A maximal chain in a tree is called a path, and a tree is called bounded when all its paths contain leaves. This paper concerns itself with first-order theories of bounded trees. We establish some sufficient conditions for the existence of bounded end-extensions that are also partial elementary extensions of a given tree. As an application of tree boundedness, we obtain a conditional axiomatisation

We will give a two-sort system which axiomatizes winning strategies for the combinatorial game Node Kayles. It is shown that our system captures alternating polynomial time reasonings in the sense that the provably total functions of the theory corresponds to those computable in APTIME. We will also show that our system is equivalently axiomatized by Sprague–Grundy theorem which states that any Node

Hjorth (Variations on Scott, 1998; The fine structure and Borel complexity of orbits, 2010) introduced a Scott analysis for general Polish group actions, and asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank—namely, if the orbit equivalence relation is Borel, then Hjorth ranks are bounded. We answer Hjorth’s question positively. As a corollary we prove

The goal of this paper is to define and study the notion of infinity in the framework of finitely supported structures, presenting new properties of infinite cardinalities. Some of these properties are extended from the non-atomic Zermelo–Fraenkel set theory to the world of atomic objects with finite support, while other properties are specific to finitely supported structures. We compare alternative

We are interested in the possible sets of cardinalities of branches of Kurepa trees in models of ZFC \(+\) CH. In this paper we present a sufficient condition (for sets of cardinals) to be consistently the set of cardinalities of branches of Kurepa trees.

In strategic situations, agents base actions on knowledge and beliefs. This includes knowledge about others’ strategies and preferences over strategy profiles, but also about other external factors. Bernheim and Pearce in 1984 independently defined the game theoretic solution concept of rationalizability, which is built on the premise that rational agents will only take actions that are the best response

There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jónsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect

We analyze the expressive resources of \(\mathrm {IF}\) logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular \(\mathrm {IF}\) sentences, this amounts to the study of the fragment of \(\mathrm {IF}\) logic which is individuated by the game-theoretical property of action recall (AR). We prove that the fragment of prenex AR sentences can express

Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism (advocated by Heine), game formalism (advocated by Thomae) and instrumental formalism (advocated by Hilbert)

In this paper we study the variety of order Hilbert algebras, which is the equivalent algebraic semantics of the order implicational calculus of Bull (J Symb Logic, 29:33–34, 1964).

We continue the study of Rosenthal families initiated by Damian Sobota. We show that every Rosenthal filter is the intersection of a finite family of ultrafilters that are pairwise incomparable in the Rudin-Keisler partial ordering of ultrafilters. We introduce a property of filters, called an \(r\)-filter, properly between a selective filter and a \(p\)-filter. We prove that every \(r\)-ultrafilter

This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal

We present a relatively simple description of binary, definable subsets of models of weakly quasi-o-minimal theories. In particular, we closely describe definable linear orders and prove a weak version of the monotonicity theorem. We also prove that weak quasi-o-minimality of a theory with respect to one definable linear order implies weak quasi-o-minimality with respect to any other such order.

We prove a compactness theorem for pseudopower operations of the form \({{\,\mathrm{pp}\,}}_{\Gamma (\mu ,\sigma )}(\mu )\) where \(\aleph _0<\sigma ={{\,\mathrm{cf}\,}}(\sigma )\le {{\,\mathrm{cf}\,}}(\mu )\). Our main tool is a result that has Shelah’s cov versus pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory

We show that the tree property, stationary reflection and the failure of approachability at \(\kappa ^{++}\) are consistent with \(\mathfrak {u}(\kappa )= \kappa ^+ < 2^\kappa \), where \(\kappa \) is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if \(\lambda \) is a regular cardinal, then stationary reflection at \(\lambda ^+\) is indestructible

We show: (1) for every noncomputable c.e. incomplete c-degree, there exists a nonspeedable c-degree incomparable with it; (2) The c-degree of a hypersimple set includes an infinite collection of \(c_1\)-degrees linearly ordered under \(\le _{c_1}\) with order type of the integers and consisting entirely of hypersimple sets; (3) there exist two c.e. sets having no c.e. least upper bound in the \(c_1\)-reducibility

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.