A finitely based, finitely generated variety with finitely many subvarieties is a Cross variety. In the present article, it is shown that a variety of J-trivial monoids is Cross if and only if it excludes 14 specific almost Cross varieties. Consequently, these 14 varieties exhaust all almost Cross varieties of J-trivial monoids.

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.

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 characterize sums of normal ultrafilters after the Magidor iteration of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower Axiom. We also construct a nonrigid ultrapower and two uniform ultrafilters on different cardinals that have the same ultrapower.

We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is obtained by replacing the classical ‘least’ function associated with ideals by a two-dimensional ‘last’ function associated with walks on ordinals. Consequently, we obtain conditions under which a measure admits not just infinite pairwise disjoint families of positive sets, but in fact families of maximum possible size. As an application we solve a problem left open in Shelah’s Cardinal Arithmetic book, proving that for every weakly inaccessible cardinal $\kappa $, if there exists a stationary subset of $\kappa $ that does not reflect at regulars, then the strong Ramsey relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ holds.

In this article, we deal with the classification complexity of continuous (Devaney) chaotic systems in dimensions $0,1,$ and $\infty $ using the framework of invariant descriptive set theory. We identify the complexity in dimensions $0$ and $\infty $, while in dimension $1$ we get some partial results.

More precisely, we prove the topological conjugacy relation of invertible chaotic systems on the Hilbert cube (resp. on all compact metric spaces) has the same complexity as (i.e., is Borel bireducible with) the universal orbit relation induced by a Polish group. As a consequence, this answers a recent question asked by L. Ding. We also prove that the topological conjugacy relation of invertible chaotic systems on the Cantor space has the same complexity as the universal relation induced by the group $S_\infty $. This answers a recent question by M. Foreman. Some non-trivial bounds on the classification complexity of chaotic systems on the interval and on the circle are also obtained. Namely, the lower bound is the Vitali equivalence relation, and the upper bound is the equality of countable sets of reals. This especially implies that the relation is Borel. However, the exact complexity remains unknown.

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 the correctness of the AKS algorithm [1] within the bounded arithmetic theory $T^{\text {count}}_2$ or, equivalently, the first-order consequences of the theory $\text {VTC}^0$ expanded by the smash function, which we denote by $\text {VTC}^0_2$. Our approach initially demonstrates the correctness within the theory $S^1_2 + \mathrm {iWPHP}$ augmented by two algebraic axioms and then shows that they are provable in $\text {VTC}^0_2$. The two axioms are: a generalized version of Fermat’s Little Theorem and an axiom adding a new function symbol which injectively maps roots of polynomials over a definable finite field to numbers bounded by the degree of the given polynomial. To obtain our main result, we also give new formalizations of parts of number theory and algebra:

• • In $\mathrm {PV}_1$: We formalize Legendre’s Formula on the prime factorization of $n!$, key properties of the Combinatorial Number System and the existence of cyclotomic polynomials over the finite fields $\mathbb {Z}/p$.

• • In $S^1_2$: We prove the inequality $\text {lcm}(1,\dots , 2n) \geq 2^n$.

• • In $\text {VTC}^0$: We verify the correctness of the Kung–Sieveking algorithm for polynomial division.

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 $\omega _{4}$.

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.

We prove the consistency of the inequality $\mathfrak {r}_{\mathsf {nwd}}<\mathfrak {irr}$, which in turn implies the consistency of $\mathfrak {r}_{\mathsf {nwd}}<\mathfrak {i}$ and $\mathfrak {r}_{\mathsf {scatt}}<\mathfrak {irr}$. This answers one question from [1] and one question from [2]. We also prove the consistency of the inequality $\mathfrak {r}_{\mathbb {Q}}<\mathfrak {u}_{\mathbb {Q}}$.

This paper presents a unified algebraic study of a family of logics related to Abelian logic (Ab), the logic of Abelian lattice-ordered groups. We treat Ab as the base system and refer to its expansions as superabelian logics. The paper focuses on two main families of expansions. First, we investigate the rich landscape of infinitary extensions of Ab, providing an axiomatization for the infinitary logic of real numbers and showing that there exist $2^{2^\omega }$ distinct logics in this family. Second, we introduce pointed Abelian logic (${\text {pAb}}$), the logic of pointed Abelian lattice-ordered groups, by adding a new constant to the language. This framework includes Łukasiewicz unbound logic. We provide axiomatizations for its finitary and infinitary versions as extensions of ${\text {pAb}}$ and establish their precise relationship with standard Łukasiewicz logic via a formal translation. Finally, the methods developed for this analysis are generalized to axiomatize the logics of other prominent pointed groups.

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 $\mathfrak {b}$.