We study randomizations of definable groups. Whenever the underlying theory is stable or NIP and the group is definably amenable, we show its randomization is definably connected.

In this note we show that proof-theoretic uniform boundedness or bounded collection principles which allow one to formalize certain instances of countable Heine–Borel compactness in proofs using abstract metric structures must be carefully distinguished from an unrestricted use of countable Heine–Borel compactness.

In finite type arithmetic, the real numbers are represented by rapidly converging Cauchy sequences of rational numbers. Ulrich Kohlenbach introduced abstract types for certain structures such as metric spaces, normed spaces, Hilbert spaces, etc. With these types, the elements of the spaces are given directly, not through the mediation of a representation. However, these abstract spaces presuppose the

We characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.

We explore the relationship between analytic behavior of a computable real valued function and the computability-theoretic complexity of the individual values of its derivative (the function’s slopes) almost-everywhere. Given a computable function f, the values of its derivative \(f'(x)\), where they are defined, are uniformly computable from \(x'\), the Turing jump of the input. It is known that when

It was proved recently that Telgársky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of \(\mathsf {GCH}+\square \). The proof introduces a combinatorial principle that is shown to follow from \(\mathsf {GCH}+\square \), namely: \(\bigtriangledown \):: Every separative poset \({\mathbb {P}}\) with the \(\kappa \)-cc contains a dense sub-poset \({\mathbb

We show that the quantifier elimination result for the Shelah-Spencer almost sure theories of sparse random graphs \(G(n,n^{-\alpha })\) given by Laskowski (Isr J Math 161:157–186, 2007) extends to their various analogues. The analogues will be obtained as theories of generic structures of certain classes of finite structures with a notion of strong substructure induced by rank functions and we will

In the present paper, we show the first-order definability of the jump operator in the upper semi-lattice of the \(\omega \)-enumeration degrees. As a consequence, we derive the isomorphicity of the automorphism groups of the enumeration and the \(\omega \)-enumeration degrees.

Monotone modal logics have emerged in several application areas such as computer science and social choice theory. Since many of the most studied selfextensional logics have a conjunction, in this paper we study some distributive extensions obtained from a semilattice based deductive system with monotonic modal operators, and we give them neighborhood and algebraic semantics. For each logic defined

It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. The purpose of the paper is to make a first step in understanding

We extend natural deduction for first-order logic (FOL) by introducing diagrams as components of formal proofs. From the viewpoint of FOL, we regard a diagram as a deductively closed conjunction of certain FOL formulas. On the basis of this observation, we first investigate basic heterogeneous logic (HL) wherein heterogeneous inference rules are defined in the styles of conjunction introduction and

We introduce the notion of hereditary G-compactness (with respect to interpretation). We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact. Assuming that a long-standing conjecture about unstable NIP theories holds, this implies that an NIP theory is hereditarily G-compact if and only if it is stable

This short note is on the question whether the intersection of all fixed points of a positive arithmetic operator and the intersection of all its closed points can proved to be equivalent in a weak fragment of second order arithmetic.

We study ultraproducts of finite residue rings \(\prod \nolimits _{n\in {\mathbb {N}}} {\mathbb {Z}}/n{\mathbb {Z}}\diagup {\mathcal {U}} \) where \({\mathcal {U}}\) is a non-principal ultrafilter. We find sufficient conditions of the ultrafilter \({\mathcal {U}}\) to determine if the resulting ultraproduct \(\prod \nolimits _{n\in {\mathbb {N}}} {\mathbb {Z}}/n{\mathbb {Z}}\diagup {\mathcal {U}}\)

We present a construction of a truth class (an interpretation of a compositional truth predicate) in an arbitrary countable recursively saturated model of first-order arithmetic. The construction is fully classical in that it employs nothing more than the classical techniques of formal proof theory.

Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is \(\aleph _2\). The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the

We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model \(\mathcal {A}\) of a computably enumerable, model complete theory, the entire elementary diagram \(E(\mathcal {A})\) must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a uniform procedure

We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal J extending the nonstationary ideal on a regular uncountable (non necessarily successor) cardinal \(\kappa \), our goal being to witness the nonsaturation of J by the existence of towers (of length

We show that for every sofic chunk E there is a bijective homomorphism \(f:E_c \rightarrow E\), where \(E_c\) is a chunk of the group of computable permutations of \(\mathbb {N}\) so that the approximating morphisms of E can be viewed as restrictions of permutations of \(E_c\) to finite subsets of \(\mathbb {N}\). Using this we study some relevant effectivity conditions associated with sofic chunks

We present a new method of constructing a model of \(\lnot \)SCH+\(\lnot \)AP.

In this paper, we show how to construct for a given consistent theory U a \(\varSigma ^0_1\)-predicate that both satisfies the Löb Conditions and the Kreisel Condition—even if U is unsound. We do this in such a way that U itself can verify satisfaction of an internal version of the Kreisel Condition.

In this paper, we provide a new characterization of Keisler’s order in terms of saturation of Boolean ultrapowers. To do so, we apply and expand the framework of ‘separation of variables’ recently developed by Malliaris and Shelah. We also show that good ultrafilters on Boolean algebras are precisely the ones which capture the maximum class in Keisler’s order, answering a question posed by Benda in

We prove an analogue of the fixed-point theorem for the case of definably amenable groups.

We reimplement the creature forcing construction used by Fischer et al. (Arch Math Log 56(7–8):1045–1103, 2017. https://doi.org/10.1007/S00153-017-0553-8. arXiv:1402.0367 [math.LO]) to separate Cichoń’s diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our construction by adding uncountably many additional cardinal characteristics

We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature \((\vee ,\perp ,0,1)\), where ‘\(\perp \)’ is the orthogonality relation. Our main result is that already its quasi-identities are undecidable: there is no algorithm to decide whether an implication between equations and orthogonality relations implies another equation. This is a corollary of a recent

Let \(\widetilde{{\mathcal {M}}}=\langle {{{\mathcal {M}}}}, G\rangle \) be an expansion of a real closed field \({{{\mathcal {M}}}}\) by a dense subgroup G of \(\langle M^{>0}, \cdot \rangle \) with the Mann property. We prove that the induced structure on G by \({{{\mathcal {M}}}}\) eliminates imaginaries. As a consequence, every small set X definable in \({{{\mathcal {M}}}}\) can be definably embedded

The present work determines the arithmetic complexity of the index sets of u.c.e. families which are learnable according to various criteria of algorithmic learning. Specifically, we prove that the index set of codes for families that are TxtFex\(^a_b\)-learnable is \(\Sigma _4^0\)-complete and that the index set of TxtFex\(^*_*\)-learnable and the index set of TxtFext\(^*_*\)-learnable families are

Within ZFC, we develop a general technique to topologize trees that provides a uniform approach to topological completeness results in modal logic with respect to zero-dimensional Hausdorff spaces. Embeddings of these spaces into well-known extremally disconnected spaces then gives new completeness results for logics extending S4.2.

We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space \({{\mathbb {N}}}^{{\mathbb {N}}}\) to the natural numbers \({\mathbb {N}}\) which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction

Betweenness as a relation between three individual points has been widely studied in geometry and axiomatized by several authors in different contexts. The article proposes a more general notion of betweenness as a relation between three sets of points. The main technical result is a sound and complete logical system describing universal properties of this relation between sets of vertices of a graph

Answering a question raised by Shavrukov and Visser (Notre Dame J Form Log 55(4):569–582, 2014), we show that the lattice of \(\exists \Sigma ^\mathsf {b}_1\)-sentences (in the language of Buss’ weak arithmetical system \(\mathsf {S}^1_2\)) over any computable enumerable consistent extension T of \(\mathsf {S}^1_2\) is uniformly dense (in the sense of Definition 2). We also show that for every \(\mathcal

The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality \(\mathfrak {c}\). We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality \(\mathfrak {c}\), under different set-theoretic assumptions

We prove that, for any natural number \(n\ge 1\), we can find a finite alphabet \(\Sigma \) and a finitary language L over \(\Sigma \) accepted by a one-counter automaton, such that the \(\omega \)-power $$\begin{aligned} L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$ is \({\varvec{\Pi }}^0_n\)-complete. We prove a similar result for the class \({\varvec{\Sigma

Schur’s Lemma says that the endomorphism ring of a simple left R-module is a division ring. It plays a fundamental role to prove classical ring structure theorems like the Jacobson Density Theorem and the Wedderburn–Artin Theorem. We first define the endomorphism ring of simple left R-modules by their \(\Pi ^{0}_{1}\) subsets and show that Schur’s Lemma is provable in \(\mathrm RCA_{0}\). A ring R

Strong cell decomposition property has been proved in non-valuational weakly o-minimal expansions of ordered groups. In this note, we show that all o-minimal traces have strong cell decomposition property. Also after introducing the notion of irrational nonvaluational cut in arbitrary o-minimal structures, we show that every expansion of o-minimal structures by irrational nonvaluational cuts is an

We prove that every usco multimap \(\varPhi :X\rightarrow Y\) from a metrizable separable space X to a GO-space Y has an \(F_\sigma \)-measurable selection. On the other hand, for the split interval \({\ddot{\mathbb I}}\) and the projection \(P:{{\ddot{\mathbb I}}}^2\rightarrow \mathbb I^2\) of its square onto the unit square \(\mathbb I^2\), the usco multimap \({P^{-1}:\mathbb I^2\multimap {{\ddot{\mathbb

We use the compactness theorem of continuous logic to give a new proof that \(L^r([0,1]; {\mathbb {R}})\) isometrically embeds into \(L^p([0,1]; {\mathbb {R}})\) whenever \(1 \le p \le r \le 2\). We will also give a proof for the complex case. This will involve a new characterization of complex \(L^p\) spaces based on Banach lattices.

We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.

We introduce the Easton collapse and show that the two-stage iteration of Easton collapses gives a model in which the successor of a regular cardinal carries a strongly saturated filter. This allows one to get a model in which many successor cardinals carry saturated filters just by iterating Easton collapses.

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic \(\vdash \). We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic \(\vdash \) is related to the construction of Płonka sums of the matrix models of \(\vdash \). This observation allows to obtain a Hilbert-style axiomatization

This note concerns the model theoretic properties of logics extending the first-order logic with monadic (weak) second-order variables equipped with the stationarity quantifier. The eight variations of the strong downward Löwenheim–Skolem Theorem (SDLS) down to \(<\aleph _2\) for this logic with the interpretation of second-order variables as countable subsets of the structures are classified into

We show that assuming the determinacy of all games on reals, every set of reals is \(\Theta \) universally baire.

The interplay between ultrafilters and unbounded subsets of \({}^\omega \omega \) with the order \(<^*\) of strict eventual domination is studied. Among the tools are special kinds of non-principal (“free”) ultrafilters on \(\omega \). These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \(\subset ^*\) of almost inclusion. It

The model companion of the theory of fields with valuation and automorphism (of the pure field structure) exists. A counterexample shows that the theory of models of ACFA equipped with valuation is not this model companion.

In this note we study several topics related to the schema of local reflection \(\textsf {Rfn} (T)\) and its partial and relativized variants. Firstly, we introduce the principle of uniform reflection with \(\varSigma _n\)-definable parameters, establish its relationship with relativized local reflection principles and corresponding versions of induction with definable parameters. Using this schema

We investigate the classes of ideals for which the Egoroff’s theorem or the generalized Egoroff’s theorem holds between ideal versions of pointwise and uniform convergences. The paper is motivated by considerations of Korch (Real Anal Exchange 42(2):269–282, 2017).

We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a single \({\mathcal {L}}_{\omega _1,\omega }\)-sentence \(\psi \) that codes Kurepa trees to prove the following statements: (1) The spectrum of \(\psi \) is consistently equal to \([\aleph _0,\aleph _{\omega _1}]\) and also consistently equal to \([\aleph _0,2^{\aleph _1})\), where \(2^{\aleph _1}\) is

Anderson and Csima (Notre Dame J Form Log 55:245–264, 2014) defined a bounded jump operator for bounded-Turing reduction, and studied its basic properties. Anderson et al. (Arch Math Logic 56:507–521, 2017) constructed a low bounded-high set and conjectured that such sets cannot be computably enumerable (c.e. for short). Ng and Yu (Notre Dame J Form Log, to appear) proved that bounded-high c.e. sets

The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent

Answering a question of Sakai (Arch Math Logic 52(1–2):29–45, 2013), we show that the existence of an \(\omega _1\)-Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with \(\square _{\omega _1, 2}\). By a result of Donder (In: Set theory and model theory (Bonn, 1979), volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give

We consider a version of \(\kappa \)-Miller forcing on an uncountable cardinal \(\kappa \). We show that under \(2^{<\kappa } = \kappa \) this forcing collapses \(2^\kappa \) to \(\omega \) and adds a \(\kappa \)-Cohen real. The same holds under the weaker assumptions that \({{\,\mathrm{cf}\,}}(\kappa ) > \omega \), \(2^{2^{<\kappa }}= 2^\kappa \), and forcing with \(([\kappa ]^\kappa , \subseteq )\)

Tanaka (Ann Pure Appl Log 84:41–49, 1997) proved a powerful generalization of Friedman’s self-embedding theorem that states that given a countable nonstandard model \(({\mathcal {M}}, {\mathcal {A}})\) of the subsystem \(\mathrm {WKL}_{0}\) of second order arithmetic, and any element m of \({\mathcal {M}}\), there is a self-embedding j of \(({\mathcal {M}},{\mathcal {A}})\) onto a proper initial segment

Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility \(\leqslant _c\). This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the \(\Delta ^0_2\) case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable

In this paper, we prove results concerning the existence of proper end extensions of arbitrary models of fragments of Peano arithmetic (PA). In particular, we give alternative proofs that concern (a) a result of Clote (Fundam Math 127(2):163–170, 1986); (Fundam Math 158(3):301–302, 1998), on the end extendability of arbitrary models of \(\Sigma _n\)-induction, for \(n{\ge } 2\), and (b) the fact that

In this paper, we deal with a relationship among the law of excluded middle, the double negation elimination and the independence of premiss rule (\(\mathrm {IPR}\)) for (many-sorted) intuitionistic predicate logic. After giving a general machinery, we give, as corollaries, several examples of extensions of \(\mathbf {HA}\) and \(\mathbf {HA}^\omega \) which are closed under \(\mathrm {IPR}\) but do

After showing that \(AD +DC \) refutes \(\lozenge ^+_\kappa \) for all regular cardinals \(\kappa \ge \omega _1\), we present a diamond-plus principle \(\lozenge _{{{\mathbb {R}}} }^+\) concerning all subsets of \(\varTheta \). Using a forcing argument, we prove that \(\lozenge _{{{\mathbb {R}}} }^+\) holds in Steel’s core model \({{{{\mathbf {K}}}({{{{\mathbb {R}}} }})}}\), an inner model in which

In the paper Randomizations of Scattered Sentences, Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every \(L_{\omega _1\omega }\)-sentence with few separable

Let \(f(z)=\Sigma _{k\ge 0}a_{k}z^{k}\) be a transcendental entire function with real coefficients. The main purpose of this paper is to show that the restriction of f to \(\mathbb {R}\) is not definable in the ordered field of real numbers with restricted analytic functions, \(\mathbb {R}_{\text {an}}\). Furthermore, we show that there is \(\theta \in \mathbb {R}\) such that the function \(f(xe^{i\theta

Shoenfield’s completeness theorem (1959) states that every true first order arithmetical sentence has a recursive \(\omega \)-proof encodable by using recursive applications of the \(\omega \)-rule. For a suitable encoding of Gentzen style \(\omega \)-proofs, we show that Shoenfield’s completeness theorem applies to cut free \(\omega \)-proofs encodable by using primitive recursive applications of