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Preservation of NATP J. Math. Log. (IF 0.9) Pub Date : 2023-12-20 Jinhoo Ahn, Joonhee Kim, Hyoyoon Lee, Junguk Lee
We prove the preservation theorems for NATP; many of them extend the previously established preservation results for other model-theoretic tree properties. Using them, we also furnish proper examples of NATP theories which are simultaneously TP2 and SOP. First, we show that NATP is preserved by the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation
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How far is almost strong compactness from strong compactness J. Math. Log. (IF 0.9) Pub Date : 2023-11-21 Zhixing You, Jiachen Yuan
Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg gives a positive answer in the case SCH holds from below and the least almost strongly compact cardinal has uncountable cofinality. In this paper, we give a negative
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Enriching a predicate and tame expansions of the integers J. Math. Log. (IF 0.9) Pub Date : 2023-11-21 Gabriel Conant, Christian d’Elbée, Yatir Halevi, Léo Jimenez, Silvain Rideau-Kikuchi
Given a structure ℳ and a stably embedded ∅-definable set Q, we prove tameness preservation results when enriching the induced structure on Q by some further structure 𝒬. In particular, we show that if T=Th(ℳ) and Th(𝒬) are stable (respectively, superstable, ω-stable), then so is the theory T[𝒬] of the enrichment of ℳ by 𝒬. Assuming simplicity of T, elimination of hyperimaginaries and a further
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Turing independence and Baire category J. Math. Log. (IF 0.9) Pub Date : 2023-11-18 Ashutosh Kumar, Saharon Shelah
We show that it is relatively consistent with ZFC that there is a non-meager set of reals X such that for every non-meager Y⊆X, there exist distinct x,y,z∈Y such that z is computable from the Turing join of x and y.
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Halin’s infinite ray theorems: Complexity and reverse mathematics J. Math. Log. (IF 0.9) Pub Date : 2023-11-11 James S. Barnes, Jun Le Goh, Richard A. Shore
Halin in 1965 proved that if a graph has n many pairwise disjoint rays for each n then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those
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The Biggest Five of Reverse Mathematics J. Math. Log. (IF 0.9) Pub Date : 2023-10-06 Dag Normann, Sam Sanders
The aim of Reverse Mathematics (RM for short) is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable
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Games with filters I J. Math. Log. (IF 0.9) Pub Date : 2023-08-29 Matthew Foreman, Menachem Magidor, Martin Zeman
This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length ω on κ is equivalent to weak compactness. Winning the game of length 2κ is equivalent to κ being measurable. We show that for games of intermediate length γ, II winning implies the existence of
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The variety of projections of a tree Prikry forcing J. Math. Log. (IF 0.9) Pub Date : 2023-08-29 Tom Benhamou, Moti Gitik, Yair Hayut
We study which κ-distributive forcing notions of size κ can be embedded into tree Prikry forcing notions with κ-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a κ-distributive forcing notion of size κ be extended to a κ-complete ultrafilter.
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Valued fields with a total residue map J. Math. Log. (IF 0.9) Pub Date : 2023-08-24 Konstantinos Kartas
When k is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory of Algebra and Arithmetic, Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that the total residue map res:k((t))→k, which picks out the constant term of the Laurent series, is definable in the language
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MAω1(S)[S] does not imply 𝒦2 J. Math. Log. (IF 0.9) Pub Date : 2023-08-17 Yinhe Peng, Liuzhen Wu
We construct a model in which MAω1(S)[S] holds and 𝒦2 fails. This shows that MAω1(S)[S] does not imply 𝒦2 and answers an old question of Larson and Todorcevic in [Katetov’s problem, Trans. Amer. Math. Soc.354(5) (2002) 1783–1791]. We also investigate different strong colorings in models of MAω1(S)[S].
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Degrees of categoricity and treeable degrees J. Math. Log. (IF 0.9) Pub Date : 2023-08-11 Barbara F. Csima, Dino Rossegger
In this paper, we give a characterization of the strong degrees of categoricity of computable structures greater or equal to 0″. They are precisely the treeable degrees — the least degrees of paths through computable trees — that compute 0″. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree d with 0(α)≤d≤0(α+1) for α a computable ordinal
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Hensel minimality: Geometric criteria for ℓ-h-minimality J. Math. Log. (IF 0.9) Pub Date : 2023-08-01 Floris Vermeulen
Recently, Cluckers et al. developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this paper, we continue the study of Hensel minimality, in particular focusing on ω-h-minimality and ℓ-h-minimality
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Coloring closed Noetherian graphs J. Math. Log. (IF 0.9) Pub Date : 2023-07-06 Jindřich Zapletal
If Γ is a closed Noetherian graph on a σ-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF+DC that Γ is countably chromatic and there is no Vitali set.
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Few new reals J. Math. Log. (IF 0.9) Pub Date : 2023-06-29 David Asperó, Miguel Angel Mota
We introduce a new method for building models of CH, together with Π2 statements over H(ω2), by forcing. Unlike other forcing constructions in the literature, our construction adds new reals, although only ℵ1-many of them. Using this approach, we build a model in which a very strong form of the negation of Club Guessing at ω1 known as Measuring holds together with CH, thereby answering a well-known
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Ramsey’s theorem for pairs, collection, and proof size J. Math. Log. (IF 0.9) Pub Date : 2023-05-31 Leszek Aleksander Kołodziejczyk, Tin Lok Wong, Keita Yokoyama
We prove that any proof of a ∀Σ20 sentence in the theory WKL0+RT22 can be translated into a proof in RCA0 at the cost of a polynomial increase in size. In fact, the proof in RCA0 can be obtained by a polynomial-time algorithm. On the other hand, RT22 has nonelementary speedup over the weaker base theory RCA0∗ for proofs of Σ1 sentences. We also show that for n≥0, proofs of Πn+2 sentences in BΣn+1+exp
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Logical metatheorems for accretive and (generalized) monotone set-valued operators J. Math. Log. (IF 0.9) Pub Date : 2023-05-22 Nicholas Pischke
Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of certain set-valued mappings between function spaces. This paper deals with the computational properties of these accretive and (generalized) monotone set-valued operators. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a
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Maximal models up to the first measurable in ZFC J. Math. Log. (IF 0.9) Pub Date : 2023-05-23 John T. Baldwin, Saharon Shelah
Theorem: There is a complete sentence ϕ of Lω1,ω such that ϕ has maximal models in a set of cardinals λ that is cofinal in the first measurable μ while ϕ has no maximal models in any χ≥μ.
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Strongly compact cardinals and ordinal definability J. Math. Log. (IF 0.9) Pub Date : 2023-05-15 Gabriel Goldberg
This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis
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Henselian expansions of NIP fields J. Math. Log. (IF 0.9) Pub Date : 2023-04-29 Franziska Jahnke
Let K be an NIP field and let v be a Henselian valuation on K. We ask whether (K,v) is NIP as a valued field. By a result of Shelah, we know that if v is externally definable, then (K,v) is NIP. Using the definability of the canonical p-Henselian valuation, we show that whenever the residue field of v is not separably closed, then v is externally definable. In the case of separably closed residue field
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Coarse computability, the density metric, Hausdorff distances between Turing degrees, perfect trees, and reverse mathematics J. Math. Log. (IF 0.9) Pub Date : 2023-04-12 Denis R. Hirschfeldt, Carl G. Jockusch, Paul E. Schupp
For A⊆ω, the coarse similarity class of A, denoted by [A], is the set of all B⊆ω such that the symmetric difference of A and B has asymptotic density 0. There is a natural metric δ on the space 𝒮 of coarse similarity classes defined by letting δ([A],[B]) be the upper density of the symmetric difference of A and B. We study the metric space of coarse similarity classes under this metric, and show in
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Higher indescribability and derived topologies J. Math. Log. (IF 0.9) Pub Date : 2023-04-06 Brent Cody
We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of Lκ+,κ+-indescribability and Πξ1-indescribability of a cardinal κ for all ξ<κ+. In this context, universal Πξ1 formulas exist, there
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Hindman’s theorem in the hierarchy of choice principles J. Math. Log. (IF 0.9) Pub Date : 2023-03-16 David Fernández-Bretón
In the context of ZF, we analyze a version of Hindman’s finite unions theorem on infinite sets, which normally requires the Axiom of Choice to be proved. We establish the implication relations between this statement and various classical weak choice principles, thus precisely locating the strength of the statement as a weak form of the AC.
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Exactly two and exactly three near-coherence classes J. Math. Log. (IF 0.9) Pub Date : 2023-03-16 Heike Mildenberger
We prove that for n=2 and n=3 there is a forcing extension with exactly n near-coherence classes of non-principal ultrafilters. We introduce localized versions of Matet forcing and we develop Ramsey spaces of names. The evaluation of some of the new forcings is based on a relative of Hindman’s theorem due to Blass 1987.
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Covering at limit cardinals of K J. Math. Log. (IF 0.9) Pub Date : 2023-03-16 William J. Mitchell, Ernest Schimmerling
Assume that there is no transitive class model of ZFC with a Woodin cardinal. Let ν be a singular ordinal such that ν>ω2 and cf(ν)<|ν|. Suppose ν is a regular cardinal in K. Then ν is a measurable cardinal in K. Moreover, if cf(ν)>ω, then oK(ν)≥cf(ν).
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Corrigendum to Reducing ω-model reflection to iterated syntactic reflection J. Math. Log. (IF 0.9) Pub Date : 2023-03-06 Fedor Pakhomov, James Walsh
We fix a gap in a proof in our paper Reducing ω-model reflection to iterated syntactic reflection.
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Actions of tame abelian product groups J. Math. Log. (IF 0.9) Pub Date : 2023-02-28 Shaun Allison, Assaf Shani
A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. When G=∏nΓn for countable abelian Γn, Solecki [Equivalence relations induced by actions of Polish groups, Trans. Amer. Math. Soc. 347 (1995) 4765–4777] gave a characterization for when G is tame. In [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math
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The degree of nonminimality is at most 2 J. Math. Log. (IF 0.9) Pub Date : 2023-01-31 James Freitag, Rémi Jaoui, Rahim Moosa
In this paper, it is shown that if p∈S(A) is a complete type of Lascar rank at least 2, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations a1, a2 such that p has a nonalgebraic forking extension over Aa1a2. Moreover, if A is contained in the field of constants then p already has a nonalgebraic forking extension over Aa1. The results are also
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More definable combinatorics around the first and second uncountable cardinals J. Math. Log. (IF 0.9) Pub Date : 2023-01-05 William Chan, Stephen Jackson, Nam Trang
Assume ZF+AD. If 𝜖 is an ordinal and X is a set of ordinals, then [X]∗𝜖 is the collection of order-preserving functions f:𝜖→X which have uniform cofinality ω and discontinuous everywhere. The weak partition properties on ω1 and ω2 yield partition measures on [ω1]∗𝜖 when 𝜖<ω1 and [ω2]∗𝜖 when 𝜖<ω2. The following almost everywhere continuity properties for functions on partition spaces with respect
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Co-theory of sorted profinite groups for PAC structures J. Math. Log. (IF 0.9) Pub Date : 2023-01-05 Daniel Max Hoffmann, Junguk Lee
We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the “Weak Independence Theorem” for pseudo-algebraically closed (PAC) substructures of an ambient
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The Ramsey theory of Henson graphs J. Math. Log. (IF 0.9) Pub Date : 2022-12-17 Natasha Dobrinen
Analogues of Ramsey’s Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden
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Simultaneously vanishing higher derived limits without large cardinals J. Math. Log. (IF 0.9) Pub Date : 2022-12-17 Jeffrey Bergfalk, Michael Hrušák, Chris Lambie-Hanson
A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits limn (n>0) of a certain inverse system A indexed by ωω to simultaneously
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On the antichain tree property J. Math. Log. (IF 0.9) Pub Date : 2022-12-17 JinHoo Ahn, Joonhee Kim, Junguk Lee
In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and k-ATP, and provide a criterion for theories to have
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Borel combinatorics fail in HYP J. Math. Log. (IF 0.9) Pub Date : 2022-12-17 Henry Towsner, Rose Weisshaar, Linda Westrick
We characterize the completely determined Borel subsets of HYP as exactly the Δ1(Lω1ck) subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics
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Compactness and guessing principles in the Radin extensions J. Math. Log. (IF 0.9) Pub Date : 2022-12-17 Omer Ben-Neria, Jing Zhang
We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on κ, if κ is weakly compact, then ♢(κ) holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model
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Invariant measures in simple and in small theories J. Math. Log. (IF 0.9) Pub Date : 2022-12-17 Artem Chernikov, Ehud Hrushovski, Alex Kruckman, Krzysztof Krupiński, Slavko Moconja, Anand Pillay, Nicholas Ramsey
We give examples of (i) a simple theory with a formula (with parameters) which does not fork over ∅ but has μ-measure 0 for every automorphism invariant Keisler measure μ and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G. We also discuss paradoxical decompositions both in the setting of discrete groups
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The two halves of disjunctive correctness J. Math. Log. (IF 0.9) Pub Date : 2022-12-17 Cezary Cieśliński, Mateusz Łełyk, Bartosz Wcisło
Ali Enayat had asked whether two halves of Disjunctive Correctness (DC) for the compositional truth predicate are conservative over Peano Arithmetic (PA). In this paper, we show that the principle “every true disjunction has a true disjunct” is equivalent to bounded induction for the compositional truth predicate and thus it is not conservative. On the other hand, the converse implication “any disjunction
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On piecewise hyperdefinable groups J. Math. Log. (IF 0.9) Pub Date : 2022-12-17 A. Rodriguez Fanlo
The aim of this paper is to generalize and improve two of the main model-theoretic results of “Stable group theory and approximate subgroups” by Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the Stabilizer Theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular
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Complexity of distances: Theory of generalized analytic equivalence relations J. Math. Log. (IF 0.9) Pub Date : 2022-09-27 Marek Cúth, Michal Doucha, Ondřej Kurka
We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that
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Compactness versus hugeness at successor cardinals J. Math. Log. (IF 0.9) Pub Date : 2022-09-27 Sean Cox, Monroe Eskew
If κ is regular and 2<κ≤κ+, then the existence of a weakly presaturated ideal on κ+ implies □κ∗. This partially answers a question of Foreman and Magidor about the approachability ideal on ω2. As a corollary, we show that if there is a presaturated ideal I on ω2 such that 𝒫(ω2)∕I is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously
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Decomposing Aronszajn lines J. Math. Log. (IF 0.9) Pub Date : 2022-09-27 Keegan Dasilva Barbosa
We show that under the proper forcing axiom the class of all Aronszajn lines behave like σ-scattered orders under the embeddability relation. In particular, we are able to show that the class of better-quasi-order labeled fragmented Aronszajn lines is itself a better-quasi-order. Moreover, we show that every better-quasi-order labeled Aronszajn line can be expressed as a finite sum of labeled types
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Exact saturation in pseudo-elementary classes for simple and stable theories J. Math. Log. (IF 0.9) Pub Date : 2022-09-27 Itay Kaplan, Nicholas Ramsey, Saharon Shelah
We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable
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Sigma-Prikry forcing II: Iteration Scheme J. Math. Log. (IF 0.9) Pub Date : 2022-09-12 Alejandro Poveda, Assaf Rinot, Dima Sinapova
In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call Σ-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are Σ-Prikry. We showed that given a Σ-Prikry poset ℙ and a ℙ-name
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New jump operators on equivalence relations J. Math. Log. (IF 0.9) Pub Date : 2022-08-11 John D. Clemens, Samuel Coskey
We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group Γ we introduce the Γ-jump. We study the elementary properties of the Γ-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups Γ, the Γ-jump is proper in the sense that for any Borel equivalence relation E the Γ-jump
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Every Δ20 degree is a strong degree of categoricity J. Math. Log. (IF 0.9) Pub Date : 2022-08-11 Barbara F. Csima, Keng Meng Ng
A strong degree of categoricity is a Turing degree d such that there is a computable structure 𝒜 that is d-computably categorical (there is a d-computable isomorphism between any two computable copies of 𝒜), and such that there exist two computable copies of 𝒜 between which every isomorphism computes d. The question of whether every Δ20 degree is a strong degree of categoricity has been of interest
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Definable completeness of P-minimal fields and applications J. Math. Log. (IF 0.9) Pub Date : 2022-06-22 Pablo Cubides Kovacsics, Françoise Delon
We show that every definable nested family of closed and bounded subsets of a P-minimal field K has nonempty intersection. As an application we answer a question of Darnière and Halupczok showing that P-minimal fields satisfy the “extreme value property”: for every closed and bounded subset U⊆K and every interpretable continuous function f:U→ΓK (where ΓK denotes the value group), f(U) admits a maximal
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Strong compactness and the ultrapower axiom I: the least strongly compact cardinal J. Math. Log. (IF 0.9) Pub Date : 2022-06-22 Gabriel Goldberg
The Ultrapower Axiom is a combinatorial principle concerning the structure of large cardinals that is true in all known canonical inner models of set theory. A longstanding test question for inner model theory is the equiconsistency of strongly compact and supercompact cardinals. In this paper, it is shown that under the Ultrapower Axiom, the least strongly compact cardinal is supercompact. A number
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Incompatible bounded category forcing axioms J. Math. Log. (IF 0.9) Pub Date : 2022-06-22 David Asperó, Matteo Viale
We introduce bounded category forcing axioms for well-behaved classes Γ. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe HλΓ+ modulo forcing in Γ, for some cardinal λΓ naturally associated to Γ. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation λΓ=ω — to classes Γ with λΓ>ω
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Forcing the Σ31-separation property J. Math. Log. (IF 0.9) Pub Date : 2022-06-22 Stefan Hoffelner
We generically construct a model in which the Σ31-separation property is true, i.e. every pair of disjoint Σ31-sets can be separated by a Δ31-definable set. This answers an old question from the problem list “Surrealist landscape with figures” by A. Mathias from 1968. We also construct a model in which the (lightface) Σ31-separation property is true.
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Reduction games, provability and compactness J. Math. Log. (IF 0.9) Pub Date : 2022-06-22 Damir D. Dzhafarov, Denis R. Hirschfeldt, Sarah Reitzes
Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between Π21 principles over ω-models of RCA0. They also introduced a version of this game that similarly captures provability over RCA0. We generalize and extend this game-theoretic framework to other formal systems, and establish
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Approximate counting and NP search problems J. Math. Log. (IF 0.9) Pub Date : 2022-06-22 Leszek Aleksander Kołodziejczyk, Neil Thapen
We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory APC2 of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829–860]. In particular, the Ramsey and weak pigeonhole search problems lie in the
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A negative solution of Kuznetsov’s problem for varieties of bi-Heyting algebras J. Math. Log. (IF 0.9) Pub Date : 2022-06-22 Guram Bezhanishvili, David Gabelaia, Mamuka Jibladze
In this paper, we show that there exist (continuum many) varieties of bi-Heyting algebras that are not generated by their complete members. It follows that there exist (continuum many) extensions of the Heyting–Brouwer logic HB that are topologically incomplete. This result provides further insight into the long-standing open problem of Kuznetsov by yielding a negative solution of the reformulation
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Abraham–Rubin–Shelah open colorings and a large continuum J. Math. Log. (IF 0.9) Pub Date : 2022-03-11 Thomas Gilton, Itay Neeman
We show that the Abraham–Rubin–Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with 2ℵ0=ℵ3. This answers one of the main open questions from [U. Abraham, M. Rubin and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ℵ1-dense real order types, Ann. Pure Appl. Logic 325(29) (1985) 123–206]. As in [U. Abraham
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Equivalence relations and determinacy J. Math. Log. (IF 0.9) Pub Date : 2022-02-05 Logan Crone, Lior Fishman, Stephen Jackson
We introduce the notion of (Γ,E)-determinacy for Γ a pointclass and E an equivalence relation on a Polish space X. A case of particular interest is the case when E = EG is the (left) shift-action of G on SG where S = 2 = {0, 1} or S = ω. We show that for all shift actions by countable groups G, and any “reasonable” pointclass Γ, that (Γ,EG)-determinacy implies Γ-determinacy. We also prove a corresponding
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Structural reflection, shrewd cardinals and the size of the continuum J. Math. Log. (IF 0.9) Pub Date : 2022-02-05 Philipp Lücke
Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle SR− introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy
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Model theory of differential fields with finite group actions J. Math. Log. (IF 0.9) Pub Date : 2021-12-31 Daniel Max Hoffmann, Omar León Sánchez
Let G be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in m commuting derivations equipped with a G-action by differential field automorphisms. In the language of G-differential rings (i.e. the language of rings with added symbols for derivations and automorphisms), we prove that this class has a model-companion — denoted G −DCF0
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The domination monoid in o-minimal theories J. Math. Log. (IF 0.9) Pub Date : 2021-12-09 Rosario Mennuni
We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of 1-types. We show this to hold in Real Closed Fields, where generators of this monoid correspond to invariant convex subrings of the monster model. Combined with [C. Ealy, D. Haskell and J. Maríková, Residue
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Reducing ω-model reflection to iterated syntactic reflection J. Math. Log. (IF 0.9) Pub Date : 2021-12-09 Fedor Pakhomov, James Walsh
In mathematical logic there are two seemingly distinct kinds of principles called “reflection principles.” Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic reflection principles assert that every provable sentence from some complexity class is true. In this paper, we study connections between these two kinds of reflection
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Coarse groups, and the isomorphism problem for oligomorphic groups J. Math. Log. (IF 0.9) Pub Date : 2021-07-30 André Nies, Philipp Schlicht, Katrin Tent
Let S∞ denote the topological group of permutations of the natural numbers. A closed subgroup G of S∞ is called oligomorphic if for each n, its natural action on n-tuples of natural numbers has only finitely many orbits. We study the complexity of the topological isomorphism relation on the oligomorphic subgroups of S∞ in the setting of Borel reducibility between equivalence relations on Polish spaces
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A generalization of the 𝔾0 dichotomy and a strengthening of the 𝔼0ℕ dichotomy J. Math. Log. (IF 0.9) Pub Date : 2021-07-05 Benjamin D. Miller
We generalize the 𝔾0 dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi Polish groups on Polish spaces that can be decomposed into countably-many Borel actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of the identity. We also show
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Maximal almost disjoint families, determinacy, and forcing J. Math. Log. (IF 0.9) Pub Date : 2021-05-10 Karen Bakke Haga, David Schrittesser, Asger Törnquist
We study the notion of 𝒥-MAD families where 𝒥 is a Borel ideal on ω. We show that if 𝒥 is any finite or countably iterated Fubini product of the ideal of finite sets Fin, then there are no analytic infinite 𝒥-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective 𝒥-MAD families; and under the full Axiom of Determinacy +V=L(ℝ) or under AD+ there