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

We show that for every intermediate \(\Sigma ^0_2\) s-degree (i.e. a nonzero s-degree strictly below the s-degree of the complement of the halting set) there exists an incomparable \(\Pi ^0_1\) s-degree. (The same proof yields a similar result for other positive reducibilities as well, including enumeration reducibility.) As a consequence, for every intermediate \(\Pi ^0_2\) Q-degree (i.e. a nonzero

We compute the model-theoretic Grothendieck ring, \(K_0({\mathcal {Q}})\), of a dense linear order (DLO) with or without end points, \({\mathcal {Q}}=(Q,<)\), as a structure of the signature \(\{<\}\), and show that it is a quotient of the polynomial ring over \({\mathbb {Z}}\) generated by \({\mathbb {N}}_+\times (Q\sqcup \{-\infty \})\) by an ideal that encodes multiplicative relations of pairs of

We discuss the (non)effectivity of Arslanov’s completeness criterion. In particular, we show that a parameterized version, similar to the recursion theorem with parameters, fails. We also discuss the effectivity of another extension of the recursion theorem, namely Visser’s ADN theorem, as well as that of a joint generalization of the ADN theorem and Arslanov’s completeness criterion.

We introduce and investigate the notion of uniform Lyndon interpolation property (ULIP) which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including \(\mathbf{K}\), \(\mathbf{KB}\), \(\mathbf{GL}\) and \(\mathbf{Grz}\) enjoy ULIP. Our proofs are modifications of Visser’s proofs of uniform interpolation property

We introduce a new formulation of the axiom of dependent choice, which can be viewed as an abstract termination principle that in particular generalises recursive path orderings, the latter being fundamental tools used to establish termination of rewrite systems. We consider several variants of our termination principle, and relate them to general termination theorems in the literature.

In this paper, we consider, for a set \(\mathcal {A}\) of natural numbers, the following notions of finiteness FIN1: There are a natural number l and a bijection f between \(\{ x\in \mathbb {N}:xy)(x\in \mathcal {A})\); FIN5: It is not the case that \(\forall l\exists \mathcal {B}\subseteq \mathcal {A}(|\mathcal {B}|=l)\), and infiniteness INF1: There are not a natural number l and a bijection f between

For an integer \(n\ge 2\), Ramsey Choice\(\mathsf {RC}_{n}\) is the weak choice principle “every infinite setxhas an infinite subset y such that\([y]^{n}\) (the set of alln-element subsets of y) has a choice function”, and \(\mathsf {C}_{n}^{-}\) is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for

A relational structure \({{\mathbb {X}}}\) is called reversible iff each bijective homomorphism from \({{\mathbb {X}}}\) onto \({{\mathbb {X}}}\) is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations

A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are

We study variants of Buss’s theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on \(\hat{\varPi }^{b}_i\) induction schemes, which were so far neglected in the literature. We present inclusions and conservation results between the systems (including a witnessing theorem for \(T^i_2\)

For a computable structure \({\mathcal {A}}\), if there is a computable infinitary Scott sentence, then the complexity of this sentence gives an upper bound for the complexity of the index set \(I({\mathcal {A}})\). If we can also show that \(I({\mathcal {A}})\) is m-complete at that level, then there is a correspondence between the complexity of the index set and the complexity of a Scott sentence

We prove that Martin’s Axiom implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while \(\mathfrak b=\mathfrak c\) is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel’s conjecture.

Suppose \(p \ge 1\) is a computable real. We extend previous work of Clanin, Stull, and McNicholl by determining the degrees of categoricity of the separable \(L^p\) spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we ascertain the complexity of associated projection maps.

In his seminal paper introducing the fine structure of L, Jensen (Ann Math Log 4:229–308, 1972) proved that under \(V=L\) any regular cardinal that reflects stationary sets is weakly compact. In this paper we give a new proof of Jensen’s result that is straight-forward and accessible to those without a knowledge of Jensen’s fine structure theory. The proof here instead uses hyperfine structure, a very

We show that if the structural rules are admissible over a set \(\mathcal{R}\) of atomic rules, then they are admissible in the sequent calculus obtained by adding the rules in \(\mathcal{R}\) to the multisuccedent minimal and intuitionistic \(\mathbf{G3}\) calculi as well as to the classical one. Two applications to pure logic and to the sequent calculus with equality are presented.

We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \(\mathsf {cov}(\mathcal {M})\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are

Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the \(\limsup \) of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the \(\limsup \) rule with an ‘eventually constant’ rule: at each

It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on \({\aleph }_{\omega +1}\) which is generated by fewer than \({2}^{{\aleph }_{\omega +1}}\) sets.

Let T be a simple L-theory and let \(T^-\) be a reduct of T to a sublanguage \(L^-\) of L. For variables x, we call an \(\emptyset \)-invariant set \(\Gamma (x)\) in \({\mathcal {C}}\) a universal transducer if for every formula \(\phi ^-(x,y)\in L^-\) and every a,$$\begin{aligned} \phi ^-(x,a)\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma (x)\wedge \phi ^-(x,a)\ L\text{-forks }

We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we say the property is recursively recognizable within some class of descriptions. When there is not, we ask how difficult it is to detect the property in an algorithmic

We consider two variants of transfinite induction, one with monotonicity assumption on the predicate and one with the induction hypothesis only for cofinally many below. The latter can be seen as a transfinite analogue of the successor induction, while the usual transfinite induction is that of cumulative induction. We calculate the supremum of ordinals along which these schemata for \(\varDelta _0\)

In this note the proof-theoretic ordinal of the well-ordering principle for the normal functions \(\mathsf {g}\) on ordinals is shown to be equal to the least fixed point of \(\mathsf {g}\). Moreover corrections to the previous paper (Arai in Arch Math Log 57:649–664, 2017) are made.