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  1. Indestructibility When the First Two Measurable Cardinals Are Strongly Compact.Arthur W. Apter - 2022 - Journal of Symbolic Logic 87 (1):214-227.
    We prove two theorems concerning indestructibility properties of the first two strongly compact cardinals when these cardinals are in addition the first two measurable cardinals. Starting from two supercompact cardinals $\kappa _1 < \kappa _2$, we force and construct a model in which $\kappa _1$ and $\kappa _2$ are both the first two strongly compact and first two measurable cardinals, $\kappa _1$ ’s strong compactness is fully indestructible, and $\kappa _2$ ’s strong compactness is indestructible under $\mathrm {Add}$ for any (...)
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  2. Reconstruction of Non--Categorical Theories.Itaï Ben Yaacov - 2022 - Journal of Symbolic Logic 87 (1):159-187.
    We generalise the correspondence between $\aleph _0$ -categorical theories and their automorphism groups to arbitrary complete theories in classical logic, and to some theories in continuous logic.
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  3.  5
    Combinatorics of Ultrafilters on Cohen and Random Algebras.Jörg Brendle & Francesco Parente - 2022 - Journal of Symbolic Logic 87 (1):109-126.
    We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect this analysis with a cardinal invariant of Boolean algebras, the ultrafilter number, and prove consistency results concerning its possible values on Cohen and random algebras.
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  4.  5
    Model Theory and Combinatorics of Banned Sequences.Hunter Chase & James Freitag - 2022 - Journal of Symbolic Logic 87 (1):1-20.
    We set up a general context in which one can prove Sauer-Shelah type lemmas. We apply our general results to answer a question of Bhaskar [1] and give a slight improvement to a result of Malliaris and Terry [7]. We also prove a new Sauer-Shelah type lemma in the context of op-rank, a notion of Guingona and Hill [4].
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  5.  7
    Ramsey’s Coheirs.Eugenio Colla & Domenico Zambella - 2022 - Journal of Symbolic Logic 87 (1):377-391.
    We use the model theoretic notion of coheir to give short proofs of old and new theorems in Ramsey Theory. As an illustration we start from Ramsey's theorem itself. Then we prove Hindman's theorem and the Hales-Jewett theorem. Finally, we prove the two Ramsey theoretic principles that have among their consequences partition theorems due to Carlson and to Gowers.
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  6.  4
    Relationships Between Computability-Theoretic Properties of Problems.Rod Downey, Noam Greenberg, Matthew Harrison-Trainor, Ludovic Patey & Dan Turetsky - 2022 - Journal of Symbolic Logic 87 (1):47-71.
    A problem is a multivalued function from a set of instances to a set of solutions. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a noncomputable set A and a computable instance of a problem ${\mathsf {P}}$, to find a solution relative to which A (...)
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  7.  4
    On Extensions of Partial Isomorphisms.Mahmood Etedadialiabadi & Su Gao - 2022 - Journal of Symbolic Logic 87 (1):416-435.
    In this paper we study a notion of HL-extension for a structure in a finite relational language $\mathcal {L}$. We give a description of all finite minimal HL-extensions of a given finite $\mathcal {L}$ -structure. In addition, we study a group-theoretic property considered by Herwig–Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal {L}$ -structures and show that every countable $\mathcal {L}$ -structure can be extended to a countable (...)
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  8.  11
    The Tree of Tuples of a Structure.Matthew Harrison-Trainor & Antonio Montalbán - 2022 - Journal of Symbolic Logic 87 (1):21-46.
    Our main result is that there exist structures which cannot be computably recovered from their tree of tuples. This implies that there are structures with no computable copies which nevertheless cannot code any information in a natural/functorial way.
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  9.  1
    Contributions to the Theory of F-Automatic Sets.Christopher Hawthorne - 2022 - Journal of Symbolic Logic 87 (1):127-158.
    Fix an abelian group $\Gamma $ and an injective endomorphism $F\colon \Gamma \to \Gamma $. Improving on the results of [2], new characterizations are here obtained for the existence of spanning sets, F-automaticity, and F-sparsity. The model theoretic status of these sets is also investigated, culminating with a combinatorial description of the F-sparse sets that are stable in $$, and a proof that the expansion of $$ by any F-sparse set is NIP. These methods are also used to show for (...)
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  10. On Non-Compact P-Adic Definable Groups.Will Johnson & Ningyuan Yao - 2022 - Journal of Symbolic Logic 87 (1):188-213.
    In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is not definably compact, then G contains a definable torsion-free subgroup of dimension 1. We prove here a p-adic analogue of the Peterzil–Steinhorn theorem, in the special case of abelian groups. Let G be an abelian group definable in a p-adically closed field M. If G is not definably compact then there is a definable subgroup H of dimension 1 which is not definably compact. (...)
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  11. Density-Like and Generalized Density Ideals.Adam Kwela & Paolo Leonetti - 2022 - Journal of Symbolic Logic 87 (1):228-251.
    We show that there exist uncountably many pairwise nonisomorphic density-like ideals on $\omega $ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja et al. in [this Journal, vol. 80, pp. 1268–1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.
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  12. The Reverse Mathematics of the Thin Set and Erdős–Moser Theorems.Lu Liu & Ludovic Patey - 2022 - Journal of Symbolic Logic 87 (1):313-346.
    The thin set theorem for n-tuples and k colors states that every k-coloring of $[\mathbb {N}]^n$ admits an infinite set of integers H such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\operatorname {\mathrm {\sf {TS}}}^n_k$, nor the free set theorem imply the Erdős–Moser theorem whenever k is sufficiently large. Given a problem $\mathsf {P}$, a computable instance of $\mathsf {P}$ is universal (...)
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  13. Pseudo-Finite Sets, Pseudo-o-Minimality—Erratum.Nadav Meir - 2022 - Journal of Symbolic Logic 87 (1):436-436.
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  14. Htp-Complete Rings of Rational Numbers.Russell Miller - 2022 - Journal of Symbolic Logic 87 (1):252-272.
    For a ring R, Hilbert’s Tenth Problem $HTP$ is the set of polynomial equations over R, in several variables, with solutions in R. We view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP$, which is naturally viewed as a set of polynomials in $\mathbb {Z}[X_1,X_2,\ldots ]$. It is known that for almost all W, the jump $W'$ does not $1$ -reduce to $HTP$. In contrast, we show that every Turing degree contains a set W (...)
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  15.  3
    Mitchell-Inspired Forcing, with Small Working Parts and Collections of Models of Uniform Size as Side Conditions, and Gap-One Simplified Morasses.Charles Morgan - 2022 - Journal of Symbolic Logic 87 (1):392-415.
    We show that a $$ -simplified morass can be added by a forcing with working parts of size smaller than $\kappa $. This answers affirmatively the question, asked independently by Shelah and Velleman in the early 1990s, of whether it is possible to do so.Our argument use a modification of a technique of Mitchell’s for adding objects of size $\omega _2$ in which collections of models – all of equal, countable size – are used as side conditions. In our modification, (...)
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  16. Notes on the Dprm Property for Listable Structures.Hector Pasten - 2022 - Journal of Symbolic Logic 87 (1):273-312.
    A celebrated result by Davis, Putnam, Robinson, and Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of arithmetic. We investigate analogues of this result over structures endowed with a listable presentation. When such an analogue holds, the structure is said to have the DPRM property. We prove several results addressing foundational aspects around this problem, such as uniqueness of the listable presentation, transference of the DPRM property under (...)
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  17.  3
    Ramsey-Like Theorems and Moduli of Computation.Ludovic Patey - 2022 - Journal of Symbolic Logic 87 (1):72-108.
    Ramsey’s theorem asserts that every k-coloring of $[\omega ]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$, there exists a computable k-coloring of $[\omega ]^n$ whose solutions compute the halting set. On the other hand, for every computable k-coloring of $[\omega ]^2$ and every noncomputable set C, there is an infinite monochromatic set H such that $C \not \leq _T H$. The latter property is known as cone avoidance.In this article, we design a natural class of Ramsey-like theorems encompassing (...)
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  18.  1
    Non-Classical Foundations of Set Theory.Sourav Tarafder - 2022 - Journal of Symbolic Logic 87 (1):347-376.
    In this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also study a few basic (...)
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