In this article, we study a degenerate version of Ramsey’s theorem for pairs and two colors (${\mathsf {RT}}^2_2$), in which the homogeneous sets for color 1 are of bounded size. By ${\mathsf {RT}}^2_2$, it follows that every such coloring admits an infinite homogeneous set for color 0. This statement, called ${\mathsf {BRT}}^2_2$, is known to be computably true, that is, every computable instance admits a computable solution, but the known proofs use $\Sigma ^0_2$-induction ($\mathsf {I}\Sigma ^0_2$). We prove that ${\mathsf {BRT}}^2_2$ follows from the Erdős–Moser theorem but not from the ascending descending sequence principle, and that its computably true version is equivalent to $\mathsf {I}\Sigma ^0_2$ over ${\mathsf {RCA}}_0$.

We study the uniform computational content of Ramsey’s theorem using both Weihrauch reducibility and a new variant of Weihrauch reducibility, where the functions in the reduction are required to be total. In the latter setting, we show that the strength of Ramsey’s theorem varies significantly depending on how one represents its solutions, for example, using characteristic functions or using enumerations. Some of our results extend beyond variants of Ramsey’s theorem. In particular, we show that $\mathsf {RT}^{2}_{2}$ where solutions are represented using characteristic functions is not totally Weihrauch reducible to any computational problem whose solutions are represented using enumerations. Next, we study the computational problems $\mathsf {RT}^{n}_{\infty }$ which take as input a colouring of n-tuples which has some infinite homogeneous set and asks for any such set. The problem $\mathsf {RT}^{1}_{\infty }$ is fairly well-studied, as it is Weihrauch equivalent to the cluster point problem on $\mathbb {N}$. Our main result shows that $\mathsf {RT}^{1}_{\infty }$ is not Weihrauch reducible to $\mathsf {RT}^{2}_{\mathbb {N}}$, strengthening a result of Soldà and Valenti. We also show that the jump of $\mathsf {RT}^{1}_{\infty }$ is Weihrauch equivalent to $\mathsf {SRT}^{2}_{\infty }$.

Tennenbaum’s theorem states that PA does not admit any nonstandard computable model. In 2022, Pakhomov proved that this theorem is fragile in regards to how PA is expressed, by constructing a theory that is definitionally equivalent to PA (roughly: “it’s PA but with a different choice of signature”) for which there is a computable nonstandard model. He showed that this fragility does not extend to true arithmetic (any nonstandard model of a theory definitionally equivalent to Th(N) is not computable), but the question of whether this fragility extends to fragments of PA of intermediate strength was left open. We show that it does, by constructing a sequence of theories $T^n$ which are definitionally equivalent to: “PA plus all $\Pi _n$ truths,” all of which admit computable nonstandard models. In the process, we produce a general-purpose theorem for strong jump inversion. Besides applying this theorem to obtain our novel result, we show that several known results from the literature can be seen as direct applications of our theorem.

We study the reverse mathematics of characterization theorems of regular countable second countable (CSC) spaces. We prove that arithmetic comprehension is equivalent over $\mathbf {RCA}_0$ to every $T_3$ CSC space being metrizable, and we characterize the $T_3$ spaces which are metrizable over $\mathbf {RCA}_0$. We show that Lynn’s theorem for CSC spaces can be carried out in $\mathbf {ACA}_0$, namely that every zero-dimensional separable space is homeomorphic to the order topology of a linear order. We also show that arithmetic comprehension is equivalent to every $T_2$ compact CSC space being well-orderable. From general topology, we know that the locally compact $T_2$ CSC spaces are the well-orderable CSC spaces, and that the $T_3$ scattered CSC spaces are the completely metrizable CSC spaces. We show that these characterizations and a few others are equivalent to arithmetic transfinite recursion over $\mathbf {RCA}_0$. We also find a few statements that are equivalent to $\Pi ^1_1$ comprehension. In particular we show that every $T_3$ CSC space has a Cantor–Bendixson rank and that every $T_3$ CSC space is the disjoint union of a scattered space and dense in itself space are equivalent to $\Pi ^1_1$ comprehension over $\mathbf {RCA}_0$.

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.

We study the existence and the distribution of “long” chains in the Weihrauch degrees, mostly focusing on chains with uncountable cofinality. We characterize when such chains have an upper bound and prove that there are no cofinal chains (of any order type) in the Weihrauch degrees. Furthermore, we show that the existence of coinitial sequences of non-zero degrees is equivalent to $\mathrm {CH}$. Finally, we explore the extendibility of antichains, providing some necessary conditions for maximality.

$\mu $-Abstract Elementary Classes are a model theoretic framework introduced in [4] to encompass classes axiomatized by $\mathbb {L}_{\infty , \infty }$. We show that the framework extends beyond these logics by showing classes axiomatized in $\mathbb {L}(aa)$ with just the $aa$ quantifier are an $\aleph _1$-Abstract Elementary Class.

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.

We study metric valued fields in continuous logic, following Ben Yaacov’s approach, thus working in the metric space given by the projective line. As our main result, we obtain an approximate Ax–Kochen–Ershov principle in this framework, completely describing elementary equivalence in equicharacteristic 0 in terms of the residue field and value group. Moreover, we show that, in any characteristic, the theory of metric valued difference fields does not admit a model-companion. This answers a question of Ben Yaacov.

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 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].

Hyperenumeration reducibility was first introduced by Sanchis [11]. The relationship between hyperenumeration and hyperarithmetic reducibility shares many parallels with the relationship between enumeration and Turing reducibility. We ask if this relationship can be pushed to prove and analog of Selman’s Theorem for hyperenumeration reducibility. By studying e-pointed trees in Baire space we are able to get a counter example. An e-pointed tree T is a tree with no dead ends and the property that every path in T enumerates T. We prove that if T is an e-pointed tree then for all X if T is $\Pi ^1_1$ in X then $\overline {T}$ is $\Pi ^1_1$ in X. We build an e-pointed tree T such that $\overline {T}$ is not hyperenumeration reducible to T.

In the product $L_1\times L_2$ of two Kripke complete consistent logics, local tabularity of $L_1$ and $L_2$ is necessary for local tabularity of $L_1\times L_2$. However, it is not sufficient: the product of two locally tabular logics may not be locally tabular. We provide extra semantic and axiomatic conditions that give criteria of local tabularity of the product of two locally tabular logics, and apply them to identify new families of locally tabular products. We show that the product of two locally tabular logics may lack the product finite model property. We give an axiomatic criterion of local tabularity for all extensions of . Finally, we describe a new prelocally tabular extension of .

We prove a Thomason-style duality for the category of instantial neighbourhood frames and instantial neighbourhood morphisms, use it to define and investigate ultrafilter extensions, and show that it restricts to Thomason duality for Kripke frames.

In an important (yet unpublished) research note, Ben Yaacov describes how to turn global Keisler measures into types over a monster model of the randomization. These transfer methods allow one to turn questions involving measures into those involving types (in continuous logic). Assuming that T is NIP, we show that the Morley product commutes with the transfer map for finitely satisfiable measures. We characterize when the Morley product commutes with the restriction map for pairs of global finitely satisfiable types in the randomization. We end by making some brief observations about the Ellis semigroup in this context.

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 $\omega _{4}$.