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].

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 .

In an important (yet unpublished) research note, Ben Yaacov describes how to turn global Keisler measures into types over a monster model of the randomization.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 $\omega _{4}$.

This article is a contribution to the study of extensions of arbitrary models of $\mathsf {ZF}$ (Zermelo–Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. Our main results include the theorems below; in Theorems A and B, ${\mathcal {N}}$ is said to be a conservative elementary extension of $\mathcal {M}$ if $\mathcal { N}$ elementarily extends $\mathcal {M}$, and the intersection of every $ {\mathcal {N}}$-definable set with the universe of $\mathcal {M}$ is $\mathcal {M} $-definable (parameters allowed). In Theorem B, $\mathsf {ZFC}$ is the result of augmenting $\mathsf {ZF}$ with the axiom of choice.

Theorem A. Every model $\mathcal {M}$ of $\mathsf {ZF}+\exists p\left ( \mathrm {V}=\mathrm {HOD}(p)\right ) $ has a conservative elementary extension ${\mathcal {N}}$ that contains an ordinal above all of the ordinals of $\mathcal {M}$.

Theorem B. If ${\mathcal {N}}$ is a conservative elementary extension of a model $\mathcal {M}$ of $ \mathsf {ZFC}$, and ${\mathcal {N}}$ has the same natural numbers as $\mathcal {M}$, then $\mathcal {M}$ is cofinal in ${\mathcal {N}}$.

Theorem C. Every consistent extension of $ \mathrm {ZF}$ has a model $\mathcal {M}$ of power $\aleph _{1}$ such that $\mathcal {M}$ has no proper end extension to a model of $\mathsf {ZF}$.

It was proved by Maksimova in 1977 that exactly eight varieties of Heyting algebras have the amalgamation property, and hence exactly eight axiomatic extensions of intuitionistic propositional logic have the deductive interpolation property. The prevalence of these properties for substructural logics and varieties of pointed residuated lattices (their algebraic semantics) is far less well understood. Taking as our starting point a formulation of intuitionistic propositional logic as the full Lambek calculus with exchange, weakening, and contraction, we investigate the role of the exchange rule—algebraically, the commutativity law—in determining the scope of these properties. First, we show that there are continuum-many varieties of idempotent semilinear residuated lattices that have the amalgamation property and contain non-commutative members, and hence continuum-many axiomatic extensions of the corresponding logic that have the deductive interpolation property in which exchange is not derivable. We then show that, in contrast, exactly 60 varieties of commutative idempotent semilinear residuated lattices have the amalgamation property, and hence exactly 60 axiomatic extensions of the corresponding logic with exchange have the deductive interpolation property. From this latter result, it follows also that there are exactly 60 varieties of commutative idempotent semilinear residuated lattices whose first-order theories have a model completion.

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

We prove that every functor from the category of Hilbert spaces and linear isometric embeddings to the category of sets which preserves directed colimits must be essentially constant on all infinite-dimensional spaces. In other words, every finitary set-valued imaginary over the theory of Hilbert spaces, in a broad signature-independent sense, must be essentially trivial. This extends a result and answers a question by Lieberman–Rosický–Vasey, who showed that no such functor on the supercategory of Hilbert spaces and injective linear contractions can be faithful.

Gödel algebras are the Heyting algebras satisfying the axiom $(x \to y) \vee (y \to x)=1$. We utilize Priestley and Esakia dualities to dually describe free Gödel algebras and coproducts of Gödel algebras. In particular, we realize the Esakia space dual to a Gödel algebra free over a distributive lattice as the, suitably topologized and ordered, collection of all nonempty closed chains of the Priestley dual of the lattice. This provides a tangible dual description of free Gödel algebras without any restriction on the number of free generators, which generalizes known results for the finitely generated case. A similar approach allows us to characterize the Esakia spaces dual to coproducts of arbitrary families of Gödel algebras. We also establish analogous dual descriptions of free algebras and coproducts in every variety of Gödel algebras. As consequences of these results, we obtain a formula to compute the depth of coproducts of Gödel algebras and show that all free Gödel algebras are bi-Heyting algebras.

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 $2^\lambda>\lambda ^+$.

Nies and Scholz formalized the notion of an infinite qubitstring and referred to it as a ‘state’. They defined ‘quantum Martin-Löf randomness’ for states. We give a notion of measurement of a state in a computable basis and introduce ‘quantum measurement randomness’, a randomness notion for states. A state is quantum measurement random if measuring it in any computable basis yields a Martin-Löf random bitstring with probability one. Our main result is that quantum Martin-Löf randomness strictly implies quantum measurement randomness. This uses the construction of a quantum measurement random state which is not quantum Martin-Löf random. We prove two general results on which this construction relies: The first concerns Martin-Löf randomness relative to computable measures and extends a result of V. Vovk. The second is a combinatorial result about Kronecker products.

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 $A\in \mathcal {A}$.

A computable topological presentation of a space is given by an effective list of a countable basis of non-empty open sets so that the intersection of the basic sets is uniformly effectively enumerable. We show that every countably-based $T_0$-space has a computable topological presentation, and that, conversely, every (formal) computable topological presentation represents some Polish space. In the compact case, we give a computable uniform list of computable topological presentations such that every compact Polish space is represented by exactly one presentation from the list. Note that none of these results assume that the Polish (or $T_0$) spaces are effective. Quite surprisingly, the effectively compact topological presentations turn out to be rather well behaved. Not only do such presentations allow one to construct a $\Delta ^0_2$ (complete) metric compatible with the topology, but also, under a mild extra condition, they can be turned into a computably compact Polish presentation of the space.

Free Steiner triple systems (STSs) are infinite structures that are naturally characterised by a universal property. We consider the class of free STSs from a model-theoretic viewpoint. We show that free STSs on any number of generators are elementarily equivalent. We axiomatise their theory and show that it is stable.

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].

Many logical properties are known to be undecidable for normal modal logics, with few exceptions such as the consistency and the coincidence with $\mathsf {K}$. This article shows that the property of being a union-splitting in $\mathop {\mathsf {NExt}}{\mathsf {K}}$, the lattice of normal modal logics, is decidable, thus answering Problem 2 in [F. Wolter and M. Zakharyaschev, 2007]. This is done by providing a semantic characterization of union-splittings in terms of finite modal algebras. Moreover, by clarifying the connection to union-splittings, we show that in $\mathop {\mathsf {NExt}}{\mathsf {K}}$, having a decidable axiomatization problem and being a (un)decidable formula are also decidable. The latter answers Problem 17.3 in [A. Chagrov and M. Zakharyaschev, 1997] for $\mathop {\mathsf {NExt}}{\mathsf {K}}$.1

By a result of Scott [16], each countable structure for a countable language L is described up to isomorphism by an $L_{\omega _1\omega }$-sentence, called a Scott sentence. We consider structures that are countably infinite. By a result of A. Miller [11], no such structure has a $\Sigma _2$ Scott sentence, so having a $\Pi _2$ Scott sentence is as simple as possible. A result of Montalbán [12] yields a nice characterization of the structures that (for a fixed countable language) have a $\Pi _2$ Scott sentence. Computable infinitary formulas involve c.e. disjunctions and conjunctions, so they are in a sense comprehensible. Therefore, we set out to characterize the structures that (for a fixed computable language) have a computable $\Pi _2$ Scott sentence. We found some examples and some partial results. However, it turns out that (for most languages) there is no nice characterization of the class. The index set is $\Pi ^1_1$-complete.

We construct a model in which all C-sequences are trivial, yet there exists a $\kappa $-Souslin tree all of whose limit levels are vanishing levels. This provides an optimal combination of compactness and incompactness. It is obtained by incorporating a so-called mutually exclusive ascent path to Kunen’s 1978 forcing construction, and by analyzing a gallery of $\kappa $-cc forcing extensions of the outcome model.