For cardinals $\mathfrak {a}$ and $\mathfrak {b}$, we write $\mathfrak {a}=^\ast \mathfrak {b}$ if there are sets A and B of cardinalities $\mathfrak {a}$ and $\mathfrak {b}$, respectively, such that there are partial surjections from A onto B and from B onto A. $=^\ast $-equivalence classes are called surjective cardinals. In this article, we show that $\mathsf {ZF}+\mathsf {DC}_\kappa $, where $\kappa $ is a fixed aleph, cannot prove that surjective cardinals form a cardinal algebra, which gives a negative solution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27, 165–207 (1984)]. Nevertheless, we show that surjective cardinals form a “surjective cardinal algebra”, whose postulates are almost the same as those of a cardinal algebra, except that the refinement postulate is replaced by the finite refinement postulate. This yields a smoother proof of the cancellation law for surjective cardinals, which states that $m\cdot \mathfrak {a}=^\ast m\cdot \mathfrak {b}$ implies $\mathfrak {a}=^\ast \mathfrak {b}$ for all cardinals $\mathfrak {a},\mathfrak {b}$ and all nonzero natural numbers m.

We show, assuming PD, that every complete finitely axiomatized second-order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second-order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g., supercompact) cardinals does not imply the categoricity of all finitely axiomatizable complete second-order theories. More exactly, we show that a non-categorical complete finitely axiomatized second-order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second-order theories with a model of a certain singular cardinality $\kappa $ of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming $V=L$, that every complete finitely axiomatized second-order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second-order theory with a countable model which is non-categorical.

We prove a rigidity result for maps between Čech–Stone remainders under fairly mild forcing axioms.

The Fregean ontology can be naturally interpreted within set theory with urelements, where objects correspond to sets and urelements, and concepts to classes. Consequently, Fregean abstraction principles can be formulated as set-theoretic principles. We investigate how the size of reality—i.e., the number of urelements—interacts with these principles. We show that Basic Law V implies that for some well-ordered cardinal $\kappa $, there is no set of urelements of size $\kappa $. Building on recent work by Hamkins [10], we show that, under certain additional axioms, Basic Law V holds if and only if the urelements form a set. We construct models of urelement set theory in which the Reflection Principle holds while Hume’s Principle fails for sets. Additionally, assuming the consistency of an inaccessible cardinal, we produce a model of Kelley–Morse class theory with urelements that has a global well-ordering but lacks a definable map satisfying Hume’s Principle for classes.

Denote by $\mathcal {NA}$ and $\mathcal {MA}$ the ideals of null-additive and meager-additive subsets of $2^{\omega }$, respectively. We prove in ZFC that $\mathrm {add}(\mathcal {NA})=\mathrm {non}(\mathcal {NA})$ and introduce a new (Polish) relational system to reformulate Bartoszyński’s and Judah’s characterization of the uniformity of $\mathcal {MA}$, which is helpful to understand the combinatorics of $\mathcal {MA}$ and to prove consistency results. As for the latter, we prove that $\mathrm {cov}(\mathcal {MA})<\mathfrak {c}$ (even $\mathrm {cov}(\mathcal {MA})<\mathrm {non}(\mathcal {N})$) is consistent with ZFC, as well as several constellations of Cichoń’s diagram with $\mathrm {non}(\mathcal {NA})$, $\mathrm {non}(\mathcal {MA}),$ and $\mathrm {add}(\mathcal {SN})$, which include $\mathrm {non}(\mathcal {NA})<\mathfrak {b}< \mathrm {non}(\mathcal {MA})$ and $\mathfrak {b}< \mathrm {add}(\mathcal {SN})<\mathrm {cov}(\mathcal {M})<\mathfrak {d}=\mathfrak {c}$.

We study a family of variants of Jensen’s subcomplete forcing axiom, $\mathsf {SCFA,}$ and subproper forcing axiom, $\mathsf {SubPFA}$. Using these, we develop a general technique for proving nonimplications of $\mathsf {SCFA}$, $\mathsf {SubPFA}$ and their relatives and give several applications. For instance, we show that $\mathsf {SCFA}$ does not imply $\mathsf {MA}^+(\sigma $-closed) and $\mathsf {SubPFA}$ does not imply Martin’s Maximum.

We prove two compactness theorems for HOD. First, if $\kappa $ is a strong limit singular cardinal with uncountable cofinality and for stationarily many $\delta <\kappa $, $(\delta ^+)^{\mathrm {HOD}}=\delta ^+$, then $(\kappa ^+)^{\mathrm {HOD}}=\kappa ^+$. Second, if $\kappa $ is a singular cardinal with uncountable cofinality and stationarily many $\delta <\kappa $ are singular in $\operatorname {\mathrm {HOD}}$, then $\kappa $ is singular in $\operatorname {\mathrm {HOD}}$. We also discuss the optimality of these results and show that the first theorem does not extend from $\operatorname {\mathrm {HOD}}$ to other $\omega $-club amenable inner models.

We produce, relative to a $\textsf {ZFC}$ model with a supercompact cardinal, a $\textsf {ZFC}$ model of the Proper Forcing Axiom in which the nonstationary ideal on $\omega _1$ is $\Pi _1$-definable in a parameter from $H_{\aleph _2}$.

We define several notions of a limit point on sequences with domain a barrier in $[\omega ]^{<\omega }$ focusing on the two dimensional case $[\omega ]^2$. By exploring some natural candidates, we show that countable compactness has a number of generalizations in terms of limits of high dimensional sequences and define a particular notion of $\alpha $-countable compactness for $\alpha \leq \omega _1$. We then focus on dimension 2 and compare 2-countable compactness with notions previously studied in the literature. We present a number of counterexamples showing that these classes are different. In particular assuming the existence of a Ramsey ultrafilter, a subspace of $\beta \omega $ which is doubly countably compact whose square is not countably compact, answering a question of T. Banakh, S. Dimitrova, and O. Gutik [3]. The analysis of this construction leads to some possibly new types of ultrafilters related to discrete, P-points and Ramsey ultrafilters.

It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles on a given Euclidean space has countable chromatic number, while the hypergraph of isosceles triangles on $\mathbb {R}^2$ does not.

Using iterated Sacks forcing and topological games, we prove that the existence of a totally imperfect Menger set in the Cantor cube with cardinality continuum is independent from ZFC. We also analyze the structure of Hurewicz and consonant subsets of the Cantor cube in the Sacks model.

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 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.

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.

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.

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 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.