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.

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.

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.

This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor–Bendixson theorem, and various problems arising from them. In the framework of reverse mathematics, these theorems are equivalent, respectively, to $\mathsf {ATR}_0$ and $\boldsymbol {\Pi }^1_1{-}\mathsf{CA}_0$, the two strongest subsystems of second order arithmetic among the so-called big five. As far as we know, this is the first systematic study of problems at the level of $\boldsymbol {\Pi }^1_1{-}\mathsf{CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.

Definable stationary sets, and specifically, ordinal definable ones, play a significant role in the study of canonical inner models of set theory and the class HOD of hereditarily ordinal definable sets. Fixing a certain notion of definability and an uncountable cardinal, one can consider the associated family of definable closed unbounded sets. In this paper, we study the extent to which such families can approximate the full closed unbounded filter and their dependence on the defining complexity. Focusing on closed unbounded subsets of a cardinal $\kappa $ which are $\Sigma _1$-definable in parameters from H${}_\kappa $ and ordinal parameters, we show that the ability of such closed unbounded sets to well approximate the closed unbounded filter on $\kappa $ can highly vary and strongly depends on key properties of the underlying universe of set theory.

We investigate the tower spectrum in the generalized Baire space, i.e., the set of lengths of towers in $\kappa ^\kappa $. We show that both small and large tower spectra at all regular cardinals simultaneously are consistent. Furthermore, based on previous work by Bağ, the first author and Friedman, we prove that globally, a small tower spectrum is consistent with an arbitrarily large spectrum of maximal almost disjoint families. Finally, we show that any non-trivial upper bound on the tower spectrum in $\kappa ^\kappa $ is consistent.

In [15] we defined and proved the consistency of the principle $\mathrm {GM}^+(\omega _3,\omega _1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $\omega _2$ and $\omega _3$. In this paper we formulate a strengthening of $\mathrm {GM}^+(\omega _3,\omega _1)$ that we call $\mathrm {SGM}^+(\omega _3,\omega _1)$. We also prove, modulo the consistency of two supercompact cardinals, that $\mathrm {SGM}^+(\omega _3,\omega _1)$ is consistent with ZFC. In addition to all the consequences of $\mathrm {GM}^+(\omega _3,\omega _1)$, the principle $\mathrm {SGM}^+(\omega _3,\omega _1)$, together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of $\omega _2$ either adds a real or collapses some cardinal. This gives a partial answer to a question of Abraham [1] and extends a previous result of Todorčević [16] in this direction.

We give an update on the problem list of Erdős and Hajnal.

The generic multiverse was introduced in [74] and [81] to explicate the portion of mathematics which is immune to our independence techniques. It consists, roughly speaking, of all universes of sets obtainable from a given universe by forcing extension. Usuba recently showed that the generic multiverse contains a unique definable universe, assuming strong large cardinal hypotheses. On the basis of this theorem, a non-pluralist about set theory could dismiss the generic multiverse as irrelevant to what set theory is really about, namely that unique definable universe. Whatever one’s attitude towards the generic multiverse, we argue that certain impure proofs ensure its ongoing relevance to the foundations of set theory. The proofs use forcing-fragile theories and absoluteness to prove ${\mathrm {ZFC}}$ theorems about simple “concrete” objects.

We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman–Stanley embedding on Scott sentence complexity and show that it only preserves $\Pi ^{\mathrm {in}}_{\alpha }$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott sentence complexities except $\Sigma ^{\mathrm {in}}_{3}$ and $\Sigma ^{\mathrm {in}}_{\lambda +1}$ for $\lambda $ a limit ordinal. We show that the former cannot be the Scott sentence complexity of a linear ordering. In the process we develop new techniques which appear to be helpful to calculate the Scott sentence complexities of structures.

We introduce a generalization of sequential compactness using barriers on $\omega $ extending naturally the notion introduced in [W. Kubiś and P. Szeptycki, On a topological Ramsey theorem, Canad. Math. Bull., 66 (2023), 156–165]. We improve results from [C. Corral and O. Guzmán and C. López-Callejas, High dimensional sequential compactness, Fund. Math.] by building spaces that are ${\mathcal {B}}$-sequentially compact but not ${\mathcal {C}}$-sequentially compact when the barriers ${\mathcal {B}}$ and ${\mathcal {C}}$ satisfy certain rank assumption which turns out to be equivalent to a Katětov-order assumption. Such examples are constructed under the assumption ${\mathfrak {b}} ={\mathfrak {c}}$. We also exhibit some classes of spaces that are ${\mathcal {B}}$-sequentially compact for every barrier ${\mathcal {B}}$, including some classical classes of compact spaces from functional analysis, and as a byproduct, we obtain some results on angelic spaces. Finally, we introduce and compute some cardinal invariants naturally associated to barriers.

We study generic properties of topological groups in the sense of Baire category.

First, we investigate countably infinite groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed groups and the word problem. Recently, I. Goldbring, S. Kunnawalkam Elayavalli, and Y. Lodha proved that every isomorphism class is meager among countably infinite groups. In contrast, it follows from the work of W. Hodges on model-theoretic forcing that there exists a comeager isomorphism class among countably infinite abelian groups. We present a new elementary proof of this result.

Then, we turn to compact metrizable abelian groups. We use Pontryagin duality to show that there is a comeager isomorphism class among compact metrizable abelian groups. We discuss its connections to the countably infinite case.

Finally, we study compact metrizable groups. We prove that the generic compact metrizable group is neither connected nor totally disconnected; also it is neither torsion-free nor a torsion group.

We give a short proof of a theorem of Beleznay asserting that the set $L2$ of reals coding linear orders of the form $I + I$ is complete analytic.

Much of the theory of large cardinals beyond a measurable cardinal concerns the structure of elementary embeddings of the universe of sets into inner models. This paper seeks to answer the question of whether the inner model uniquely determines the elementary embedding.

We answer a question of Woodin [3] by showing that “$\mathrm {NS}_{\omega _1}$ is $\omega _1$-dense” holds in a stationary set preserving extension of any universe with a cardinal $\kappa $ which is a limit of ${<}\kappa $-supercompact cardinals. We introduce a new forcing axiom $\mathrm {Q}$-Maximum, prove it consistent from a supercompact limit of supercompact cardinals, and show that it implies the version of Woodin’s $(*)$-axiom for $\mathbb Q_{\mathrm {max}}$. It follows that $\mathrm {Q}$-Maximum implies “$\mathrm {NS}_{\omega _1}$ is $\omega _1$-dense.” Along the way we produce a number of other new instances of Asperó–Schindler’s $\mathrm {MM}^{++}\Rightarrow (*)$ (see [1]).

To force $\mathrm {Q}$-Maximum, we develop a method which allows for iterating $\omega _1$-preserving forcings which may destroy stationary sets, without collapsing $\omega _1$. We isolate a new regularity property for $\omega _1$-preserving forcings called respectfulness which lies at the heart of the resulting iteration theorem.

In the second part, we show that the $\kappa $-mantle, i.e., the intersection of all grounds which extend to V via forcing of size ${<}\kappa $, may fail to be a model of $\mathrm {AC}$ for various types of $\kappa $. Most importantly, it can be arranged that $\kappa $ is a Mahlo cardinal. This answers a question of Usuba [2].

Abstract prepared by Andreas Lietz

E-mail: andreas.lietz@tuwien.ac.at.

URL: https://andreas-lietz.github.io/resources/PDFs/AJourneyGuidedByThe Stars.pdf.

Traditionally, the role of general topology in model theory has been mainly limited to the study of compacta that arise in first-order logic. In this context, the topology tends to be so trivial that it turns into combinatorics, motivating a widespread approach that focuses on the combinatorial component while usually hiding the topological one. This popular combinatorial approach to model theory has proved to be so useful that it has become rare to see more advanced topology in model-theoretic articles. Prof. Franklin D. Tall has led the re-introduction of general topology as a valuable tool to push the boundaries of model theory. Most of this thesis is directly influenced by and builds on this idea.

The first part of the thesis will answer a problem of T. Gowers on the undefinability of pathological Banach spaces such as Tsirelson space. The topological content of this chapter is centred around Grothendieck spaces.

In a similar spirit, the second part will show a new connection between the notion of metastability introduced by T. Tao and the topological concept of pseudocompactness. We shall make use of this connection to show a result of X. Caicedo, E. Dueñez, J. Iovino in a much simplified manner.

The third part of the thesis will carry a higher set-theoretic content as we shall use forcing and descriptive set theory to show that the well-known theorem of M. Morley on the trichotomy concerning the number of models of a first-order countable theory is undecidable if one considers second-order countable theories instead.

The only part that did not originate from model-theoretic questions will be the fourth one. We show that $\operatorname {ZF} + \operatorname {DC} +$“all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations. This result provides evidence in favour of a long-standing conjecture asking whether Turing determinacy implies the axiom of determinacy.

Abstract prepared by Clovis Hamel

E-mail: chamel@math.toronto.edu

In the first part of this paper, we consider several natural axioms in urelement set theory, including the Collection Principle, the Reflection Principle, the Dependent Choice scheme and its generalizations, as well as other axioms specifically concerning urelements. We prove that these axioms form a hierarchy over $\text {ZFCU}_{\text {R}}$ (ZFC with urelements formulated with Replacement) in terms of direct implication. The second part of the paper studies forcing over countable transitive models of $\text {ZFU}_{\text {R}}$. We propose a new definition of ${\mathbb P}$-names to address an issue with the existing approach. We then prove the fundamental theorem of forcing with urelements regarding axiom preservation. Moreover, we show that forcing can destroy and recover certain axioms within the previously established hierarchy. Finally, we demonstrate how ground model definability may fail when the ground model contains a proper class of urelements.

We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation $\mathbb {E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

All spaces are assumed to be Tychonoff. Given a realcompact space X, we denote by $\mathsf {Exp}(X)$ the smallest infinite cardinal $\kappa $ such that X is homeomorphic to a closed subspace of $\mathbb {R}^\kappa $. Our main result shows that, given a cardinal $\kappa $, the following conditions are equivalent:

• • There exists a countable crowded space X such that $\mathsf {Exp}(X)=\kappa $.

• • $\mathfrak {p}\leq \kappa \leq \mathfrak {c}$.

$\mathfrak {d}\leq \kappa \leq \mathfrak {c}$

$2^\kappa $

$\kappa $

$\kappa $

$\mathsf {Exp}(X)$

For a space X let $\mathcal {K}(X)$ be the set of compact subsets of X ordered by inclusion. A map $\phi :\mathcal {K}(X) \to \mathcal {K}(Y)$ is a relative Tukey quotient if it carries compact covers to compact covers. When there is such a Tukey quotient write $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$, and write $(X,\mathcal {K}(X)) =_T (Y,\mathcal {K}(Y))$ if $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ and vice versa.

We investigate the initial structure of pairs $(X,\mathcal {K}(X))$ under the relative Tukey order, focussing on the case of separable metrizable spaces. Connections are made to Menger spaces.

Applications are given demonstrating the diversity of free topological groups, and related free objects, over separable metrizable spaces. It is shown a topological group G has the countable chain condition if it is either $\sigma $-pseudocompact or for some separable metrizable M, we have $\mathcal {K}(M) \ge _T (G,\mathcal {K}(G))$.