group $G$ has a definably amenable radical $D$, with quotient map $π:G\to A$ onto a centreless semisimple group $A$. In $A$, a definably amenable component $V$ is selected: a maximal definably amenable subgroup containing a maximal definably solvable subgroup. Its preimage $B=π^{-1}(V)$ is a definably amenable component of $G$, contains $D$, and is the carrier for the computation of the Ellis group. Our main result is that, for every $M\succ M_0$, the Ellis group of the universal definable flow of $G$ over $M$ is isomorphic to that of $B$ over $M$, and hence to $B/B^{00}$. In particular, the Ellis groups of $G$ are model-independent. As a consequence, when $G$ is definable over a $p$-adically closed field, Newelski's conjecture holds for $G$ if and only if $G$ is definably amenable. The amenable direction is the known result of \cite{CS-Definably-Amenable-NIP-Groups}, while the converse is new.
A key motivation for Heyting's intuitionistic logic was to gain a formal notion of Brouwer's idea of mathematics as a "construction of the mind". One might thus argue that Heyting's Calculus should also correspond to a notion of provability. Inspired by this idea, Gödel formalised this connection via an embedding into a modal calculus, which is now known as the modal logic S4. While in his original publication, he only proved soundness for the embedding from intuitionistic propositional logic into S4, the converse was proved fifteen years later by McKinsey and Tarski. Although, it was later discovered that Gödel also had obtained a proof of the faithfulness of his embedding in unpublished notes in 1941. Rasioa and Sikorski later extended Gödel's embedding to first-order intuitionistic logic. In 1954, Maehara independently obtained the same results using proof-theoretic methods, even extending the embedding to one from intuitionistic first-order logic into intuitionistic first-order modal logic. This document presents a faithful English translation of Maehara's 1954 paper.