In this paper we first consider hyperfinite Borel equivalence relations with a pair of Borel $\mathbb {Z}$-orderings. We define a notion of compatibility between such pairs, and prove a dichotomy theorem which characterizes exactly when a pair of Borel $\mathbb {Z}$-orderings are compatible with each other. We show that, if a pair of Borel $\mathbb {Z}$-orderings are incompatible, then a canonical incompatible pair of Borel $\mathbb {Z}$-orderings of $E_0$ can be Borel embedded into the given pair. We then consider hyperfinite-over-finite equivalence relations, which are countable Borel equivalence relations admitting Borel $\mathbb {Z}^2$-orderings. We show that if a hyperfinite-over-hyperfinite equivalence relation E admits a Borel $\mathbb {Z}^2$-ordering which is self-compatible, then E is hyperfinite.

The distinction of the semantic spaces of elements and types is common practice in practically all type systems. A few type systems, including some early ones, have been proposed whose semantic space has functions only, i.e., depending on the context functions may play element roles as well as type roles. All of these systems are either lacking expressive power, in particular, polymorphism, or they violate uniqueness of types. This work presents for the first time a function-based type system in which typing is a relation between functions and which is using an ordering of functions to introduce bounded polymorphism. The ordering is based on an infinite set of top objects, itself strictly linearly ordered, each of which characterizes a certain function space. These top objects are predicative in the sense that a function using some top object cannot be smaller than this object. The interpretation of proposition as types and elements as proofs remains valid and is extended by viewing the ordering between types as logical implication. The proposed system can be shown to satisfy confluence and subject reduction. Furthermore one can show that the ordering is a partial order, every set of expressions has a maximal element, and there is a (unique) minimal, logically strongest, type among all types of an element. The latter result implies an alternative notion of uniqueness of types. Strong normalisation is the deepest property and its proof is based on a well-founded relation defined over a subsystem of expressions without eliminators. Semantic abstraction of the objects involved in typing, i.e., to use functions in element as well as type roles in a relational setting, is the major contribution of function-based type systems. This work shows that dependent products are not necessary for defining type systems with bounded polymorphism, rather it presents a consistent system with bounded polymorphism and minimal types where typing is a relation between partially ordered functions.

We study H-structures associated with $SU$-rank 1 measurable structures. We prove that the $SU$-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ${}^\omega \omega $-bounding forcing notions, 2) the class of subproper, T-preserving forcing notions (where T is a fixed Souslin tree) and 3) the class of subproper, $[T]$-preserving forcing notions (where T is an $\omega _1$-tree) are iterable with revised countable support. In the second part, we adopt Miyamoto’s theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve $\omega _1$, and, in the case of subcompleteness, don’t add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.

We investigate Steel’s conjecture in ‘The Core Model Iterability Problem’ [10], that if $\mathcal {W}$ and $\mathcal {R}$ are $\Omega +1$-iterable, $1$-small weasels, then $\mathcal {W}\leq ^{*}\mathcal {R}$ iff there is a club $C\subset \Omega $ such that for all $\alpha \in C$, if $\alpha $ is regular, then $\alpha ^{+\mathcal {W}}\leq \alpha ^{+\mathcal {R}}$. We will show that the conjecture fails, assuming that there is an iterable premouse M which models KP and which has a -Woodin cardinal. On the other hand, we show that assuming there is no transitive model of KP with a Woodin cardinal the conjecture holds. In the course of this we will also show that if M is a premouse which models KP with a largest, regular, uncountable cardinal $\delta $, and $\mathbb {P} \in M$ is a forcing poset such that $M\models "\mathbb {P}\text { has the }\delta \text {-c.c.}"$, and $g\subset \mathbb {P}$ is M-generic, then $M[g]\models \text {KP}$. Additionally, we study the preservation of admissibility under iteration maps. At last, we will prove a fact about the closure of the set of ordinals at which a weasel has the S-hull property. This answers another question implicit in remarks in [10].

Homotopy type theory (HoTT) enables reasoning about groups directly as the types of symmetries (automorphisms) of mathematical structures. The HoTT approach to groups—first put forward by Buchholtz, van Doorn, and Rijke—identifies a group with the type of objects of which it is the symmetries. This type is called the “delooping” of the group, taking a term from algebraic topology. This approach naturally extends the group theory to higher groups which have symmetries between symmetries, and so on. In this paper, we formulate and prove a higher version of Schreier’s classification of all group extensions of a given group. Specifically, we prove that extensions of a group G by a group K are classified by actions of G on a delooping of K. Our proof is formalized in Cubical Agda, a dependently typed programming language and proof assistant which implements HoTT.

In this paper, we show that the existence of certain first-countable compact-like extensions is equivalent to the equality between corresponding cardinal characteristics of the continuum. For instance, $\mathfrak b=\mathfrak s=\mathfrak c$ if and only if every regular first-countable space of weight $< \mathfrak c$ can be densely embedded into a regular first-countable countably compact space.

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. The classical zero-one law (independently proved by Fagin and Glebskiĭ et al.) states that every sentence in a purely relational language is almost surely false or almost surely true, meaning that the probability that the formula is true in a randomly chosen finite structures of cardinal n is asymptotically $0$ or $1$ as n grows to infinity. We obtain generalizations of this result for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp’s generalization of the zero-one law. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete (generalizing Grandjean’s result for the classical case), and for some logics we describe completely the set of truth-values that can be taken by sentences almost surely.

We study versions of the tree pigeonhole principle, $\mathsf {TT}^1$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics, an outstanding question of which investigation is whether $\mathsf {TT}^1$ is $\Pi ^1_1$-conservative over the ordinary pigeonhole principle, $\mathsf {RT}^1$. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate the analog of this question for Weihrauch reducibility, and give an affirmative answer. In combination with other results, we use this to show that unlike $\mathsf {RT}^1$, the problem $\mathsf {TT}^1$ is not Weihrauch requivalent to any first-order problem. Our proofs develop new combinatorial machinery for constructing and understanding solutions to instances of $\mathsf {TT}^1$.

We characterize the epimorphisms in homotopy type theory (HoTT) as the fiberwise acyclic maps and develop a type-theoretic treatment of acyclic maps and types in the context of synthetic homotopy theory as developed in univalent foundations. We present examples and applications in group theory, such as the acyclicity of the Higman group, through the identification of groups with 0-connected, pointed 1-types. Many of our results are formalized as part of the agda-unimath library.

We investigate natural variations of behaviourally correct learning and explanatory learning—two learning paradigms studied in algorithmic learning theory—that allow us to “learn” equivalence relations on Polish spaces. We give a characterization of the learnable equivalence relations in terms of their Borel complexity and show that the behaviourally correct and explanatory learnable equivalence relations coincide both in uniform and non-uniform versions of learnability and provide a characterization of the learnable equivalence relations in terms of their Borel complexity. We also show that the set of uniformly learnable equivalence relations is $\boldsymbol {\Pi }^1_1$-complete in the codes and study the learnability of several equivalence relations arising naturally in logic as a case study.

Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^\omega /c(G)$, where $c(G)$ is the group of all convergent sequences in G. We first established two results:

1. (1) Let $G,H$ be two Polish groups. If H is TSI but G is not, then $E(G)\not \le _BE(H)$.

2. (2) Let G be a Polish group. Then the following are equivalent: (a) G is TSI non-archimedean; (b)$E(G)\leq _B E_0^\omega $; and (c) $E(G)\leq _B {\mathbb {R}}^\omega /c_0$. In particular, $E(G)\sim _B E_0^\omega $ iff G is TSI uncountable non-archimedean.

A critical theorem presented in this article is as follows: Let G be a TSI Polish group, and let H be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If $E(G)\le _BE(H)$, then there exists a continuous homomorphism $S:G_0\rightarrow H$ such that $\ker (S)$ is non-archimedean, where $G_0$ is the connected component of the identity of G. The converse holds if G is connected, $S(G)$ is closed in H, and the interval $[0,1]$ can be embedded into H.

As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively.

We show that in the Silver model the inequality $\mathrm {cov}(\mathfrak {C} _2) < \mathrm {cov}(\mathfrak {P}_2)$ holds true, where $\mathfrak {C}_2$ and $\mathfrak {P}_2$ are the two-dimensional Mycielski ideals.

We analyze a countable support product of a free Suslin tree which turns it into a highly rigid Kurepa tree with no Aronszajn subtree. In the process, we introduce a new rigidity property for trees, which says roughly speaking that any non-trivial strictly increasing function from a section of the tree into itself maps into a cofinal branch.

In [2], Pillay introduced definable Galois cohomology, a model-theoretic generalization of Galois cohomology. Let M be an atomic and strongly $\omega $-homogeneous structure over a set of parameters A. Let B be a normal extension of A in M. We show that a short exact sequence of automorphism groups $1 \to \operatorname {\mathrm {Aut}}(M/B) \to \operatorname {\mathrm {Aut}}(M/A) \to \operatorname {\mathrm {Aut}}(B/A) \to 1$ induces a short exact sequence in definable Galois cohomology. We also discuss compatibilities with [3]. Our result complements the long exact sequence in definable Galois cohomology developed in [4].

We show that the evasion number $\mathfrak {e}$ can be added to Cichoń’s maximum with a distinct value. More specifically, it is consistent that $\aleph _1<\operatorname {\mathrm {add}}(\mathcal {N})<\operatorname {\mathrm {cov}}(\mathcal {N})<\mathfrak {b}<\mathfrak {e}<\operatorname {\mathrm {non}}(\mathcal {M})<\operatorname {\mathrm {cov}}(\mathcal {M})<\mathfrak {d}<\operatorname {\mathrm {non}}(\mathcal {N})<\operatorname {\mathrm {cof}}(\mathcal {N})<2^{\aleph _0}$ holds.

We contribute to the study of generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set ${\mathcal {P}}({\lambda })$ of a singular cardinal $\lambda $ of countable cofinality or products $\prod _{i<\omega }\lambda _i$ for a strictly increasing sequence $\langle {\lambda _i}~\vert ~{i<\omega }\rangle $ of cardinals. We consider the question under which large cardinal hypothesis classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms and classes of sets definable by $\Sigma _1$-formulas with parameters from various collections of sets. We prove that $\omega $-many measurable cardinals, while sufficient to prove the perfect set property of all $\Sigma _1$-definable sets with parameters in $V_\lambda \cup \{V_\lambda \}$, are not enough to prove it if there is a cofinal sequence in $\lambda $ in the parameters. For this conclusion, the existence of an I2-embedding is enough, but there are parameters in $V_{\lambda +1}$ for which I2 is still not enough. The situation is similar for the Baire property: under I2 all sets that are $\Sigma _1$-definable using elements of $V_\lambda $ and a cofinal sequence as parameters have the Baire property, but I2 is not enough for some parameter in $V_{\lambda +1}$. Finally, the existence of an I0-embedding implies that all sets that are $\Sigma ^1_n$-definable with parameters in $V_{\lambda +1}$ have the Baire property.

In this article, we propose a new classification of $\Sigma ^0_2$ formulas under the realizability interpretation of many-one reducibility (i.e., Levin reducibility). For example, $\mathsf {Fin}$, the decision of being eventually zero for sequences, is many-one/Levin complete among $\Sigma ^0_2$ formulas of the form $\exists n\forall m\geq n.\varphi (m,x)$, where $\varphi $ is decidable. The decision of boundedness for sequences $\mathsf {BddSeq}$ and for width of posets $\mathsf {FinWidth}$ are many-one/Levin complete among $\Sigma ^0_2$ formulas of the form $\exists n\forall m\geq n\forall k.\varphi (m,k,x)$, where $\varphi $ is decidable. However, unlike the classical many-one reducibility, none of the above is $\Sigma ^0_2$-complete. The decision of non-density of linear orders $\mathsf {NonDense}$ is truly $\Sigma ^0_2$-complete.

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065–1092]. For an ideal $\mathcal {I}$ on $\omega $ we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ and write $f\leq _{\mathcal {I}} g$ if $\{n\in \omega :f(n)>g(n)\}\in \mathcal {I}$, where $f,g\in \omega ^{\omega }$.

We study the cardinal numbers $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ describing the smallest sizes of subsets of $\mathcal {D}_{\mathcal {I}}$ that are unbounded from below with respect to $\leq _{\mathcal {I}}$.

In particular, we examine the relationships of $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ with the dominating number $\mathfrak {d}$. We show that, consistently, $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))>\mathfrak {d}$ for some ideal $\mathcal {I}$, however $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))\leq \mathfrak {d}$ for all analytic ideals $\mathcal {I}$. Moreover, we give example of a Borel ideal with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\operatorname {\mathrm {add}}(\mathcal {M})$.

In $\mathsf {ZF}$ (i.e., Zermelo–Fraenkel set theory minus the axiom of choice ($\mathsf {AC}$)), we investigate the open problem of the deductive strength of the principle

which was introduced by Herzberg, Kanovei, Katz, and Lyubetsky [(2018), Journal of Symbolic Logic, 83(1), 385–391]. Typical results are:

1. (1) “$\aleph _{1}\leq 2^{\aleph _{0}}$” is strictly weaker than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$.

2. (2) “There exists a free ultrafilter on $\omega $” does not imply “$\aleph _{1}\leq 2^{\aleph _{0}}$” in $\mathsf {ZF}$, and thus (by (1)) neither does it imply $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$. This fills the gap in information in Howard and Rubin [Mathematical Surveys and Monographs, American Mathematical Society, 1998], as well as in Herzberg et al. (2018).

3. (3) Martin’s Axiom ($\mathsf {MA}$) implies “no free ultrafilter on $\omega $ has a well-orderable base of cardinality $<2^{\aleph _{0}}$”, and the latter principle is not implied by $\aleph _{0}$-Martin’s Axiom ($\mathsf {MA}(\aleph _{0})$) in $\mathsf {ZF}$.

4. (4) $\mathsf {MA} + \mathsf {UF_{wob}}(\omega )$ implies $\mathsf {AC}(\mathbb {R})$ (the axiom of choice for non-empty sets of reals), which in turn implies $\mathsf {UF_{wob}}(\omega )$. Furthermore, $\mathsf {MA}$ and $\mathsf {UF_{wob}}(\omega )$ are mutually independent in $\mathsf {ZF}$.

5. (5) For any infinite linearly orderable set X, each of “every filter base on X can be well ordered” and “every filter on X has a well-orderable base” is equivalent to “$\wp (X)$ can be well ordered”. This yields novel characterizations of the principle “every linearly ordered set can be well ordered” in $\mathsf {ZFA}$ (i.e., Zermelo–Fraenkel set theory with atoms), and of $\mathsf {AC}$ in $\mathsf {ZF}$.

6. (6) “Every filter on $\mathbb {R}$ has a well-orderable base” implies “every filter on $\omega $ has a well-orderable base”, which in turn implies $\mathsf {UF_{wob}}(\omega )$, and none of these implications are reversible in $\mathsf {ZF}$.

7. (7) “Every filter on $\omega $ can be extended to an ultrafilter with a well-orderable base” is equivalent to $\mathsf {AC}(\mathbb {R}),$ and thus is strictly stronger than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$.

8. (8) “Every filter on $\omega $ can be extended to an ultrafilter” implies “there exists a free ultrafilter on $\omega $ which has no well-orderable base of cardinality ${<2^{\aleph _{0}}}$”. The former principle does not imply “there exists a free ultrafilter on $\omega $ which has no well-orderable base” in $\mathsf {ZF}$, and the latter principle is true in the Basic Cohen Model.