The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers and that in an arbitrary supersoluble skew brace B many relevant skew brace-theoretical properties are easier to identify: For example, a centrally nilpotent ideal of B is B-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, B has finite multipermutational level if and only if $(B,+)$ is nilpotent.

Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite nondegenerate solution of the Yang–Baxter equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau )$ be the set of countable structures with universe $\omega $ in vocabulary $\tau $ topologized by the Scott topology. We show that an invariant set $X\subseteq Mod(\tau )$ is $\Pi ^0_\alpha $ in the Borel hierarchy of this topology if and only if it is definable by a $\Pi ^p_\alpha $-formula, a positive $\Pi ^0_\alpha $ formula in the infinitary logic $L_{\omega _1\omega }$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $\mathcal {K}$ be positively computably embeddable in $\mathcal {K}'$ by $\Phi $, then for every $\Pi ^p_\alpha $ formula $\xi $ in the vocabulary of $\mathcal {K}'$ there is a $\Pi ^p_\alpha $ formula $\xi ^{*}$ in the vocabulary of $\mathcal {K}$ such that for all $\mathcal {A}\in \mathcal {K}$, $\mathcal {A}\models \xi ^{*}$ if and only if $\Phi (\mathcal {A})\models \xi $. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ n-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures.

Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.

We introduce a natural two-cardinal version of Bagaria’s sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Holy and White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.

We prove that, for any countable acylindrically hyperbolic group G, there exists a generating set S of G such that the corresponding Cayley graph $\Gamma (G,S)$ is hyperbolic, $|\partial \Gamma (G,S)|>2$, the natural action of G on $\Gamma (G,S)$ is acylindrical and the natural action of G on the Gromov boundary $\partial \Gamma (G,S)$ is hyperfinite. This result broadens the class of groups that admit a non-elementary acylindrical action on a hyperbolic space with a hyperfinite boundary action.

We investigate the notion of a semi-retraction between two first-order structures (in typically different signatures) that was introduced by the second author as a link between the Ramsey property and generalized indiscernible sequences. We look at semi-retractions through a new lens establishing transfers of the Ramsey property and finite Ramsey degrees under quite general conditions that are optimal as demonstrated by counterexamples. Finally, we compare semi-retractions to the category theoretic notion of a pre-adjunction.

We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over $\mathsf {Set}$. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski duality for modal algebras beyond the zero-dimensional setting.

This paper initiates the reverse mathematics of social choice theory, studying Arrow’s impossibility theorem and related results including Fishburn’s possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in ${\mathsf {RCA}}_0$. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in ${\mathsf {RCA}}_0$. This approach yields a proof of Arrow’s theorem in ${\mathsf {RCA}}_0$, and thus in $\mathrm {PRA}$, since Arrow’s theorem can be formalised as a $\Pi ^0_1$ sentence. Finally we show that Fishburn’s possibility theorem for countable societies is equivalent to ${\mathsf {ACA}}_0$ over ${\mathsf {RCA}}_0$.

We develop the theory of cofinal types of ultrafilters over measurable cardinals and establish its connections to Galvin’s property. We generalize fundamental results from the countable to the uncountable, but often in surprisingly strengthened forms, and present models with varying structures of the cofinal types of ultrafilters over measurable cardinals.

We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on $\mathsf {IKP}$ and $\mathsf {CZF}$. Most previously studied large set axioms, notably, the constructive analogues of large cardinals below $0^\sharp $, have proof-theoretic strength weaker than full Second-Order Arithmetic. On the other hand, the situation is dramatically different for those defined via elementary embeddings. We show that by adding to $\mathsf {IKP}$ the basic properties of an elementary embedding $j\colon V\to M$ for $\Delta _0$-formulas, which we will denote by $\Delta _0\text {-}\mathsf {BTEE}_M$, we obtain the consistency of $\mathsf {ZFC}$ and more. We will also see that the consistency strength of a Reinhardt set exceeds that of $\mathsf {ZF+WA}$. Furthermore, we will define super Reinhardt sets and $\mathsf {TR}$, which is a constructive analogue of V being totally Reinhardt, and prove that their proof-theoretic strength exceeds that of $\mathsf {ZF}$ with choiceless large cardinals.

We show that the conceptual distance between any two theories of first-order logic is the same as the generator distance between their Lindenbaum–Tarski algebras of concepts. As a consequence of this, we show that, for any two arbitrary mathematical structures, the generator distance between their meaning algebras (also known as cylindric set algebras) is the same as the conceptual distance between their first-order logic theories. As applications, we give a complete description for the distances between meaning algebras corresponding to structures having at most three elements and show that this small network represents all the possible conceptual distances between complete theories. As a corollary of this, we will see that there are only two non-trivial structures definable on three-element sets up to conceptual equivalence (i.e., up to elementary plus definitional equivalence).

For many years, there have been conducting research (e.g., by Bergelson, Furstenberg, Kojman, Kubiś, Shelah, Szeptycki, Weiss) into sequentially compact spaces that are, in a sense, topological counterparts of some combinatorial theorems, for instance, Ramsey’s theorem for coloring graphs, Hindman’s finite sums theorem, and van der Waerden’s arithmetical progressions theorem. These spaces are defined with the aid of different kinds of convergences: IP-convergence, R-convergence, and ordinary convergence.

The first aim of this paper is to present a unified approach to these various types of convergences and spaces. Then, using this unified approach, we prove some general theorems about existence of the considered spaces and show that all results obtained so far in this subject can be derived from our theorems.

The second aim of this paper is to obtain new results about the specific types of these spaces. For instance, we construct a Hausdorff Hindman space that is not an $\mathcal {I}_{1/n}$-space and a Hausdorff differentially compact space that is not Hindman. Moreover, we compare Ramsey spaces with other types of spaces. For instance, we construct a Ramsey space that is not Hindman and a Hindman space that is not Ramsey.

The last aim of this paper is to provide a characterization that shows when there exists a space of one considered type that is not of the other kind. This characterization is expressed in purely combinatorial manner with the aid of the so-called Katětov order that has been extensively examined for many years so far.

This paper may interest the general audience of mathematicians as the results we obtain are on the intersection of topology, combinatorics, set theory, and number theory.

We introduce a family of local ranks $D_Q$ depending on a finite set Q of pairs of the form $(\varphi (x,y),q(y)),$ where $\varphi (x,y)$ is a formula and $q(y)$ is a global type. We prove that in any NSOP$_1$ theory these ranks satisfy some desirable properties; in particular, $D_Q(x=x)<\omega $ for any finite tuple of variables x and any Q, if $q\supseteq p$ is a Kim-forking extension of types, then $D_Q(q)<D_Q(p)$ for some Q, and if $q\supseteq p$ is a Kim-non-forking extension, then $D_Q(q)=D_Q(p)$ for every Q that involves only invariant types whose Morley powers are -stationary. We give natural examples of families of invariant types satisfying this property in some NSOP$_1$ theories.

We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory $T_\infty $ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in $T_\infty $, strengthening an earlier observation that $T_\infty $ satisfies the existence axiom for forking independence.

Finally, we slightly modify our definitions and go beyond NSOP$_1$ to find out that our local ranks are bounded by the well-known ranks: the inp-rank (burden), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example, if T is dp-minimal. Hence, our notion of rank identifies a non-trivial class of theories containing all NSOP$_1$ and NTP$_2$ theories.

In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by $\alpha $-CLI and L-$\alpha $-CLI where $\alpha $ is a countable ordinal. We establish three results:

1. (1) G is $0$-CLI iff $G=\{1_G\}$;

2. (2) G is $1$-CLI iff G admits a compatible complete two-sided invariant metric; and

3. (3) G is L-$\alpha $-CLI iff G is locally $\alpha $-CLI, i.e., G contains an open subgroup that is $\alpha $-CLI.

Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups $G_\alpha $ and $H_\alpha $ for $\alpha <\omega _1$, such that:

1. (1) $H_\alpha $ is $\alpha $-CLI but not L-$\beta $-CLI for $\beta <\alpha $; and

2. (2) $G_\alpha $ is $(\alpha +1)$-CLI but not L-$\alpha $-CLI.

Wilkie proved in 1977 that every countable model ${\mathcal M}$ of Peano Arithmetic has an elementary end extension ${\mathcal N}$ such that the interstructure lattice $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M})$ is the pentagon lattice ${\mathbf N}_5$. This theorem implies that every countable nonstandard ${\mathcal M}$ has an elementary cofinal extension ${\mathcal N}$ such that $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$. It is proved here that whenever ${\mathcal M} \prec {\mathcal N} \models \mathsf {PA}$ and $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$, then ${\mathcal N}$ must be either an end or a cofinal extension of ${\mathcal M}$. In contrast, there are ${\mathcal M}^* \prec {\mathcal N}^* \models \mathsf {PA}^*$ such that $\operatorname {\mathrm {Lt}}({\mathcal N}^* / {\mathcal M}^*) \cong {\mathbf N}_5$ and ${\mathcal N}^*$ is neither an end nor a cofinal extension of ${\mathcal M}^*$.

This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.

The following summarizes the main results proved under suitable partition hypotheses.

• • If $\kappa $ is a cardinal, $\epsilon < \kappa $, ${\mathrm {cof}}(\epsilon ) = \omega $, $\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ and $\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then $\Phi $ satisfies the almost everywhere short length continuity property: There is a club $C \subseteq \kappa $ and a $\delta < \epsilon $ so that for all $f,g \in [C]^\epsilon _*$, if $f \upharpoonright \delta = g \upharpoonright \delta $ and $\sup (f) = \sup (g)$, then $\Phi (f) = \Phi (g)$.

• • If $\kappa $ is a cardinal, $\epsilon $ is countable, $\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ holds and $\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then $\Phi $ satisfies the strong almost everywhere short length continuity property: There is a club $C \subseteq \kappa $ and finitely many ordinals $\delta _0, ..., \delta _k \leq \epsilon $ so that for all $f,g \in [C]^\epsilon _*$, if for all $0 \leq i \leq k$, $\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$, then $\Phi (f) = \Phi (g)$.

• • If $\kappa $ satisfies $\kappa \rightarrow _* (\kappa )^\kappa _2$, $\epsilon \leq \kappa $ and $\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then $\Phi $ satisfies the almost everywhere monotonicity property: There is a club $C \subseteq \kappa $ so that for all $f,g \in [C]^\epsilon _*$, if for all $\alpha < \epsilon $, $f(\alpha ) \leq g(\alpha )$, then $\Phi (f) \leq \Phi (g)$.

• • Suppose dependent choice ($\mathsf {DC}$), ${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$ and the almost everywhere short length club uniformization principle for ${\omega _1}$ hold. Then every function $\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$ satisfies a finite continuity property with respect to closure points: Let $\mathfrak {C}_f$ be the club of $\alpha < {\omega _1}$ so that $\sup (f \upharpoonright \alpha ) = \alpha $. There is a club $C \subseteq {\omega _1}$ and finitely many functions $\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$ so that for all $f \in [C]^{\omega _1}_*$, for all $g \in [C]^{\omega _1}_*$, if $\mathfrak {C}_g = \mathfrak {C}_f$ and for all $i < n$, $\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$, then $\Phi (g) = \Phi (f)$.

• • Suppose $\kappa $ satisfies $\kappa \rightarrow _* (\kappa )^\epsilon _2$ for all $\epsilon < \kappa $. For all $\chi < \kappa $, $[\kappa ]^{<\kappa }$ does not inject into ${}^\chi \mathrm {ON}$, the class of $\chi $-length sequences of ordinals, and therefore, $|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$. As a consequence, under the axiom of determinacy $(\mathsf {AD})$, these two cardinality results hold when $\kappa $ is one of the following weak or strong partition cardinals of determinacy: ${\omega _1}$, $\omega _2$, $\boldsymbol {\delta }_n^1$ (for all $1 \leq n < \omega $) and $\boldsymbol {\delta }^2_1$ (assuming in addition $\mathsf {DC}_{\mathbb {R}}$).

We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on $\mathbb {N}$ obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc. and represent the canonical way of speaking about large finite numbers. The second one is ordinal collapsing functions, which represent the best-known method of describing large computable ordinals.

We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. The isomorphism theorem asserts that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by cardinal collapsing functions. As an application of this fact, we obtain a restatement of the subsystem $\Pi ^1_1$-${\mathsf {CA_0}}$ of analysis as a higher-type well-ordering principle asserting that binary fast-growing hierarchies preserve well-foundedness.

Due to Gödel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments—Warren [43] and Murzi and Topey [30]—for the idea that the natural deduction rules for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s [19] local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e., non-compact. Consequently, I reconsider the use of the $\omega $-rule and I show that the addition of the $\omega $-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement.

We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n < 2^{-n}$ for infinitely many ${n \in \mathbb {N}}$. We also call a set $A\subseteq \mathbb {N}$ regainingly approximable if it is c.e. and the strongly left-computable number $2^{-A}$ is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. K-trivial, we construct such an $\alpha $ such that ${K(\alpha \restriction n)>n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.

On the basis of Poincaré and Weyl’s view of predicativity as invariance, we develop an extensive framework for predicative, type-free first-order set theory in which $\Gamma _0$ and much bigger ordinals can be defined as von Neumann ordinals. This refutes the accepted view of $\Gamma _0$ as the “limit of predicativity”.