Schmidt’s game and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, \mathsf {AD}_{\mathbb R}$\mathsf {AD}_{\mathbb R}$ , which is a much stronger axiom than that asserting all integer games are determined, \mathsf {AD}$\mathsf {AD}$ . One of our main results is a general theorem which under the hypothesis \mathsf {AD}$\mathsf {AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt’s (\alpha ,\beta ,\rho )$(\alpha ,\beta ,\rho )$ game on \mathbb R$\mathbb R$ is determined from \mathsf {AD}$\mathsf {AD}$ alone, but on \mathbb R^n$\mathbb R^n$ for n \geq 3$n \geq 3$ we show that \mathsf {AD}$\mathsf {AD}$ does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt’s (\alpha , \beta , \rho )$(\alpha , \beta , \rho )$ game on \mathbb {R}$\mathbb {R}$ has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt’s game. These results highlight the obstacles in obtaining the determinacy of Schmidt’s game from \mathsf {AD}$\mathsf {AD}$ .

Following the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function \#$\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank \omega $\omega$, of Kripke–Feferman truth theory and of Friedman–Sheard truth theory, and show that all of these have the same strength as the finitist arithmetic of one higher level along Grzegorczyk hierarchy. On the other hand, we also show that adding Burgess-style groundedness schema, adjusted to the finitist setting, makes Kripke–Feferman truth theory as strong as primitive recursive arithmetic. Meanwhile, we obtain some basic results on finitist theories of (full and hat) inductive definitions and on the second order axiom of hat inductive definitions for positive operators.

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form \forall \exists ! \mathop{\boldsymbol {\bigwedge }}\limits p = q$\forall \exists ! \mathop{\boldsymbol {\bigwedge }}\limits p = q$ . For a logic L algebraized by a quasivariety \mathcal {Q}$\mathcal {Q}$ we show that the AE-subclasses of \mathcal {Q}$\mathcal {Q}$ correspond to certain natural expansions of L, which we call algebraic expansions. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of abelian \ell $\ell$ -groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.

Fisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [25, 2.12], it is asked if the same is true for any abstract elementary class (K, \leq _p)$(K, \leq _p)$ such that K is a class of modules and \leq _p$\leq _p$ is the pure submodule relation. In this paper we give some instances where this is true:Theorem.

Assume R is an associative ring with unity. Let (K, \leq _p)$(K, \leq _p)$ be an AEC such that K \subseteq R\text {-Mod}$K \subseteq R\text {-Mod}$

and K is closed under finite direct sums, then:

• • If K is closed under pure-injective envelopes, then \mathbf {K}$\mathbf {K}$ is \lambda $\lambda$ -stable for every \lambda \geq \operatorname {LS}(\mathbf {K})$\lambda \geq \operatorname {LS}(\mathbf {K})$ such that \lambda ^{|R| + \aleph _0}= \lambda $\lambda ^{|R| + \aleph _0}= \lambda$ .

• • If K is closed under pure submodules and pure epimorphic images, then \mathbf {K}$\mathbf {K}$ is \lambda $\lambda$ -stable for every \lambda $\lambda$ such that \lambda ^{|R| + \aleph _0}= \lambda $\lambda ^{|R| + \aleph _0}= \lambda$ .

• • Assume R is Von Neumann regular. If \mathbf {K}$\mathbf {K}$ is closed under submodules and has arbitrarily large models, then \mathbf {K}$\mathbf {K}$ is \lambda $\lambda$ -stable for every \lambda $\lambda$ such that \lambda ^{|R| + \aleph _0}= \lambda $\lambda ^{|R| + \aleph _0}= \lambda$ .

As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, Dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy.

Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of \mathfrak {s}$\mathfrak {s}$ -torsion modules.

We give a new approach to the failure of the Canonical Base Property (CBP) in the so far only known counterexample, produced by Hrushovski, Palacín and Pillay. For this purpose, we will give an alternative presentation of the counterexample as an additive cover of an algebraically closed field. We isolate two fundamental weakenings of the CBP, which already appeared in work of Chatzidakis and Moosa-Pillay and show that they do not hold in the counterexample. In order to do so, a study of imaginaries in additive covers is developed. As a by-product of the presentation, we observe that a pure binding-group-theoretic account of the CBP is unlikely.

We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( \mathrm {WVP}$\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular, we show that \mathrm {WVP}$\mathrm {WVP}$ for \Sigma _2$\Sigma _2$ -definable classes is equivalent to the existence of a strong cardinal. The main theorem (Theorem 5.11) shows, more generally, that \mathrm {WVP}$\mathrm {WVP}$ for \Sigma _n$\Sigma _n$ -definable classes is equivalent to the existence of a \Sigma _n$\Sigma _n$ -strong cardinal (Definition 5.1). Hence, \mathrm {WVP}$\mathrm {WVP}$ is equivalent to the existence of a \Sigma _n$\Sigma _n$ -strong cardinal for all n<\omega $n<\omega$ .

Let P be a forcing notion and G\subseteq P$G\subseteq P$ its generic subset. Suppose that we have in V[G]$V[G]$ a \kappa{-}$\kappa{-}$ complete ultrafilter1 , 2 W over \kappa $\kappa$ . Set U=W\cap V$U=W\cap V$ .

For given Boolean algebras \mathbb {A}$\mathbb {A}$ and \mathbb {B}$\mathbb {B}$ we endow the space \mathcal {H}(\mathbb {A},\mathbb {B})$\mathcal {H}(\mathbb {A},\mathbb {B})$ of all Boolean homomorphisms from \mathbb {A}$\mathbb {A}$ to \mathbb {B}$\mathbb {B}$ with various topologies and study convergence properties of sequences in \mathcal {H}(\mathbb {A},\mathbb {B})$\mathcal {H}(\mathbb {A},\mathbb {B})$ . We are in particular interested in the situation when \mathbb {B}$\mathbb {B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on \mathbb {A}$\mathbb {A}$ in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin’s result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on \mathcal {H}(\mathbb {A},\mathbb {B})$\mathcal {H}(\mathbb {A},\mathbb {B})$ for a Boolean algebra \mathbb {B}$\mathbb {B}$ carrying a strictly positive measure and convergence properties of sequences of measures on \mathbb {A}$\mathbb {A}$ .

We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. A decomposition theorem into quasi-special submanifolds is also demonstrated.

We introduce type-theoretic algebraic weak factorisation systems and show how they give rise to homotopy-theoretic models of Martin-Löf type theory. This is done by showing that the comprehension category associated with a type-theoretic algebraic weak factorisation system satisfies the assumptions necessary to apply a right adjoint method for splitting comprehension categories. We then provide methods for constructing several examples of type-theoretic algebraic weak factorisation systems, encompassing the existing groupoid and cubical sets models, as well as new models based on normal fibrations.

We prove two general results about the preservation of extendible and C^{(n)}$C^{(n)}$ -extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka’s Principle and C^{(n)}$C^{(n)}$ -extendible cardinals under Jensen’s iteration for forcing the GCH [17], previously obtained in [8, 27], respectively. We prove that C^{(n)}$C^{(n)}$ -extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible \Delta _2$\Delta _2$ -definable behaviour of the power-set function on regular cardinals. We show that one can force proper class-many disagreements between the universe and HOD with respect to the calculation of successors of regular cardinals, while preserving C^{(n)}$C^{(n)}$ -extendible cardinals. We also show, assuming the GCH, that the class forcing iteration of Cummings–Foreman–Magidor for forcing \diamondsuit _{\kappa ^+}^+$\diamondsuit _{\kappa ^+}^+$ at every \kappa $\kappa$ [10] preserves C^{(n)}$C^{(n)}$ -extendible cardinals. We give an optimal result on the consistency of weak square principles and C^{(n)}$C^{(n)}$ -extendible cardinals. In the last section prove another preservation result for C^{(n)}$C^{(n)}$ -extendible cardinals under very general (not necessarily definable or weakly homogeneous) class forcing iterations. As applications we prove the consistency of C^{(n)}$C^{(n)}$ -extendible cardinals with \mathrm {{V}}=\mathrm {{HOD}}$\mathrm {{V}}=\mathrm {{HOD}}$ , and also with \mathrm {GA}$\mathrm {GA}$ (the Ground Axiom) plus \mathrm {V}\neq \mathrm {HOD}$\mathrm {V}\neq \mathrm {HOD}$ , the latter being a strengthening of a result from [14].

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.

A permutation group G on a set A is {\kappa }${\kappa }$ -homogeneous iff for all X,Y\in \bigl [ {A} \bigr ]^ {\kappa } $X,Y\in \bigl [ {A} \bigr ]^ {\kappa }$ with |A\setminus X|=|A\setminus Y|=|A|$|A\setminus X|=|A\setminus Y|=|A|$ there is a g\in G$g\in G$ with g[X]=Y$g[X]=Y$ . G is {\kappa }${\kappa }$ -transitive iff for any injective function f with \operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }} $\operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }}$ and |A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$|A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$ there is a g\in G$g\in G$ with f\subset g$f\subset g$ .

Giving a partial answer to a question of P. M. Neumann [6] we show that there is an {\omega }${\omega }$ -homogeneous but not {\omega }${\omega }$ -transitive permutation group on a cardinal {\lambda }${\lambda }$ provided

1. (i) {\lambda }<{\omega }_{\omega }${\lambda }<{\omega }_{\omega }$ , or

2. (ii) 2^{\omega }<{\lambda }$2^{\omega }<{\lambda }$ , and {\mu }^{\omega }={\mu }^+${\mu }^{\omega }={\mu }^+$ and \Box _{\mu }$\Box _{\mu }$ hold for each {\mu }\le {\lambda }${\mu }\le {\lambda }$ with {\omega }=\operatorname {cf}({\mu })<{{\mu }}${\omega }=\operatorname {cf}({\mu })<{{\mu }}$ , or

3. (iii) our model was obtained by adding (2^{\omega })^+$(2^{\omega })^+$ many Cohen generic reals to some ground model.

For {\kappa }>{\omega }${\kappa }>{\omega }$ we give a method to construct large {\kappa }${\kappa }$ -homogeneous, but not {\kappa }${\kappa }$ -transitive permutation groups. Using this method we show that there exist {\kappa }^+${\kappa }^+$ -homogeneous, but not {\kappa }^+${\kappa }^+$ -transitive permutation groups on {\kappa }^{+n}${\kappa }^{+n}$ for each infinite cardinal {\kappa }${\kappa }$ and natural number n\ge 1$n\ge 1$ provided V=L$V=L$ .

Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies that the variety has equationally definable principal congruences. This result is then used to provide necessary and sufficient conditions for the existence of a model completion for theories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes lattice-ordered abelian groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of pointed residuated lattices has a model completion, it must have equationally definable principal congruences. In particular, the theories of lattice-ordered abelian groups and MV-algebras do not have a model completion, as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuated lattices generated by their linearly ordered members, including lattice-ordered abelian groups and MV-algebras, can be extended with a binary operation to obtain theories that do have a model completion.

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete one-types, then it has a Borel complete reduct. Similarly, if Th(M)$Th(M)$ is not small, then M^{eq}$M^{eq}$ has a Borel complete reduct, and if a theory T is not \omega $\omega$ -stable, then the elementary diagram of some countable model of T has a Borel complete reduct.

The Gödel translation provides an embedding of the intuitionistic logic \mathsf {IPC}$\mathsf {IPC}$ into the modal logic \mathsf {Grz}$\mathsf {Grz}$ , which then embeds into the modal logic \mathsf {GL}$\mathsf {GL}$ via the splitting translation. Combined with Solovay’s theorem that \mathsf {GL}$\mathsf {GL}$ is the modal logic of the provability predicate of Peano Arithmetic \mathsf {PA}$\mathsf {PA}$ , both \mathsf {IPC}$\mathsf {IPC}$ and \mathsf {Grz}$\mathsf {Grz}$ admit provability interpretations. When attempting to ‘lift’ these results to the monadic extensions \mathsf {MIPC}$\mathsf {MIPC}$ , \mathsf {MGrz}$\mathsf {MGrz}$ , and \mathsf {MGL}$\mathsf {MGL}$ of these logics, the same techniques no longer work. Following a conjecture made by Esakia, we add an appropriate version of Casari’s formula to these monadic extensions (denoted by a ‘+’), obtaining that the Gödel translation embeds \mathsf {M^{+}IPC}$\mathsf {M^{+}IPC}$ into \mathsf {M^{+}Grz}$\mathsf {M^{+}Grz}$ and the splitting translation embeds \mathsf {M^{+}Grz}$\mathsf {M^{+}Grz}$ into \mathsf {MGL}$\mathsf {MGL}$ . As proven by Japaridze, Solovay’s result extends to the monadic system \mathsf {MGL}$\mathsf {MGL}$ , which leads us to a provability interpretation of both \mathsf {M^{+}IPC}$\mathsf {M^{+}IPC}$ and \mathsf {M^{+}Grz}$\mathsf {M^{+}Grz}$ .