This note presents some new results from [1] about the Suszko operator and truth-equational logics, following the works of Czelakowski [11] and Raftery [17]. It is proved that the Suszko operator relative to a truth-equational logic S preserves suprema and commutes with endomorphisms. Together with injectivity, proved by Raftery in [17], the Suszko operator relative to a truth-equational logic is a

We define an axiom schema I Δ 0 S for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic, I Δ 0 , and use some of its basic model theory to establish some independence results for various axioms of set theory over I Δ 0 S . Then we ask: given a model M of I Δ 0 , is there a model of I Δ 0 S whose ordinal arithmetic is isomorphic to M? We show that the answer

We study expressive power of continuous logic in classes of metric groups defined by properties of their actions. We concentrate on unbounded continuous actions on metric spaces. For example, we consider the properties non-OB, non-FH and non-FR.

We provide a fine-grained analysis on the relation between König's lemma, weak König's lemma, and the decidable fan theorem in the context of constructive reverse mathematics. In particular, we show that double negated variants of König's lemma and weak König's lemma are equivalent to double negated variants of the general decidable fan theorem and the binary decidable fan theorem, respectively, over

We study modal completeness and incompleteness of several sublogics of the interpretability logic IL . We introduce the sublogic IL − , and prove that IL − is sound and complete with respect to Veltman prestructures which are introduced by Visser. Moreover, we prove the modal completeness of twelve logics between IL − and IL with respect to Veltman prestructures. On the other hand, we prove that eight

The ultraproduct construction is generalized to p-ultramean constructions ( 1 ⩽ p < ∞ ) by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments L p of continuous logic and are very close to the L p ( R ) constructions in real analysis. A powermean variant of the Keisler-Shelah isomorphism theorem is proved for L p . It is then proved that L

In our paper [3], a sequent system GRW + ∘ for the contraction-less positive relevant logic with co-tenability, RW + ∘ , is considered. Its sequent system with the rule of cut, which does not involve the truth constant t, is presented, and the proof that it admits the elimination of cut is given. However, in the proof of the cut-elimination theorem some forms of proofs, which may cause difficulties

We write fin ( m ) and S fin ( m ) for the cardinalities of the set of finite subsets and the set of permutations with finitely many non-fixed points, respectively, of a set which is of cardinality m . In this paper, we investigate relationships between fin ( m ) and S fin ( m ) for an infinite cardinal m in the absence of the Axiom of Choice. We give conditions that make fin ( m ) and S fin ( m )

We define an evolving Shelah-Spencer process as one by which a random graph grows, with at each time τ ∈ N a new node incorporated and attached to each previous node with probability τ − α , where α ∈ ( 0 , 1 ) ∖ Q is fixed. We analyse the graphs that result from this process, including the infinite limit, in comparison to Shelah-Spencer sparse random graphs discussed in [21] and throughout the model-theoretic

Our aim was to try to generalize some theorems about the saturation of ultrapowers to reduced powers. Naturally, we deal with saturation for types consisting of atomic formulas. We succeed to generalize “the theory of dense linear order (or T with the strict order property) is maximal and so is any pair ( T , Δ ) which is SOP3”, (where Δ consists of atomic or conjunction of atomic formulas). However

We consider continuous structures which are obtained from finite dimensional Hilbert spaces over C by adding some unitary operators. We consider appropriate algorithmic problems concerning continuous theories of natural classes of these structures. We connect these questions with property MF.

We study the role of meeting numbers in pcf theory. In particular, Shelah's Strong Hypothesis is shown to be equivalent to the assertion that for any singular cardinal σ of cofinality ω, there is a size σ + collection Q of countable subsets of σ with the property that for any infinite subset a of σ, there is a member of Q meeting a in an infinite set.

We continue the research of an extension ∣ ∼ of the divisibility relation to the Stone-Čech compactification β N . First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in β N and nonstandard extensions of N are answered, providing a few more equivalent conditions for divisibility in β

The uniform continuity theorem ( UCT ) states that every pointwise continuous real-valued function on the unit interval is uniformly continuous. In constructive mathematics, UCT is strictly stronger than the decidable fan theorem ( DFT ) , but Loeb [17] has shown that the two principles become equivalent by encoding continuous real-valued functions as type-one functions. However, the precise relation

We show that if M is an r-maximal set, A is a major subset of M, B is an arbitrary set and M ∖ A ≡ Q 1 , N B , then M ∖ A ≤ m B . We prove that the c.e. Q 1 , N -degrees are not dense. We also show that there exist infinite collections of Q 1 , N -degrees { a i } i ∈ ω and { b j } j ∈ ω such that the following hold: (i) for every i, j, a i < Q 1 , N a i + 1 , b j + 1 < Q 1 , N b j and a i < Q 1 , N

Audrito and Viale introduced the new large cardinal notion of an (α)-uplifting cardinal (for an ordinal α). We shall show that this notion cannot be defined (or expressed) in the standard first-order language of set theory for every tranfinite α.

In this paper we study the subdirectly irreducible algebras in the variety PCD M of pseudocomplemented De Morgan algebras by means of their De Morgan p-spaces. We introduce the notion of the body of an algebra L ∈ PCD M and determine Body ( L ) when L is subdirectly irreducible. As a consequence of this, in the case of pseudocomplemented Kleene algebras, two special subvarieties arise naturally, for

We present a theory of stratified truth ST μ with a μ‐operator, where terms representing fixed points of stratified monotone operations are available. We prove that ST μ is relatively intepretable into Quine's NF (or subsystems thereof). The motivation is to investigate a strong theory of truth, which is consistent by means of stratification, i.e., by adopting an implicit type theoretic discipline

We show that certain classes of modules have universal models with respect to pure embeddings: Let R be a ring, T a first‐order theory with an infinite model extending the theory of R‐modules and K T = ( Mod ( T ) , ≤ pp ) (where ⩽pp stands for “pure submodule”). Assume K T has the joint embedding and amalgamation properties. If λ | T | = λ or ∀ μ < λ ( μ | T | < λ ) , then K T has a universal model

We determine the number of non‐isomorphic semi‐Heyting algebras on an n‐element chain, where n is a positive integer, using a recursive method. We then prove that the numbers obtained agree with those determined in [1]. We apply the formula to calculate the number of non‐isomorphic semi‐Heyting chains of a given size in some important subvarieties of the variety of semi‐Heyting algebras that were introduced

In this paper, we are interested in parallels to the classical notions of special subsets in R defined in the generalized Cantor and Baire spaces (2κ and κ κ ). We consider generalizations of the well‐known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ‐sets, γ‐sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also

Regarding strictly monadic second‐order logic (SMSOL), which is the fragment of monadic second‐order logic in which all predicate constants are unary and there are no function symbols, we show that a standard deductive system with full comprehension is sound and complete with respect to standard semantics. This result is achieved by showing that in the case of SMSOL, the truth value of any formula

We introduce agreement reducibility and highlight its major features. Given subsets A and B of N , we write A ≤ agree B if there is a total computable function f : N → N satisfying for all e , e ′ , W e ∩ A = W e ′ ∩ A if and only if W f ( e ) ∩ B = W f ( e ′ ) ∩ B .

We study rigidity properties of linearly ordered sets (chains) under automorphisms, embeddings, epimorphisms, and endomorphisms. We focus on two main cases: dense subchains of the real numbers, and uncountable dense chains of higher regular cardinalities. We also give a Fraenkel‐Mostowski model which illustrates the role of the axiom of choice in one of the key proofs.

In the context of reverse mathematics, effective transfinite recursion refers to a principle that allows us to construct sequences of sets by recursion along arbitrary well orders, provided that each set is Δ 1 0 ‐definable relative to the previous stages of the recursion. It is known that this principle is provable in ACA 0 . In the present note, we argue that a common formulation of effective transfinite

Ishihara's problem of decidable variables asks which class of decidable propositional variables is sufficient to warrant classical theorems in intuitionistic logic. We present several refinements to the class proposed by Ishii for this problem, which also allows the class to cover Glivenko's logic. We also treat the extension of the problem to minimal logic, suggesting a couple of new classes.

We improve on results and constructions by Apter, Dimitriou, Gitik, Hayut, Karagila, and Koepke concerning large cardinals, ultrafilters, and cofinalities without the axiom of choice. In particular, we show the consistency of the following statements from certain assumptions: the first supercompact cardinal can be the first uncountable regular cardinal, all successors of regular cardinals are Ramsey

We associate Hanf numbers H ( t ) to triples t = ( T , T 1 , p ) where T and T1 are theories and p is a type. We show that the Hanf number for the property: “there is a model M1 of T 1 which omits p, but M 1 ↾ τ is saturated” is larger than the Hanf number of L λ + , κ but smaller than the Hanf number of L ( 2 λ ) + , κ when T is stable with κ = κ ( T ) . In fact, surprisingly, we even characterise

We study independent families of functions and permutations and associated cardinal characteristics i f and i p . In this paper, we show that these cardinals lie between the cardinals p and i and give some related consistency results.

The non‐forking‐instances topology (NFI topology) is a topology on the Stone space of a theory T that depends on a reduct T − of T. This topology has been used in [6] to describe the set of universal transducers for ( T , T − ) (invariants sets that translate forking‐open sets in T − to forking‐open sets in T). In this paper we show that in contrast to the stable case, the NFI topology need not be

We discuss the possibility of applying the compactness theorem to the study of finite structures. Given a class of finite structures, it is important to determine whether it can be expressed by a particular category of sentences. We are interested in this type of problem, and use nonstandard method for showing the non‐expressibility of certain classes of finite graphs by an existential monadic second

We present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem, which presents a simple condition under which the converse holds. Our approach is inspired by proof theory, and in particular G\"{o}del's functional interpretation

Michael Rathjen and the present author have shown that $\Pi^1_1$-bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in $\mathbf{ACA_0}$. In this note we show that the base theory can be weakened to $\mathbf{RCA_0}$. Our argument makes crucial use of a normal function $f$ with $f(\alpha)\leq 1+\alpha^2$ and $f'(\alpha)=\o

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called "computable reducibility"), or in the same isomorphism type (with the isomorphism

We reformulate slightly Russell’s notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first-order structure. We argue that the notion parallels Martin-Löf (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of “small” cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero

A class of Kripke frames is called modally definable if there is a set of modal formulas such that the class consists exactly of frames on which every formula from that set is valid, i. e. globally true under any valuation. Here, existential definability of Kripke frame classes is defined analogously, by demanding that each formula from a defining set is satisfiable under any valuation. This is equivalent

We discuss ``puzzles of prisoners and hats`` with infinitely many prisoners and more than two hat colors. Assuming that the set of hat colors is equipped with a commutative group structure, we prove strategic equivalence among puzzles of several protocols with countably many prisoners.

The Artin‐Schreier theorem says that every formally real field has orders. Friedman, Simpson and Smith showed in [6] that the Artin‐Schreier theorem is equivalent to WKL 0 over RCA 0 . We first prove that the generalization of the Artin‐Schreier theorem to noncommutative rings is equivalent to WKL 0 over RCA 0 . In the theory of orderings on rings, following an idea of Serre, we often show the existence

In this paper we analyse some questions concerning trees on $\kappa$, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in \cite[Question 5.2]{FKK16} about the diagram for regularity properties.

We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, `Talagrand's stability', and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove the `almost definability' and `Baire~1 definability' of coheirs

We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well-founded models of ZFC.

Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed-point logic. Inclusion logic is not axiomatizable in full, but its first-order consequences can be axiomatized. In this paper, we provide such an explicit partial axiomatization by introducing a system of natural deduction for inclusion logic that is sound and complete for first-order

Following a line of research initiated in \cite{BBNN}, I describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cicho\'n diagram for cardinal characteristics of the continuum. I show that working from relatively modest assumptions about a notion of reduction, one can construct a robust version of such a diagram. As an application

We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o-minimal structures. Most noteworthily, dense pairs of real closed fields have this property.

A quasivariety K of algebras has the joint embedding property (JEP) iff it is generated by a single algebra A. It is structurally complete iff the free countably generated algebra in K can serve as A. A consequence of this demand, called "passive structural completeness" (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it

We give a new proof of a theorem of Shelah which states that for every family of labeled trees, if the cardinality $\kappa$ of the family is much larger (in the sense of large cardinals) than the cardinality $\lambda$ of the set of labels, more precisely if the partition relation $\kappa \to (\omega)^{\mathord{<}\omega}_\lambda$ holds, then there is a homomorphism from one labeled tree in the family

We show that $ZF+DC+$"all Turing invariant sets of reals have the perfect set property" implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.

We discuss a system of strengthenings of “ ℵ ω is Jónsson” indexed by real numbers, and identify a strongest one. We give a proof of a theorem of Silver and show that there is a barrier to weakening its hypothesis.

Dear readers of Mathematical Logic Quarterly: Dr Hugo Nobrega has been the Editorial Assistant of our journal for four years, from 2016 to 2019, and has now handed over to his successor, Dr Thomas Piecha from the Eberhard Karls Universität Tübingen, who started to work for the journal on New Year's Day 2020. We thank Hugo for four years of excellent service for the journal and welcome Thomas as our

We consider topological pairs ( A , B ) , B ⊆ A , which have computable type, which means that they have the following property: if X is a computable topological space and f : A → X a topological imbedding such that f ( A ) and f ( B ) are semicomputable sets in X, then f ( A ) is a computable set in X. It is known, e.g., that ( M , ∂ M ) has computable type if M is a compact manifold with boundary

We give a model of set theory in which the strongly non‐stationary ideal over ℘ μ μ is weakly presaturated below some canonical set. Here μ is a regular uncountable cardinal. The model is the forcing extension with the Lévy collapse of a Woodin cardinal to the successor of μ. This improves on results of Goldring and of the first author.

We study the R-torsionfree part of the Ziegler spectrum of an order \Lambda over a Dedekind domain R. We underline and comment on the role of lattices over \Lambda. We describe the torsionfree part of the spectrum when \Lambda is of finite lattice representation type.

Given an inner model $W \subset V$ and a regular cardinal $\kappa$, we consider two alternatives for adding a subset to $\kappa$ by forcing: the Cohen poset $Add(\kappa,1)$, and the Cohen poset of the inner model $Add(\kappa,1)^W$. The forcing from $W$ will be at least as strong as the forcing from $V$ (in the sense that forcing with the former adds a generic for the latter) if and only if the two

Let $\widetilde{\mathbb{Q}_p}$ be the field of $p$-adic numbers in the language of rings. In this paper we consider the theory of $\widetilde{\mathbb{Q}_p}$ expanded by two predicates interpreted by multiplicative subgroups $\alpha^\mathbb{Z}$ and $\beta^\mathbb{Z}$ where $\alpha, \beta\in\mathbb{N}$ are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic

We show that the types of the witnesses in the Herbrand functional interpretation can be simplified, avoiding the use of "sets of functionals" in the interpretation of implication and universal quantification. This is done by presenting an alternative formulation of the Herbrand functional interpretation, which we show to be equivalent to the original presentation. As a result of this investigation

We show that every countable ideal of degrees that are low for isomorphism is contained in a principal ideal of degrees that are low for isomorphism by adapting an exact pair construction. We further show that within the hyperimmune-free degrees, lowness for isomorphism is entirely independent of computable traceability.

We show that the consistency of the theories “ZF + ¬ACω + GCH holds below אω + There is an injection f : אω+2 → ℘(אω) + Both אω and APאω fail” and “ZF + ¬ACω + GCH holds below אω + There is an injection f : אω+2 → ℘(אω) + אω+1 satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context to certain long-standing open questions concerning

Let E ⊆ Q p n and T be a set‐valued map from E to Q p m . We prove that if T is p‐adic semi‐algebraic, lower semi‐continuous and T ( x ) is closed for every x ∈ E , then T has a p‐adic semi‐algebraic continuous selection. In addition, we include three applications of this result. The first one is related to Fefferman's and Kollár's question on existence of p‐adic semi‐algebraic continuous solution

We introduce a two-dimensional metric (interval) temporal logic whose internal and external time flows are dense linear orderings. We provide a suitable semantics and a sequent calculus with axioms for equality and extralogical axioms. Then we prove completeness and a semantic partial cut-elimination theorem down to formulas of a certain type.

The set AC 0 ( F ) , the AC 0 closure of F, is the closure with respect to substitution and concatenation recursion on notation of a set of basic functions comprehending the set F. By improving earlier work, we show that AC 0 ( F ) is the substitution closure of a simple function set and characterize well‐known function complexity classes as the substitution closure of finite sets of simple functions