We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete for the basic positive distributive substructural logic $\mathsf {B}^+$, that collection frames on multisets are sound and complete for $\mathsf {RW}^+$ (the relevant logic $\mathsf {R}^+$, without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic $\mathsf {R}^+$. The completeness of set frames for $\mathsf {R}^+$ is, currently, an open question.

We explore the possibilities for elementary embeddings $j : M \to N$, where M and N are models of ZFC with the same ordinals, $M \subseteq N$, and N has access to large pieces of j. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line ${\mathbb R}$.

We extend the languages of both basic and graded modal logic with the infinity diamond, a modality that expresses the existence of infinitely many successors having a certain property. In both cases we define a natural notion of bisimilarity for the resulting formalisms, that we dub $\mathtt {ML}^{\infty }$ and $\mathtt {GML}^{\infty }$, respectively. We then characterise these logics as the bisimulation-invariant fragments of the naturally corresponding predicate logic, viz., the extension of first-order logic with the infinity quantifier. Furthermore, for both $\mathtt {ML}^{\infty }$ and $\mathtt {GML}^{\infty }$ we provide a sound and complete axiomatisation for the set of formulas that are valid in every Kripke frame, we prove a small model property with respect to a widened class of weighted models, and we establish decidability of the satisfiability problem.

We prove that the first-order logic of CZF is intuitionistic first-order logic. To do so, we introduce a new model of transfinite computation (Set Register Machines) and combine the resulting notion of realisability with Beth semantics. On the way, we also show that the propositional admissible rules of CZF are exactly those of intuitionistic propositional logic.

We investigate an approach for drawing logical inference from inconsistent premisses. The main idea in this approach is that the inconsistencies in the premisses should be interpreted as uncertainty of the information. We propose a mechanism, based on Kinght’s [14] study of inconsistency, for revising an inconsistent set of premisses to a minimally uncertain, probabilistically consistent one. We will then generalise the probabilistic entailment relation introduced in [15] for propositional languages to the first order case to draw logical inference from a probabilistic set of premisses. We will show how this combination can allow us to limit the effect of uncertainty introduced by inconsistent premisses to only the reasoning on the part of the premise set that is relevant to the inconsistency.

We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa $, the existence of a strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is a theorem of $\textsf{ZFC}$. Adding the requirement of subadditivity to a strongly unbounded coloring is a significant strengthening, though, and here we see that in many cases the existence of a subadditive strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is independent of $\textsf{ZFC}$. We connect the existence of subadditive strongly unbounded colorings with a number of other infinitary combinatorial principles, including the narrow system property, the existence of $\kappa $-Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is equivalent to a certain weak indexed square principle $\boxminus ^{\operatorname {\mathrm {ind}}}(\kappa , \theta )$. We conclude the paper with an application to the failure of the infinite productivity of $\kappa $-stationarily layered posets, answering a question of Cox.

We investigate Maker–Breaker games on graphs of size $\aleph _1$ in which Maker’s goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{\omega ,\omega _1}$-game under ZFC+MA+$\neg $CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{\omega _1}$-game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.

We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if, for each prime p, the p-primary part of A is either finite or it coincides with the Prüfer p-group. We also provide a model-theoretic description of the model companions we obtain.

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists ! \mathop{\boldsymbol {\bigwedge }}\limits p = q$. For a logic L algebraized by a quasivariety $\mathcal {Q}$ we show that the AE-subclasses of $\mathcal {Q}$ 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 $-groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbf {\Sigma }^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth–Kechris about discrete structures. As a different application, we also give a new proof of Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable.

Furstenberg–Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg–Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi.

Every discrete definable subset of a closed asymptotic couple with ordered scalar field ${\boldsymbol {k}}$ is shown to be contained in a finite-dimensional ${\boldsymbol {k}}$-linear subspace of that couple. It follows that the differential-valued field $\mathbb {T}$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $\mathbb {R}$ as the underlying set for the second sort.

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 study the first-order consequences of Ramsey’s Theorem for k-colourings of n-tuples, for fixed $n, k \ge 2$, over the relatively weak second-order arithmetic theory $\mathrm {RCA}^*_0$. Using the Chong–Mourad coding lemma, we show that in a model of $\mathrm {RCA}^*_0$ that does not satisfy $\Sigma ^0_1$ induction, $\mathrm {RT}^n_k$ is equivalent to its relativization to any proper $\Sigma ^0_1$-definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe.

We give a complete axiomatization of the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^n_k$ for $n \ge 3$. We show that they form a non-finitely axiomatizable subtheory of $\mathrm {PA}$ whose $\Pi _3$ fragment coincides with $\mathrm {B} \Sigma _1 + \exp $ and whose $\Pi _{\ell +3}$ fragment for $\ell \ge 1$ lies between $\mathrm {I} \Sigma _\ell \Rightarrow \mathrm {B} \Sigma _{\ell +1}$ and $\mathrm {B} \Sigma _{\ell +1}$. We also give a complete axiomatization of the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^2_k + \neg \mathrm {I} \Sigma _1$. In general, we show that the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^2_k$ form a subtheory of $\mathrm {I} \Sigma _2$ whose $\Pi _3$ fragment coincides with $\mathrm {B} \Sigma _1 + \exp $ and whose $\Pi _4$ fragment is strictly weaker than $\mathrm {B} \Sigma _2$ but not contained in $\mathrm {I} \Sigma _1$.

Additionally, we consider a principle $\Delta ^0_2$-$\mathrm {RT}^2_2$ which is defined like $\mathrm {RT}^2_2$ but with both the $2$-colourings and the solutions allowed to be $\Delta ^0_2$-sets rather than just sets. We show that the behaviour of $\Delta ^0_2$-$\mathrm {RT}^2_2$ over $\mathrm {RCA}_0 + \mathrm {B}\Sigma ^0_2$ is in many ways analogous to that of $\mathrm {RT}^2_2$ over $\mathrm {RCA}^*_0$, and that $\mathrm {RCA}_0 + \mathrm {B} \Sigma ^0_2 + \Delta ^0_2$-$\mathrm {RT}^2_2$ is $\Pi _4$- but not $\Pi _5$-conservative over $\mathrm {B} \Sigma _2$. However, the statement we use to witness failure of $\Pi _5$-conservativity is not provable in $\mathrm {RCA}_0 +\mathrm {RT}^2_2$.

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

We construct families of translationally invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lower-bounded by a constant. Moreover, this constant can be taken equal to the operator norm of the local operator that generates the Hamiltonian (the local interaction strength). The result still holds true if one restricts to arbitrarily small quantum perturbations of classical Hamiltonians. The proof combines a robustness analysis of Robinson’s aperiodic tiling, together with tools from quantum information theory: the quantum phase estimation algorithm and the history state technique mapping Quantum Turing Machines to Hamiltonians.

A low linear order with no computable copy constructed by C. Jockusch and R. Soare has Hausdorff rank equal to $2$. In this regard, the question arises, how simple can be a low linear order with no computable copy from the point of view of the linear order type? The main result of this work is an example of a low strong $\eta $-representation with no computable copy that is the simplest possible example.

We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending $\mathsf {RCA}_0$ and axiomatizable by a $\Pi ^1_{k+2}$ sentence, and for any $n\geq k+1$,

$$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \end{align*}$$

$$\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}}(T) \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}, \end{align*}$$

$T_0$

$\mathrm {TI}_{\varPi ^1_n}(\varepsilon _0)^{-}$

$\varepsilon _0$

$\varPi ^1_n$

Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

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