-
DUALITY FOR COALGEBRAS FOR VIETORIS AND MONADICITY J. Symb. Log. (IF 0.6) Pub Date : 2024-03-04 MARCO ABBADINI, IVAN DI LIBERTI
We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over $\mathsf {Set}$. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski
-
A UNIFIED APPROACH TO HINDMAN, RAMSEY, AND VAN DER WAERDEN SPACES J. Symb. Log. (IF 0.6) Pub Date : 2024-02-12 RAFAŁ FILIPÓW, KRZYSZTOF KOWITZ, ADAM KWELA
For many years, there have been conducting research (e.g., by Bergelson, Furstenberg, Kojman, Kubiś, Shelah, Szeptycki, Weiss) into sequentially compact spaces that are, in a sense, topological counterparts of some combinatorial theorems, for instance, Ramsey’s theorem for coloring graphs, Hindman’s finite sums theorem, and van der Waerden’s arithmetical progressions theorem. These spaces are defined
-
A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC J. Symb. Log. (IF 0.6) Pub Date : 2024-02-06 LONGYUN DING, XU WANG
In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by $\alpha $-CLI and L-$\alpha $-CLI where $\alpha $ is a countable ordinal. We establish three results: (1) G is $0$-CLI iff $G=\{1_G\}$; (2) G is $1$-CLI iff G admits a compatible complete two-sided invariant metric; and (3) G is
-
BETWEENNESS ALGEBRAS J. Symb. Log. (IF 0.6) Pub Date : 2024-02-06 IVO DÜNTSCH, RAFAŁ GRUSZCZYŃSKI, PAULA MENCHÓN
We introduce and study a class of betweenness algebras—Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which makes our work applicable to a wide range of betweenness structures studied in the literature. On the algebraic side, we work with two operators of possibility and
-
REGAININGLY APPROXIMABLE NUMBERS AND SETS J. Symb. Log. (IF 0.6) Pub Date : 2024-01-22 PETER HERTLING, RUPERT HÖLZL, PHILIP JANICKI
We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n < 2^{-n}$ for infinitely many ${n \in \mathbb {N}}$. We also call a set $A\subseteq \mathbb {N}$ regainingly approximable if it is c.e. and the strongly left-computable number $2^{-A}$ is regainingly approximable. We
-
POLISH SPACE PARTITION PRINCIPLES AND THE HALPERN–LÄUCHLI THEOREM J. Symb. Log. (IF 0.6) Pub Date : 2024-01-19 CHRIS LAMBIE-HANSON, ANDY ZUCKER
The Halpern–Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern–Läuchli theorem, and the same forcing from Harrington’s proof
-
NON-TRIVIAL HIGHER HOMOTOPY OF FIRST-ORDER THEORIES J. Symb. Log. (IF 0.6) Pub Date : 2024-01-11 TIM CAMPION, JINHE YE
Let T be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf {Mod}(T)$ to be homotopically equivalent to $\mathbb {CP}^\infty $. In particular, $\pi _2(\lvert \mathsf {Mod}(T)\rvert )=\mathbb {Z}$, which answers a question in our previous work. The computation is based on Connes’ cycle category $\Lambda $.
-
DEGREE OF SATISFIABILITY IN HEYTING ALGEBRAS J. Symb. Log. (IF 0.6) Pub Date : 2024-01-09 BENJAMIN MERLIN BUMPUS, ZOLTAN A. KOCSIS
We investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability
-
THE BAIRE CLOSURE AND ITS LOGIC J. Symb. Log. (IF 0.6) Pub Date : 2024-01-05 G. BEZHANISHVILI, D. FERNÁNDEZ-DUQUE
The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote $\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $\mathsf {S5}$, and prove soundness and strong completeness
-
FORBIDDEN INDUCED SUBGRAPHS AND THE ŁOŚ–TARSKI THEOREM J. Symb. Log. (IF 0.6) Pub Date : 2024-01-04 YIJIA CHEN, JÖRG FLUM
Let $\mathscr {C}$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś–Tarski Theorem from classical model theory implies that $\mathscr {C}$ is definable in first-order logic by a sentence $\varphi $ if and only if $\mathscr {C}$ has a finite set of forbidden induced finite subgraphs. This result provides a powerful tool to show nontrivial characterizations
-
NOTE ON J. Symb. Log. (IF 0.6) Pub Date : 2024-01-04 SEAN CODY
A short core model induction proof of $\mathsf {AD}^{L(\mathbb {R})}$ from $\mathsf {TD} + \mathsf {DC}_{\mathbb {R}}$.
-
STRONG MEASURE ZERO SETS ON INACCESSIBLE J. Symb. Log. (IF 0.6) Pub Date : 2024-01-03 NICK STEVEN CHAPMAN, JOHANNES PHILIPP SCHÜRZ
We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^\kappa $ for $\kappa $ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of $$\begin{align*}|2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \textrm{ is strong measure zero if and only if } |X| \leq \kappa^+. \end{align*}$$Furthermore
-
PARTITION OF LARGE SUBSETS OF SEMIGROUPS J. Symb. Log. (IF 0.6) Pub Date : 2024-01-03 TENG ZHANG
It is known that in an infinite very weakly cancellative semigroup with size $\kappa $, any central set can be partitioned into $\kappa $ central sets. Furthermore, if $\kappa $ contains $\lambda $ almost disjoint sets, then any central set contains $\lambda $ almost disjoint central sets. Similar results hold for thick sets, very thick sets and piecewise syndetic sets. In this article, we investigate
-
THE STRONG AND SUPER TREE PROPERTIES AT SUCCESSORS OF SINGULAR CARDINALS J. Symb. Log. (IF 0.6) Pub Date : 2023-12-22 WILLIAM ADKISSON
The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa $ is strongly compact if and only if the strong tree property holds at $\kappa $, and supercompact if and only if ITP holds at $\kappa $. We present several results motivated
-
INTROENUMERABILITY, AUTOREDUCIBILITY, AND RANDOMNESS J. Symb. Log. (IF 0.6) Pub Date : 2023-12-12 ANG LI
We define $\Psi $-autoreducible sets given an autoreduction procedure $\Psi $. Then, we show that for any $\Psi $, a measurable class of $\Psi $-autoreducible sets has measure zero. Using this, we show that classes of cototal, uniformly introenumerable, introenumerable, and hyper-cototal enumeration degrees all have measure zero. By analyzing the arithmetical complexity of the classes of cototal sets
-
A BOREL MAXIMAL COFINITARY GROUP J. Symb. Log. (IF 0.6) Pub Date : 2023-12-11 HAIM HOROWITZ, SAHARON SHELAH
We construct a Borel maximal cofinitary group.
-
DEGREE SPECTRA OF HOMEOMORPHISM TYPE OF COMPACT POLISH SPACES J. Symb. Log. (IF 0.6) Pub Date : 2023-12-11 MATHIEU HOYRUP, TAKAYUKI KIHARA, VICTOR SELIVANOV
A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $\mathbf {0}'$-computable low$_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $n\geq 2$, there exists a Polish space
-
DIVIDING LINES BETWEEN POSITIVE THEORIES J. Symb. Log. (IF 0.6) Pub Date : 2023-12-06 ANNA DMITRIEVA, FRANCESCO GALLINARO, MARK KAMSMA
We generalise the properties $\mathsf {OP}$, $\mathsf {IP}$, k-$\mathsf {TP}$, $\mathsf {TP}_{1}$, k-$\mathsf {TP}_{2}$, $\mathsf {SOP}_{1}$, $\mathsf {SOP}_{2}$, and $\mathsf {SOP}_{3}$ to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level
-
DIVERGENT MODELS WITH THE FAILURE OF THE CONTINUUM HYPOTHESIS J. Symb. Log. (IF 0.6) Pub Date : 2023-12-06 NAM TRANG
We construct divergent models of $\mathsf {AD}^+$ along with the failure of the Continuum Hypothesis ($\mathsf {CH}$) under various assumptions. Divergent models of $\mathsf {AD}^+$ play an important role in descriptive inner model theory; all known analyses of HOD in $\mathsf {AD}^+$ models (without extra iterability assumptions) are carried out in the region below the existence of divergent models
-
TWO EXAMPLES CONCERNING EXISTENTIAL UNDECIDABILITY IN FIELDS J. Symb. Log. (IF 0.6) Pub Date : 2023-11-23 PHILIP DITTMANN
We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe–Fehm in a strong way. Along the way, we construct an existentially decidable field of positive characteristic with an existentially undecidable finite extension, modifying a construction due to Kesavan
-
ON MODEL-THEORETIC CONNECTED GROUPS J. Symb. Log. (IF 0.6) Pub Date : 2023-11-14 JAKUB GISMATULLIN
We introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected
-
ON UNSUPERSTABLE THEORIES IN GDST J. Symb. Log. (IF 0.6) Pub Date : 2023-11-06 MIGUEL MORENO
We study the $\kappa $-Borel-reducibility of isomorphism relations of complete first-order theories by using coloured trees. Under some cardinality assumptions, we show the following: For all theories T and T’, if T is classifiable and T’ is unsuperstable, then the isomorphism of models of T’ is strictly above the isomorphism of models of T with respect to $\kappa $-Borel-reducibility.
-
ON THE ZARISKI TOPOLOGY ON ENDOMORPHISM MONOIDS OF OMEGA-CATEGORICAL STRUCTURES J. Symb. Log. (IF 0.6) Pub Date : 2023-10-31 MICHAEL PINSKER, CLEMENS SCHINDLER
The endomorphism monoid of a model-theoretic structure carries two interesting topologies: on the one hand, the topology of pointwise convergence induced externally by the action of the endomorphisms on the domain via evaluation; on the other hand, the Zariski topology induced within the monoid by (non-)solutions to equations. For all concrete endomorphism monoids of $\omega $-categorical structures
-
STATIONARY REFLECTION AND THE FAILURE OF THE SCH J. Symb. Log. (IF 0.6) Pub Date : 2023-10-27 OMER BEN-NERIA, YAIR HAYUT, SPENCER UNGER
In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu $ such that the singular cardinal hypothesis fails at $\nu $ and every collection of fewer than $\operatorname {\mathrm {cf}}(\nu )$ stationary subsets of $\nu ^{+}$ reflects simultaneously. For $\operatorname {\mathrm {cf}}(\nu )> \omega $, this situation was not previously known to
-
NOTES ON SACKS’ SPLITTING THEOREM J. Symb. Log. (IF 0.6) Pub Date : 2023-10-26 KLAUS AMBOS-SPIES, ROD G. DOWNEY, MARTIN MONATH, KENG MENG NG
We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$-c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $-c.a. as shown
-
ON EASTON SUPPORT ITERATION OF PRIKRY-TYPE FORCING NOTIONS J. Symb. Log. (IF 0.6) Pub Date : 2023-10-25 MOTI GITIK, EYAL KAPLAN
We consider of constructing normal ultrafilters in extensions are here Easton support iterations of Prikry-type forcing notions. New ways presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.
-
MAXIMAL STABLE QUOTIENTS OF INVARIANT TYPES IN NIP THEORIES J. Symb. Log. (IF 0.6) Pub Date : 2023-10-25 KRZYSZTOF KRUPIŃSKI, ADRIÁN PORTILLO
For a NIP theory T, a sufficiently saturated model ${\mathfrak C}$ of T, and an invariant (over some small subset of ${\mathfrak C}$) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from ${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main
-
PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS J. Symb. Log. (IF 0.6) Pub Date : 2023-10-20 DANIEL MAX HOFFMANN, PIOTR KOWALSKI
We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories
-
BIG IN REVERSE MATHEMATICS: MEASURE AND CATEGORY J. Symb. Log. (IF 0.6) Pub Date : 2023-10-17 SAM SANDERS
The smooth development of large parts of mathematics hinges on the idea that some sets are ‘small’ or ‘negligible’ and can therefore be ignored for a given purpose. The perhaps most famous smallness notion, namely ‘measure zero’, originated with Lebesgue, while a second smallness notion, namely ‘meagre’ or ‘first category’, originated with Baire around the same time. The associated Baire category theorem
-
UNDEFINABILITY OF MULTIPLICATION IN PRESBURGER ARITHMETIC WITH SETS OF POWERS J. Symb. Log. (IF 0.6) Pub Date : 2023-10-10 CHRIS SCHULZ
We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number
-
FINITE UNDECIDABILITY IN NIP FIELDS J. Symb. Log. (IF 0.6) Pub Date : 2023-10-04 BRIAN TYRRELL
A field K in a ring language $\mathcal {L}$ is finitely undecidable if $\mbox {Cons}(T)$ is undecidable for every nonempty finite $T \subseteq {\mathtt{Th}}(K; \mathcal {L})$. We extend a construction of Ziegler and (among other results) use a first-order classification of Anscombe and Jahnke to prove every NIP henselian nontrivially valued field is finitely undecidable. We conclude (assuming the NIP
-
WEAK INDESTRUCTIBILITY AND REFLECTION J. Symb. Log. (IF 0.6) Pub Date : 2023-10-04 JAMES HOLLAND
We establish an equiconsistency between (1) weak indestructibility for all $\kappa +2$-degrees of strength for cardinals $\kappa $ in the presence of a proper class of strong cardinals, and (2) a proper class of cardinals that are strong reflecting strongs. We in fact get weak indestructibility for degrees of strength far beyond $\kappa +2$, well beyond the next inaccessible limit of measurables (of
-
WEAK WELL ORDERS AND FRAÏSSÉ’S CONJECTURE J. Symb. Log. (IF 0.6) Pub Date : 2023-09-27 ANTON FREUND, DAVIDE MANCA
The notion of countable well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with Fraïssé’s conjecture, which has been proved by Laver. We also fill a small gap in Shore’s proof that Fraïssé’s conjecture implies arithmetic transfinite recursion
-
A PROOF COMPLEXITY CONJECTURE AND THE INCOMPLETENESS THEOREM J. Symb. Log. (IF 0.6) Pub Date : 2023-09-19 JAN KRAJÍČEK
Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of $g_T$ intersects all infinite ${\mbox {NP}}$ sets (i.e., whether it is a proof complexity generator hard for all
-
COMPUTABLE TOPOLOGICAL GROUPS J. Symb. Log. (IF 0.6) Pub Date : 2023-09-18 HEER TERN KOH, ALEXANDER G. MELNIKOV, KENG MENG NG
We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare our results with similar results in computable topology.
-
FIRST-ORDER HOMOTOPICAL LOGIC J. Symb. Log. (IF 0.6) Pub Date : 2023-09-18 JOSEPH HELFER
We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from [8] that any Grothendieck
-
THORN FORKING, WEAK NORMALITY, AND THEORIES WITH SELECTORS J. Symb. Log. (IF 0.6) Pub Date : 2023-09-11 DANIEL MAX HOFFMANN, ANAND PILLAY
We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to Corollary 4.2 from that paper, and in the process discuss “theories with selectors.”
-
COLORING ISOSCELES TRIANGLES IN CHOICELESS SET THEORY J. Symb. Log. (IF 0.6) Pub Date : 2023-09-11 YUXIN ZHOU
It is consistent relative to an inaccessible cardinal that ZF+DC holds, and the hypergraph of isosceles triangles on $\mathbb {R}^2$ has countable chromatic number while the hypergraph of isosceles triangles on $\mathbb {R}^3$ has uncountable chromatic number.
-
ON COHEN AND PRIKRY FORCING NOTIONS J. Symb. Log. (IF 0.6) Pub Date : 2023-09-11 TOM BENHAMOU, MOTI GITIK
(1) We show that it is possible to add $\kappa ^+$-Cohen subsets to $\kappa $ with a Prikry forcing over $\kappa $. This answers a question from [9]. (2) A strengthening of non-Galvin property is introduced. It is shown to be consistent using a single measurable cardinal which improves a previous result by S. Garti, S. Shelah, and the first author [5]. (3) A situation with Extender-based Prikry forcings
-
USUBA’S PRINCIPLE CAN FAIL AT SINGULAR CARDINALS J. Symb. Log. (IF 0.6) Pub Date : 2023-09-07 MOHAMMAD GOLSHANI, SAHARON SHELAH
We answer a question of Usuba by showing that the combinatorial principle $\mathrm {UB}_\lambda $ can fail at a singular cardinal. Furthermore, $\lambda $ can be taken to be $\aleph _\omega .$
-
MORE ON HALFWAY NEW CARDINAL CHARACTERISTICS J. Symb. Log. (IF 0.6) Pub Date : 2023-09-07 BARNABÁS FARKAS, LUKAS DANIEL KLAUSNER, MARC LISCHKA
We continue investigating variants of the splitting and reaping numbers introduced in [4]. In particular, answering a question raised there, we prove the consistency of and of . Moreover, we discuss their natural generalisations $\mathfrak {s}_{\rho }$ and $\mathfrak {r}_{\rho }$ for $\rho \in (0,1)$, and show that $\mathfrak {r}_{\rho }$ does not depend on $\rho $.
-
WEAK HEIRS, COHEIRS, AND THE ELLIS SEMIGROUPS J. Symb. Log. (IF 0.6) Pub Date : 2023-09-07 ADAM MALINOWSKI, LUDOMIR NEWELSKI
Assume $G\prec H$ are groups and ${\cal A}\subseteq {\cal P}(G),\ {\cal B}\subseteq {\cal P}(H)$ are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of the G-flow $S({\cal A})$ and the H-flow $S({\cal B})$ . We apply these results in the model theoretic context. Namely, assume G is a group definable
-
PROOF SYSTEMS FOR TWO-WAY MODAL MU-CALCULUS J. Symb. Log. (IF 0.6) Pub Date : 2023-09-04 BAHAREH AFSHARI, SEBASTIAN ENQVIST, GRAHAM E. LEIGH, JOHANNES MARTI, YDE VENEMA
We present sound and complete sequent calculi for the modal mu-calculus with converse modalities, aka two-way modal mu-calculus. Notably, we introduce a cyclic proof system wherein proofs can be represented as finite trees with back-edges, i.e., finite graphs. The sequent calculi incorporate ordinal annotations and structural rules for managing them. Soundness is proved with relative ease as is the
-
STABILITY RESULTS ASSUMING TAMENESS, MONSTER MODEL, AND CONTINUITY OF NONSPLITTING J. Symb. Log. (IF 0.6) Pub Date : 2023-08-07 SAMSON LEUNG
Assuming the existence of a monster model, tameness, and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>\operatorname {LS}(\mathbf {K})$ be a regular stability cardinal and let $\chi $ be the local character of $\mu $-nonsplitting. The following holds: 1. When $\mu $-nonforking is restricted to $(\mu ,\geq \chi )$-limit models ordered
-
ORTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS J. Symb. Log. (IF 0.6) Pub Date : 2023-08-01 ALESSANDRO BERARDUCCI, PANTELIS E. ELEFTHERIOU, MARCELLO MAMINO
Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove
-
FRACTAL DIMENSIONS OF k-AUTOMATIC SETS J. Symb. Log. (IF 0.6) Pub Date : 2023-07-25 ALEXI BLOCK GORMAN, CHRIS SCHULZ
This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of “k-automatic” sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy
-
COFINAL TYPES BELOW J. Symb. Log. (IF 0.6) Pub Date : 2023-07-24 ROY SHALEV
It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality $\leq \aleph _n$ is at least $c_{n+2}$ , the $(n+2)$ -Catalan number. Moreover, the class $\mathcal D_{\aleph _n}$ of directed sets of cardinality $\leq \aleph _n$ contains an isomorphic copy of the poset of Dyck $(n+2)$ -paths. Furthermore, we give a complete description whether two successive
-
ABELIAN GROUPS DEFINABLE IN p-ADICALLY CLOSED FIELDS J. Symb. Log. (IF 0.6) Pub Date : 2023-07-18 WILL JOHNSON, NINGYUAN YAO
Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable
-
SELF-DIVISIBLE ULTRAFILTERS AND CONGRUENCES IN J. Symb. Log. (IF 0.6) Pub Date : 2023-07-17 MAURO DI NASSO, LORENZO LUPERI BAGLINI, ROSARIO MENNUNI, MORENO PIEROBON, MARIACLARA RAGOSTA
We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv _w$ introduced by Šobot is an equivalence relation on $\beta {\mathbb Z}$. We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if $\equiv _w$ coincides with the strong congruence relation $\mathrel {\equiv
-
DEDEKIND-FINITE CARDINALS HAVING COUNTABLE PARTITIONS J. Symb. Log. (IF 0.6) Pub Date : 2023-07-17 SUPAKUN PANASAWATWONG, JOHN KENNETH TRUSS
We study the possible structures which can be carried by sets which have no countable subset, but which fail to be ‘surjectively Dedekind finite’, in two possible senses, that there is surjection to $\omega $, or alternatively, that there is a surjection to a proper superset.
-
CLASSICAL DETERMINATE TRUTH I J. Symb. Log. (IF 0.6) Pub Date : 2023-07-05 KENTARO FUJIMOTO, VOLKER HALBACH
We introduce and analyze a new axiomatic theory $\mathsf {CD}$ of truth. The primitive truth predicate can be applied to sentences containing the truth predicate. The theory is thoroughly classical in the sense that $\mathsf {CD}$ is not only formulated in classical logic, but that the axiomatized notion of truth itself is classical: The truth predicate commutes with all quantifiers and connectives
-
ELEMENTARY EQUIVALENCE IN POSITIVE LOGIC VIA PRIME PRODUCTS J. Symb. Log. (IF 0.6) Pub Date : 2023-07-05 TOMMASO MORASCHINI, JOHANN J. WANNENBURG, KENTARO YAMAMOTO
We introduce prime products as a generalization of ultraproducts for positive logic. Prime products are shown to satisfy a version of Łoś’s Theorem restricted to positive formulas, as well as the following variant of the Keisler Isomorphism Theorem: under the generalized continuum hypothesis, two models have the same positive theory if and only if they have isomorphic prime powers of ultrapowers.
-
COMPUTABLE VS DESCRIPTIVE COMBINATORICS OF LOCAL PROBLEMS ON TREES J. Symb. Log. (IF 0.6) Pub Date : 2023-07-04 FELIX WEILACHER
We study the position of the computable setting in the “common theory of locality” developed in [4, 5] for local problems on $\Delta $-regular trees, $\Delta \in \omega $. We show that such a problem admits a computable solution on every highly computable $\Delta $-regular forest if and only if it admits a Baire measurable solution on every Borel $\Delta $-regular forest. We also show that if such
-
ON EQUIVALENCE RELATIONS INDUCED BY LOCALLY COMPACT ABELIAN POLISH GROUPS J. Symb. Log. (IF 0.6) Pub Date : 2023-06-30 LONGYUN DING, YANG ZHENG
Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^{\omega }/c(G)$ , where $c(G)$ is the group of all convergent sequences in G. The connected component of the identity of a Polish group G is denoted by $G_0$ . Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq _B E(H)$ , then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker (S)$ is
-
THE STRUCTURAL COMPLEXITY OF MODELS OF ARITHMETIC J. Symb. Log. (IF 0.6) Pub Date : 2023-06-29 ANTONIO MONTALBÁN, DINO ROSSEGGER
We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega $ and that non-standard models of true arithmetic must have Scott rank greater than $\omega $. Other than that there are no restrictions. By giving a reduction via $\Delta ^{\mathrm {in}}_{1}$ bi-interpretability from the class of linear orderings to
-
DEFINABLE J. Symb. Log. (IF 0.6) Pub Date : 2023-06-29 PABLO ANDÚJAR GUERRERO
Let $\mathcal {S}$ be a family of nonempty sets with VC-codensity less than $2$. We prove that, if $\mathcal {S}$ has the $(\omega ,2)$-property (for any infinitely many sets in $\mathcal {S}$, at least two among them intersect), then $\mathcal {S}$ can be partitioned into finitely many subfamilies, each with the finite intersection property. If $\mathcal {S}$ is definable in some first-order structure
-
COMPLETE BIPARTITE PARTITION RELATIONS IN COHEN EXTENSIONS J. Symb. Log. (IF 0.6) Pub Date : 2023-06-29 DÁVID UHRIK
We investigate the effect of adding $\omega _2$ Cohen reals on graphs on $\omega _2$, in particular we show that $\omega _2 \to (\omega _2, \omega : \omega )^2$ holds after forcing with $\mathsf {Add}(\omega , \omega _2)$ in a model of $\mathsf {CH}$. We also prove that this result is in a certain sense optimal as $\mathsf {Add}(\omega , \omega _2)$ forces that $\omega _2 \not \to (\omega _2, \omega
-
AN EGOCENTRIC LOGIC OF KNOWING HOW TO TELL THEM APART J. Symb. Log. (IF 0.6) Pub Date : 2023-06-29 PAVEL NAUMOV, JIA TAO
Traditionally, the formulae in modal logic express properties of possible worlds. Prior introduced “egocentric” logics that capture properties of agents rather than of possible worlds. In such a setting, the article proposes the modality “know how to tell apart” and gives a complete logical system describing the interplay between this modality and the knowledge modality. An important contribution of
-
B-SYSTEMS AND C-SYSTEMS ARE EQUIVALENT J. Symb. Log. (IF 0.6) Pub Date : 2023-06-29 BENEDIKT AHRENS, JACOPO EMMENEGGER, PAIGE RANDALL NORTH, EGBERT RIJKE
C-systems were defined by Cartmell as models of generalized algebraic theories. B-systems were defined by Voevodsky in his quest to formulate and prove an initiality conjecture for type theories. They play a crucial role in Voevodsky’s construction of a syntactic C-system from a term monad. In this work, we construct an equivalence between the category of C-systems and the category of B-systems, thus
-
BIG IN REVERSE MATHEMATICS: THE UNCOUNTABILITY OF THE REALS J. Symb. Log. (IF 0.6) Pub Date : 2023-06-29 SAM SANDERS
The uncountability of $\mathbb {R}$ is one of its most basic properties, known far outside of mathematics. Cantor’s 1874 proof of the uncountability of $\mathbb {R}$ even appears in the very first paper on set theory, i.e., a historical milestone. In this paper, we study the uncountability of ${\mathbb R}$ in Kohlenbach’s higher-order Reverse Mathematics (RM for short), in the guise of the following