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 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.

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 Gödel's functional interpretation, which

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.

We prove an existential analogue of the Goldblatt‐Thomason Theorem which characterizes modal definability of elementary classes of Kripke frames using closure under model theoretic constructions. The less known version of the Goldblatt‐Thomason Theorem gives general conditions, without the assumption of first‐order definability, but uses non‐standard constructions and algebraic semantics. We present

Michael Rathjen and the present author have shown that Π 1 1 ‐bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in ACA 0 . In this note we show that the base theory can be weakened to RCA 0 . Our argument makes crucial use of a normal function f with f ( α ) ≤ 1 + α 2 and f ′ ( α ) = ω ω α . We shall also exhibit a normal

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

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

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

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 κ, 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 [6, Question 5.2] about the diagram for regularity properties.

We study and characterize stability, the negation of the independence property (NIP) and the negation of the strict order property (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 the NIP in continuous logic. Using a result of Bourgain, Fremlin

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 axiomatisable 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

Following a line of research initiated in [4], we describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cichoń diagram for cardinal characteristics of the continuum. We 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

We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o‐minimal structures. Most noteworthily, this property holds for dense pairs of real closed fields, as well as for expansions of o‐minimal structures by a dense independent set.

A quasivariety K of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A. It is structurally complete if and only if the free ℵ0‐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

We give a new proof of a theorem of Shelah which states that for every family of labeled trees, if the cardinality κ of the family is much larger (in the sense of large cardinals) than the cardinality λ of the set of labels, more precisely if the partition relation κ → ( ω ) λ < ω holds, then there is a homomorphism from one labeled tree in the family to another. Our proof uses a characterization of

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 study the R‐torsion‐free part of the Ziegler spectrum of an order Λ over a Dedekind domain R. We underline and comment on the role of lattices over Λ. We describe the torsion‐free part of the spectrum when Λ is of finite lattice representation type.

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 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 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 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 + ¬ AC ω + “ GCH holds below ℵ ω ” + “there is an injection f : ℵ ω + 2 → ℘ ( ℵ ω ) ” + “both □ ℵ ω ∗ and AP ℵ ω fail” and ZF + ¬ AC ω + “ GCH holds below ℵ ω ” + “there is an injection f : ℵ ω + 2 → ℘ ( ℵ ω ) ” + “ ℵ ω + 1 satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context

Let Q p ∼ be the field of p‐adic numbers in the language of rings. In this paper we consider the theory of Q p ∼ expanded by two predicates interpreted by multiplicative subgroups α Z and β Z where α , β ∈ N are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if α and β have positive p‐adic valuation. If either α or β has zero valuation we show that

Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a subset to κ by forcing: the Cohen poset Add(κ, 1), and the Cohen poset of the inner model Add ( κ , 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 posets have the same cardinality

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

We present a class forcing notion M ( η ) , uniformly definable for ordinals η, which forces the ground model to be the ηth inner mantle of the extension, in which the sequence of inner mantles has length at least η. This answers a conjecture of Fuchs, Hamkins, and Reitz [1] in the positive. We also show that M ( η ) forces the ground model to be the ηth iterated HOD of the extension, where the sequence

The central result of this paper is the small‐is‐very‐small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a definable property has a small witness, i.e., a witness in a sufficiently small definable cut, then it shows that the property has a very small witness: i.e., a witness below a given standard number. Which cuts are sufficiently

We show that there are models M 1 and M 2 such that M 1 elementarily embeds into M 2 but their ultrafilter extensions β β ( M 1 ) and β β ( M 2 ) are not elementarily equivalent.

In [5], examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper, we give examples of complete L ω 1 , ω ‐sentences with maximal models in more than one cardinality. From (homogeneous) characterizability of κ we construct sentences with maximal models in κ and in one

The assertion that every definable set has a definable element is equivalent over ZF to the principle V = HOD , and indeed, we prove, so is the assertion merely that every Π2‐definable set has an ordinal‐definable element. Meanwhile, every model of ZFC has a forcing extension satisfying V ≠ HOD in which every Σ2‐definable set has an ordinal‐definable element. Similar results hold for HOD ( R ) and

Work of Eagle, Farah, Goldbring, Kirchberg, and Vignati shows that the only separable C*‐algebras that admit quantifier elimination in continuous logic are C , C 2 , M 2 ( C ) , and the continuous functions on the Cantor set. We show that, among finite dimensional C*‐algebras, quantifier elimination does hold if the language is expanded to include two new predicate symbols: One for minimal projections

Robinson and Zakon gave necessary and sufficient conditions for an abelian ordered group to satisfy the same first‐order sentences as an archimedean abelian ordered group (i.e., which embeds in the group of real numbers). The present paper generalizes their work to obtain similar results for infinite subgroups of the group of unimodular complex numbers. Furthermore, the groups which satisfy the same

This paper is a compilation of results originating in the author's master thesis. We give a useful characterization of the generalized bounding and dominating numbers, b ( κ ) and d ( κ ) . We show that t ( κ ) ≤ add ( M κ ) when κ = κ < κ . And we prove a higher analogue of Bell's theorem stating that p = c is equivalent to MA ( σ ‐centered).

Let p be prime number, K be a p‐adically closed field, X ⊆ K m a semi‐algebraic set defined over K and L ( X ) the lattice of semi‐algebraic subsets of X which are closed in X. We prove that the complete theory of L ( X ) eliminates quantifiers in a certain language L ASC , the L ASC ‐structure on L ( X ) being an extension by definition of the lattice structure. Moreover it is decidable, contrary

Developing an analogue of Solovay reducibility in the higher recursion setting, we show that results from the classical computably enumerable case can be extended to the new context.

We show that the Axiom of Dependent Choice, DC , holds in countably iterable, passive premice M constructed over their reals which satisfy the Axiom of Determinacy, AD , in a ZF + DC R M background universe. This generalizes an argument of Kechris for L ( R ) using Steel's analysis of scales in mice. In particular, we show that for any n ≤ ω and any countable set of reals A so that M n ( A ) ∩ R =

In this paper, we show that for any computable ordinal α, there exists a computable tree of rank α + 1 with strong degree of categoricity 0 ( 2 α ) if α is finite, and with strong degree of categoricity 0 ( 2 α + 1 ) if α is infinite. In fact, these are the greatest possible degrees of categoricity for such trees. For a computable limit ordinal α, we show that there is a computable tree of rank α with

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as by characterising their combinatorial properties by means of their monads. This extends to arbitrary sets and properties methods previously used to study partition

We study the flow ( G ( Q p ) , S G ( Q p ) ) of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable f‐generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We shall give a description of f‐generic types of

In set theory without the Axiom of Choice ( AC ), we investigate the problem of the placement of Łoś's Theorem ( LT ) in the hierarchy of weak choice principles, and answer several open questions from the book Consequences of the Axiom of Choice by Howard and Rubin, as well as an open question by Brunner. We prove a number of results summarised in § 3.

In this paper, we consider the question of definability of types in non‐stable theories. In order to do this we introduce a notion of a relatively stable theory: a theory is stable up to Δ if any Δ‐type over a model has few extensions up to complete types. We prove that an n‐type over a model of a theory that is stable up to Δ is definable if and only if its Δ‐part is definable.

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t 1 , … , t k . A formula in this language defines a parametric set S t ⊆ Z d as t varies in Z k , and we examine the counting function | S t | as a function of t. For a single parameter, it is known that | S t | can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently

We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.

We study degree spectra of structures with respect to the bi‐embeddability relation. The bi‐embeddability spectrum of a structure is the family of Turing degrees of its bi‐embeddable copies. To facilitate our study we introduce the notions of bi‐embeddable triviality and basis of a spectrum. Using bi‐embeddable triviality we show that several known families of degrees are bi‐embeddability spectra of

We consider the notion of prediction functions (or predictors) studied before in the context of randomness and stochasticity by Ko, and later by Ambos‐Spies and others. Predictor is a total computable function which tries to predict bits of some infinite binary sequence. The prediction error is defined as the limit of the number of incorrect answers divided by the number of answers given so far. We

We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.

Yorioka introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in . We construct a matrix iteration of c.c.c. posets to force that, for many ideals in that class, their associated cardinal invariants (i.e., additivity, covering, uniformity

We show that the automorphism group of Hall's universal locally finite group has ample generics, that is, it admits comeager diagonal conjugacy classes in all dimensions. Consequently, it has the small index property, is not the union of a countable chain of non‐open subgroups, and has the automatic continuity property. Also, we discuss some algebraic and topological properties of the automorphism

A subset of a model of is called neutral if it does not change the relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non‐existence of neutral sets in various models of . We show that cofinal extensions of prime models are neutrally expandable, and ω1‐like neutrally expandable models exist, while no recursively saturated model is neutrally

Shelah considered a certain version of Strong Chang's Conjecture which we denote , and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted , and prove an analogous characterization of it. In particular, is equivalent to the assertion that the the Friedman‐Krueger poset is semiproper. This strengthens

It is proved that for every countable structure and a computable successor ordinal α there is a countable structure which is ‐least among all countable structures such that is Σ‐definable in the αth jump . We also show that this result does not hold for the limit ordinal . Moreover, we prove that there is no countable structure with the degree spectrum for .