We study saturation properties of $\sigma $-complete measures on $P_\kappa (\lambda )$, where $\lambda $ can be either regular or singular. In particular, we prove that in contrast to Galvin’s theorem, the Galvin property from [6] fails for normal, fine ultrafilters on $P_\kappa (\lambda )$, answering a question of the first author and Goldberg from [10]. We then provide several applications of our results: to ultrafilters on successors under $UA$, we generalize a result of Gitik regarding density of ground model sets in supercompact Prikry extensions, and to generating sets of $P_\kappa (\lambda )$ measures. In the second part of the article, we study variations of the Galvin property suitable for ultrafilters over $P_\kappa (\lambda )$, and generalize a result of Foreman–Magidor–Zeman [17, Theorem 1.2] on determinacy of filter games to the two-cardinal setting, answering a question of the first author and Gitman from [9].

Using creature technology, we construct families of Suslin ccc non-sweet forcing notions $\mathbb Q$ such that $ZFC$ is equiconsistent with $ZF+$ “Every set of reals equals a Borel set modulo the $(\leq \aleph _1)$-closure of the null ideal associated with $\mathbb Q$” $+$ “There is an $\omega _1$-sequence of distinct reals.” This answers a question of the second author and Kellner. As an application of independent interest, we also show how our forcing adds a new $\Pi ^1_2$ singleton over L without relying on L-combinatorics.

In classical set theory, the ordinals form a linear chain that we often think of as a very thin portion of the set-theoretic universe. In intuitionistic set theory, however, this is not the case and there can be incomparable ordinals. In this article, we shall show that starting from two incomparable ordinals, one can construct canonical bijections from any arbitrary set to an antichain of ordinals, and consequently any subset of the given set can be defined using ordinals as parameters. This implies the surprising result that in the theory “$\mathrm {IKP} + {}$there exist two incomparable ordinals,” the statements $\mathrm {Ord} \subseteq L$ and $V = L$ are equivalent.

It is widely claimed that the natural axiom systems—including the large cardinal axioms—form a well-ordered hierarchy. Yet, as is well-known, it is possible to exhibit non-linearity and ill-foundedness by means of ad hoc constructions. In this article we formulate notions of proof-theoretic strength based on set-theoretic reflection principles. We prove that they coincide with orderings on theories given by the generalized ordinal analysis of Pohlers. Accordingly, these notions of proof-theoretic strength engender genuinely well-ordered hierarchies. The reflection principles considered in this article are formulated relative to Gödel’s constructible universe; we conclude with generalizations to other inner models.

Entangled linear orders were introduced by Abraham and Shelah in [2]. Todorčević [29] showed that these linear orders exist under $\mathsf {CH}$. We prove the following results:

1. (1) If $\mathsf {CH}$ holds, then, for every $n> 1$, there is an n-entangled linear order which is not $(n+1)$-entangled.

2. (2) If $\mathsf {CH}$ holds, then there are two homeomorphic sets of reals $A,B \subseteq \mathbb {R}$ such that A is entangled but B is not $2$-entangled.

3. (3) If $\mathbb {R} \subseteq \mathsf {L}$, then there is an entangled $\Pi _1^1$ set of reals.

4. (4) If $\diamondsuit $ holds, then there is a $2$-entangled non-separable linear order.

One measure of the complexity of a first-order theory, and similarly a type, is the complexity of the formulas required to axiomatize it. We say that a theory is bounded if there is an axiomatization involving only $\forall _n$-formulas for some finite n, and unbounded otherwise. One might expect bounded theories to have only bounded types. In fact, an analogue holds in infinitary logic, where the complexity of a Scott sentence roughly agrees with the complexity of the most complicated automorphism orbit. Our main result, however, shows that this is not the case in the first-order setting: Namely, there can be a bounded theory, in fact $\forall _1$-axiomatizable, which has unbounded types.

We characterize sums of normal ultrafilters after the Magidor iteration of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower Axiom. We also construct a nonrigid ultrapower and two uniform ultrafilters on different cardinals that have the same ultrapower.

We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is obtained by replacing the classical ‘least’ function associated with ideals by a two-dimensional ‘last’ function associated with walks on ordinals. Consequently, we obtain conditions under which a measure admits not just infinite pairwise disjoint families of positive sets, but in fact families of maximum possible size. As an application we solve a problem left open in Shelah’s Cardinal Arithmetic book, proving that for every weakly inaccessible cardinal $\kappa $, if there exists a stationary subset of $\kappa $ that does not reflect at regulars, then the strong Ramsey relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ holds.

In this article, we deal with the classification complexity of continuous (Devaney) chaotic systems in dimensions $0,1,$ and $\infty $ using the framework of invariant descriptive set theory. We identify the complexity in dimensions $0$ and $\infty $, while in dimension $1$ we get some partial results.

More precisely, we prove the topological conjugacy relation of invertible chaotic systems on the Hilbert cube (resp. on all compact metric spaces) has the same complexity as (i.e., is Borel bireducible with) the universal orbit relation induced by a Polish group. As a consequence, this answers a recent question asked by L. Ding. We also prove that the topological conjugacy relation of invertible chaotic systems on the Cantor space has the same complexity as the universal relation induced by the group $S_\infty $. This answers a recent question by M. Foreman. Some non-trivial bounds on the classification complexity of chaotic systems on the interval and on the circle are also obtained. Namely, the lower bound is the Vitali equivalence relation, and the upper bound is the equality of countable sets of reals. This especially implies that the relation is Borel. However, the exact complexity remains unknown.

We show how to extend Selivanov’s fine hierarchy using descriptions of Borel Wadge classes. We give a game characterisation of containment between classes. We show that every class in the extended fine hierarchy has an admissible description, and use this to calculate heights in the hierarchy.

We show that a non-universal Polish group can induce a complete orbit equivalence relation, which answers a question of Sabok from [1].

Using a variation of Woodin’s $\mathbb {P}_{\mathrm {max}}$ forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length $\omega _{2}$ consisting of functions from $\omega $ to $\omega $. We also show that there can be no such sequence of length $\omega _{4}$.

We prove the consistency of the inequality $\mathfrak {r}_{\mathsf {nwd}}<\mathfrak {irr}$, which in turn implies the consistency of $\mathfrak {r}_{\mathsf {nwd}}<\mathfrak {i}$ and $\mathfrak {r}_{\mathsf {scatt}}<\mathfrak {irr}$. This answers one question from [1] and one question from [2]. We also prove the consistency of the inequality $\mathfrak {r}_{\mathbb {Q}}<\mathfrak {u}_{\mathbb {Q}}$.

The article is devoted to comparison of two generalizations of the bounding number $\mathfrak {b}$.

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\boldsymbol {\Sigma }^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs.

For $\lambda $ inaccessible, we show that “there is a nowhere trivial automorphism of $\mathcal P(\lambda )/[\lambda ]^{<\lambda }$” follows (in ZFC) from $2^\lambda =\lambda ^+$, and is consistent (via forcing) with $2^\lambda>\lambda ^+$.

For every natural number n we introduce a new weak choice principle $\mathrm {nRC_{fin}}$:

Given any infinite set x, there is an infinite subset $y\subseteq x$ and a selection function f that chooses an n-element subset from every finite $z\subseteq y$ containing at least n elements.

By constructing new permutation models built on a set of atoms obtained as Fraïssé limits, we will study the relation of $\mathrm {nRC_{fin}}$ to the weak choice principles $\mathrm {RC_m}$ (that has already been studied in [3] and [6]):

Given any infinite set x, there is an infinite subset $y\subseteq x$ with a choice function f on the family of all m-element subsets of y.

Moreover, we prove a stronger analogue of the results in [6] when we study the relation between $\mathrm {nRC_{fin}}$ and $\mathrm {kC_{fin}^-}$ which is defined by

Given any infinite family $\mathcal {F}$ of finite sets of cardinality greater than k, there is an infinite subfamily $\mathcal {A}\subseteq \mathcal {F}$ with a selection function f that chooses a k-element subset from each $A\in \mathcal {A}$.

We prove that the spectrum of Van Douwen families is closed under singular limits. For any maximal eventually different family Raghavan defined in [10] an associated ideal which measures how far the family is from being Van Douwen. Under CH we prove that every ideal containing $\mathrm {Fin}$ is realized as the associated ideal of some maximal eventually different family. Finally, we construct maximal eventually different families with Sacks-indestructible associated ideals to prove that in the iterated Sacks-model every $\aleph _1$-generated ideal containing $\text {Fin}$ is realized.

Let M be a tame mouse modelling $\mathrm {ZFC}$. We show that M satisfies “$V=\mathrm {HOD}_x$ for some real x”, and that the restriction $\mathbb {E}^M\!\upharpoonright \![\omega _1^M,\mathrm {OR}^M)$ of the extender sequence $\mathbb {E}^M$ of M to indices above $\omega _1^M$ is definable without parameters over the universe of M. We show that M has universe $\mathrm {HOD}^M[X]$, where $X=M|\omega _1^M$ is the initial segment of M of height $\omega _1^M$ (including $\mathbb {E}^M\!\upharpoonright \!\omega _1^M$), and that $\mathrm {HOD}^M$ is the universe of a premouse over some $t\subseteq \omega _2^M$. We also show that M has no proper grounds via strategically $\sigma $-closed forcings. We then extend some of these results partially to non-tame mice, including a proof that many natural $\varphi $-minimal mice model “$V=\mathrm {HOD}$”, assuming a certain fine structural hypothesis whose proof was almost established in Closson [1], and has since been completed in the preprint [14].

I provide several natural properties of group actions which translate into fragments of axiom of choice in the associated permutation models of choiceless set theory.