In this article, we introduce novel variations on several well-known model-theoretic tree properties (TPs), and prove several equivalences to known properties. Motivated by the study of generalized indiscernibles, we introduce the notion of the $\mathcal {I}$-tree property ($ \mathcal {I}$-TP), for an arbitrary Ramsey index structure $\mathcal {I}$. We focus attention on the colored linear order index structure c, showing that c-TP is equivalent to instability. After introducing c-$\text {TP}_1$ and c-$ \text {TP}_2$, we prove that c-$ \text {TP}_1$ is equivalent to $\text {TP}_1$, and that c-$ \text {TP}_2$ is equivalent to IP. We see that these three TPs give a dichotomy theorem, just as with TP, $\text {TP}_1$, and $\text {TP}_2$. Along the way, we observe that appropriately generalized tree index structures $\mathcal {I}^{<\omega }$ are Ramsey, allowing for the use of generalized tree indiscernibles.

In this article, we prove that the precipitousness of $\mathbb {Q}_\gamma $, the countable stationary tower of height $\gamma $, where $\gamma $ is a limit ordinal $>\omega $, is a consequence of a certain stationary reflection principle for $\mathcal {P}_{\omega _1} H_{\left \vert {V_\gamma } \right \vert {}^+}$. Using this, we prove the consistency of the precipitousness of $\mathbb {Q}_\gamma $ for some singular $\gamma $ assuming a sufficiently strong large cardinal axiom. Furthermore, assuming the existence of proper class many supercompact cardinals, we build a model where $\mathbb {Q}_\gamma $ is precipitous for every limit ordinal $>\omega $.

We show that for any class of Boolean algebras with an associative operator, if it contains the complex algebra of $(\mathcal {P}(\mathbb {N}), \cup )$, its equational theory is undecidable. Equivalently, any associative normal modal logic valid over the frame $(\mathcal {P}(\mathbb {N}), \cup )$ is undecidable. This settles a long-open question on the decidability of hyperboolean modal logic [16], and addresses several related problems.

We introduce the notion of virtual ultracategory. From a topological point of view, this notion can be seen as a categorification of relational $\beta $-modules. From a categorical point of view, virtual ultracategories generalize ultracategories in the same way that multicategories generalize monoidal categories. From a logical point of view, whereas the points of a coherent topos form an ultracategory, the points of an arbitrary topos form a virtual ultracategory. We then extend Makkai–Lurie’s conceptual completeness: a topos with enough points can be reconstructed from its virtual ultracategory of points.

We prove that it is consistent with ZFC that for every non-decreasing function $f:[0,1]\rightarrow [0,1]$, each subset of $[0,1]$ of cardinality $\mathfrak c$ contains a set of cardinality $\mathfrak c$ on which f is uniformly continuous. We show that this statement follows from the assumptions that $\mathfrak d^* <\mathfrak c$ and $\mathfrak c$ is regular, where $\mathfrak d^*\leq \mathfrak d$ is the smallest cardinality $\kappa $ such that any two disjoint countable dense sets in $2^{\mathbb {N}}$ can be separated by sets each of which is an intersection of at most $\kappa $-many open sets in $2^{\mathbb {N}}$. We establish also that $\mathfrak d^*=\min \{\mathfrak u, \mathfrak d\}=\min \{\mathfrak r, \mathfrak d\}$, thus giving an alternative proof of the latter equality established by J. Aubrey in 2004.

We show that for a Suslin ccc forcing notion $\mathbb Q$ adding a Hechler real, “$\mathrm {ZF}+\mathrm {DC}_{\omega _1}+$all sets of reals are $I_{\mathbb Q,\aleph _0}$-measurable” implies the existence of an inner model with a measurable cardinal. We also introduce a wide class of Suslin ccc forcing notions which add a Hechler real, so that the above result applies to them.

We describe the fibrational structure of sets within the predicative variant $\textbf {pEff}$ of Hyland’s Effective Topos ${\mathbf {Eff} }$, previously introduced in Feferman’s classical predicative theory of non-iterative fixpoints ${\widehat {ID_1}}$. Our structural analysis can be carried out also in constructive and predicative variants of ${\mathbf {Eff} }$ developed within extensions of Aczel’s constructive Zermelo–Fraenkel set theory. This shows that the full subcategory of discrete objects of Hyland’s Effective Topos ${\mathbf {Eff} }$ already contains a fibred predicative topos validating the Formal Church Thesis, even when both are formalized in a constructive and predicative foundational theory.

It is a long-standing open problem whether modal logics of the form $\mathbf {K}\oplus \Box ^np\to \Box ^mp$ for $n>m>1$ have the finite model property (FMP). We solve this by showing that any modal logic axiomatized by formulas of the form $\Box ^np\to \Box ^mp$ where $n>m>1$ has the FMP.

Friedman and Stanley [9] developed the notion of Borel reducibility and illustrated its use in comparing classification problems for some familiar classes of countable structures. For many embeddings, the fact that the embedding is 1–1 on isomorphism types is explained by the existence of simple formulas that, uniformly, interpret the input structure in the output structure. For the embeddings of graphs in trees, and in linear orderings, there is no uniform interpretation [16, 20]. We focus on a version of the Friedman–Stanley embedding from [16] that takes each structure $\mathcal {A}$ for the language of graphs to a labeled tree $T_{\mathcal {A}}$. Gonzalez and Rossegger [13] showed that this embedding preserves Scott complexity. We refine this result, showing that for an X-computable ordinal, if one of $\mathcal {A}$, $T_{\mathcal {A}}$ has a computable infinitary Scott sentence, then so does the other, and the complexities match. Let $\mathbb {T}$ be the class of labeled trees isomorphic to those in the range of the embedding, and let $\mathbb {T}^\alpha $ be the subclass consisting of structures of Scott rank at most $\alpha $. It follows from results of Gao [10] that $\mathbb {T}$ is not Borel. We show that for each $\alpha $, $\mathbb {T}^\alpha $ is Borel. In fact, if $\alpha $ is an X-computable ordinal, then $\mathbb {T}^\alpha $ is complete X-effective $\Pi _{2\alpha +2}$.

We continue the development of the theory of capturing schemes over $\omega _1$ by analyzing the relation between the capturing construction schemes (whose existence is implied by Jensen’s $\Diamond $-principle) and both the Continuum Hypothesis and Ostaszewski’s -principle. Formally, we show that the property of being capturing can be viewed as the conjunction of two properties, one of which is implied by and the other one by CH. We apply these principles to construct multiple gaps, entangled sets, and metric spaces without uncountable monotone subspaces.

The Robinson Splitting Theorem states that a computably enumerable (c.e.) degree $\boldsymbol {b}$ splits over any low c.e. degree $\boldsymbol {c}<\boldsymbol {b}$. We prove that a weaker version of this theorem holds in models of ${\mathsf {P}}^{-}+\mathsf {I}{\Sigma }_1$, with lowness replaced by super-lowness.

Kuroda’s translation embeds first-order classical logic into first-order intuitionistic logic. It satisfies the soundness property—if a judgment is provable in classical logic, then its translation is provable in intuitionistic logic—as well as the characterization property—a formula and its translation are equivalent in classical logic—and the reverse translation property—if the translation of a judgment is provable in intuitionistic logic then the original judgment is provable in classical logic. Brown and Rizkallah extended Kuroda’s translation to higher-order logic. However, they showed that the soundness property fails in the presence of functional extensionality, and they neither proved the characterization property nor the reverse translation property.

In this article, we emphasize different conditions under which Kuroda’s translation works in the presence of functional extensionality, including the double-negation shift. We show that the characterization property and the reverse translation property do not necessarily hold in higher-order logic. Nevertheless, we prove that it is sufficient to assume both functional extensionality and propositional extensionality.

We initiate the study of general metric lattices in the context of the model theory of metric structures. As an application we develop a theory of pseudofinite limits of partition lattices and connect this theory with the theory of continuous limits of partition lattices due to Björner and Lovász.

We obtain new characterizations of the Hobby–McKenzie varieties via compatible reflexive ternary structures. Based on our findings, we prove that in the lattice of interpretability types of varieties, the filter of the interpretability types of Hobby–McKenzie varieties is prime.

The current work includes a result announced in the year 2012 which was unproven for 10 years. The result shows that there is a co-r.e. tree with uncountably many infinite branches such that the nonisolated infinite branches of the constructed tree are all nonrecursive, generalised low, and hyperimmune-free and form a perfect tree. This article is the journal version of two conference articles from 2012 and 2022; the article contains the main result proved in 2022 and also the other major results from 2012.

In this article we answer Question 3.5 of [6] in the positive by showing that for every $2\leqslant n\in \mathbb {N}$, the statement “there is an n-entangled set, but there are no $n+1$-entangled sets” is consistent. We also prove some theorems which improve our understanding of entangled sets in relation to construction schemes: (1) The axiom FCA$^\Delta $ introduced in [8] implies the existence of n-entangled sets which are not $n+1$-entangled. (2) $\mathfrak {m}_{\mathcal {F}}>{\omega }_1$ implies the non-existence of entangled sets. Thus, $2$-capturing schemes alone are not sufficient to build these kinds of linear orders. (3) The existence of a 2-$\Delta $-capturing scheme is consistent with MA.

In this article, we introduce a translation that combines the j-translation with Kripke forcing in the internal logic of an elementary topos. First, we show that our translation is sound for intuitionistic first-order logic and Heyting arithmetic. Furthermore, its interpretation in the effective topos provides an extension of the sheaf model of realizability introduced by de Jongh and Goodman. As an application, we systematically investigate translations for semi-classical axioms. Based on this investigation, we establish a separation result on semi-classical arithmetic, which cannot be obtained using the usual j-realizability.

We examine the analyticity of the class of separable Banach spaces possessing the $\pi $-property, defined in terms of convergence along a filter. Our results establish that this class is $\Sigma ^1_3$ whenever the underlying filter is analytic (as a subset of the Cantor set $\Delta $). Furthermore, we demonstrate that if the filter is countably generated, the class of such spaces is $\Sigma ^1_2$ with respect to any admissible Polish topology on the family of closed subspaces of $C(\Delta )$.

We initiate the effective metric structure theory of Keisler randomizations. We show that a classical countable structure $\mathcal M$ has a decidable presentation if and only if its Borel randomization $\mathcal {M}^{[0,1)}$ has a computable presentation for which the constant functions are uniformly computable points. We determine a sufficient condition for which the uniform computability of the constant functions can be dropped. We show that when $\mathcal M$ is effectively $\omega $-categorical, then $\mathcal {M}^{[0,1)}$ is computably categorical, that is, has a unique computable presentation up to computable isomorphism. A special case of this result is that the unique separable atomless probability algebra is computably categorical. Finally, we show that all randomizations admit effective quantifier elimination.

Recall that an infinite subset of $\mathbb {N}$ is cohesive if it cannot be split into two infinite pieces by a c.e. set. Likewise, a set is r-cohesive if it cannot be split by a computable set, and it is p-cohesive if it cannot be split by a primitive recursive set. A cohesive power of a computable structure is an effective ultrapower of the structure, where a cohesive set plays the role of an ultrafilter. Let $\omega $, $\zeta $, and $\eta $ denote the respective order-types of the natural numbers, the integers, and the rationals. We study cohesive powers of computable copies of $\omega $ over $\Delta _2$ cohesive sets. We show that there is a computable copy $\mathcal {L}$ of $\omega $ such that, for every $\Delta _2$ cohesive set C, the cohesive power of $\mathcal {L}$ over C has order-type $\omega + \eta $. This improves an earlier result of Dimitrov, Harizanov, Morozov, Shafer, A. Soskova, and Vatev by generalizing from $\Pi _1$ cohesive sets to $\Delta _2$ cohesive sets and by computing a single copy of $\omega $ that has the desired cohesive power over all $\Delta _2$ cohesive sets. Furthermore, our result is optimal in the sense that $\Delta _2$ cannot be replaced by $\Pi _2$. More generally, we show that if $X \subseteq \mathbb {N} \setminus \{0\}$ is a Boolean combination of $\Sigma _2$ sets, thought of as a set of finite order-types, then there is a computable copy $\mathcal {L}$ of $\omega $ where the cohesive power of $\mathcal {L}$ over any $\Delta _2$ cohesive set has order-type $\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$. If X is finite and non-empty, then there is also a computable copy $\mathcal {L}$ of $\omega $ where the cohesive power of $\mathcal {L}$ over any $\Delta _2$ cohesive set has order-type $\omega + \boldsymbol {\sigma }(X)$. Here $\boldsymbol {\sigma }$ denotes the shuffle operation. An unexpected byproduct of our work is a new method for constructing infinite $\Pi _1$ sets that do not have $\Delta _2$ cohesive subsets. In fact, we construct an infinite $\Pi _1$ set that does not have a $\Delta _2$ p-cohesive subset. Infinite $\Pi _1$ sets without $\Delta _2$ r-cohesive subsets generalize D. Martin’s classic co-infinite c.e. set with no maximal superset and have appeared in the work of Lerman, Shore, and Soare.