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  1.  1
    Hanf numbers for extendibility and related phenomena.John T. Baldwin & Saharon Shelah - 2022 - Archive for Mathematical Logic 61 (3):437-464.
    This paper contains portions of Baldwin’s talk at the Set Theory and Model Theory Conference and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of \ that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.
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  2. Definable groups in dense pairs of geometric structures.Alexander Berenstein & Evgueni Vassiliev - 2022 - Archive for Mathematical Logic 61 (3):345-372.
    We study definable groups in dense/codense expansions of geometric theories with a new predicate P such as lovely pairs and expansions of fields by groups with the Mann property. We show that in such expansions, large definable subgroups of groups definable in the original language \ are also \-definable, and definably amenable \-definable groups remain amenable in the expansion. We also show that if the underlying geometric theory is NIP, and G is a group definable in a model of T, (...)
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  3.  2
    Model theory of monadic predicate logic with the infinity quantifier.Facundo Carreiro, Alessandro Facchini, Yde Venema & Fabio Zanasi - 2022 - Archive for Mathematical Logic 61 (3):465-502.
    This paper establishes model-theoretic properties of \, a variation of monadic first-order logic that features the generalised quantifier \. We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality and \, respectively). For each logic \ we will show the following. We provide syntactically defined fragments of \ characterising four different semantic properties of \-sentences: being monotone and continuous in a given set of monadic predicates; having truth preserved under (...)
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  4.  2
    Condensable models of set theory.Ali Enayat - 2022 - Archive for Mathematical Logic 61 (3):299-315.
    A model \ of ZF is said to be condensable if \\prec _{\mathbb {L}_{{\mathcal {M}}}} {\mathcal {M}}\) for some “ordinal” \, where \:=,\in )^{{\mathcal {M}}}\) and \ is the set of formulae of the infinitary logic \ that appear in the well-founded part of \. The work of Barwise and Schlipf in the 1970s revealed the fact that every countable recursively saturated model of ZF is cofinally condensable \prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}\) for an unbounded collection of \). Moreover, it (...)
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  5. A criterion for uniform finiteness in the imaginary sorts.Will Johnson - 2022 - Archive for Mathematical Logic 61 (3):583-589.
    Let T be a theory. If T eliminates \, it need not follow that \ eliminates \, as shown by the example of the p-adics. We give a criterion to determine whether \ eliminates \. Specifically, we show that \ eliminates \ if and only if \ is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of \ is unknown. As an application, we show that \ eliminates \ when (...)
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  6.  2
    Efficient elimination of Skolem functions in $$\text {LK}^\text {h}$$ LK h.Ján Komara - 2022 - Archive for Mathematical Logic 61 (3):503-534.
    We present a sequent calculus with the Henkin constants in the place of the free variables. By disposing of the eigenvariable condition, we obtained a proof system with a strong locality property—the validity of each inference step depends only on its active formulas, not its context. Our major outcomes are: the cut elimination via a non-Gentzen-style algorithm without resorting to regularization and the elimination of Skolem functions with linear increase in the proof length for a subclass of derivations with cuts.
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  7.  1
    A note on cut-elimination for classical propositional logic.Gabriele Pulcini - 2022 - Archive for Mathematical Logic 61 (3):555-565.
    In Schwichtenberg, Schwichtenberg fine-tuned Tait’s technique so as to provide a simplified version of Gentzen’s original cut-elimination procedure for first-order classical logic. In this note we show that, limited to the case of classical propositional logic, the Tait–Schwichtenberg algorithm allows for a further simplification. The procedure offered here is implemented on Kleene’s sequent system G4. The specific formulation of the logical rules for G4 allows us to provide bounds on the height of cut-free proofs just in terms of the logical (...)
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  8.  3
    Bounded inductive dichotomy: separation of open and clopen determinacies with finite alternatives in constructive contexts.Kentaro Sato - 2022 - Archive for Mathematical Logic 61 (3):399-435.
    In his previous work, the author has introduced the axiom schema of inductive dichotomy, a weak variant of the axiom schema of inductive definition, and used this schema for elementary ) positive operators to separate open and clopen determinacies for those games in which two players make choices from infinitely many alternatives in various circumstances. Among the studies on variants of inductive definitions for bounded ) positive operators, the present article investigates inductive dichotomy for these operators, and applies it to (...)
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  9.  1
    Coanalytic ultrafilter bases.Jonathan Schilhan - 2022 - Archive for Mathematical Logic 61 (3-4):567-581.
    We study the definability of ultrafilter bases on \ in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in L we can construct \ P-point and Q-point bases. We also show that the existence of a \ ultrafilter is equivalent to that of a \ ultrafilter base, for \. Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.
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  10. Multiplicative finite embeddability vs divisibility of ultrafilters.Boris Šobot - 2022 - Archive for Mathematical Logic 61 (3):535-553.
    We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations \ and \. The set of its minimal elements proves to be very rich, and the \-hierarchy is used to get a better intuition of this richness. We find the place of the set of \-maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of subsets (...)
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  11.  3
    Computability and the game of cops and robbers on graphs.Rachel D. Stahl - 2022 - Archive for Mathematical Logic 61 (3):373-397.
    Several results about the game of cops and robbers on infinite graphs are analyzed from the perspective of computability theory. Computable robber-win graphs are constructed with the property that no computable robber strategy is a winning strategy, and such that for an arbitrary computable ordinal \, any winning strategy has complexity at least \}\). Symmetrically, computable cop-win graphs are constructed with the property that no computable cop strategy is a winning strategy. Locally finite infinite trees and graphs are explored. The (...)
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  12.  3
    Combinatory logic with polymorphic types.William R. Stirton - 2022 - Archive for Mathematical Logic 61 (3):317-343.
    Sections 1 through 4 define, in the usual inductive style, various classes of object including one which is called the “combinatory terms of polymorphic type”. Section 5 defines a reduction relation on these terms. Section 6 shows that the weak normalizability of the combinatory terms of polymorphic type entails the weak normalizability of the lambda terms of polymorphic type. The entailment is not vacuous, because the combinatory terms of polymorphic type are indeed weakly normalizable, as is proven in Sect. 7 (...)
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  13.  3
    Various forms of infinity for finitely supported structures.Andrei Alexandru & Gabriel Ciobanu - 2022 - Archive for Mathematical Logic 61 (1):173-222.
    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 definitions for infinity in the world of finitely supported sets, and provide relevant examples of atomic sets which satisfy some forms (...)
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  14.  2
    The independence of $$\mathsf {GCH}$$ GCH and a combinatorial principle related to Banach–Mazur games.Will Brian, Alan Dow & Saharon Shelah - 2022 - Archive for Mathematical Logic 61 (1):1-17.
    It was proved recently that Telgársky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of \. The proof introduces a combinatorial principle that is shown to follow from \, namely: \::Every separative poset \ with the \-cc contains a dense sub-poset \ such that \ for every \. We prove this principle is independent of \ and \, in the sense that \ does not imply \, and \ does not imply \ assuming the consistency (...)
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  15.  2
    On Hilbert algebras generated by the order.J. L. Castiglioni, S. A. Celani & H. J. San Martín - 2022 - Archive for Mathematical Logic 61 (1):155-172.
    In this paper we study the variety of order Hilbert algebras, which is the equivalent algebraic semantics of the order implicational calculus of Bull.
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  16.  2
    Degree structures of conjunctive reducibility.Irakli Chitaia & Roland Omanadze - 2022 - Archive for Mathematical Logic 61 (1):19-31.
    We show: for every noncomputable c.e. incomplete c-degree, there exists a nonspeedable c-degree incomparable with it; The c-degree of a hypersimple set includes an infinite collection of \-degrees linearly ordered under \ with order type of the integers and consisting entirely of hypersimple sets; there exist two c.e. sets having no c.e. least upper bound in the \-reducibility ordering; the c.e. \-degrees are not dense.
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  17.  2
    A boundedness principle for the Hjorth rank.Ohad Drucker - 2022 - Archive for Mathematical Logic 61 (1):223-232.
    Hjorth 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 following conjecture of Hjorth—for every limit ordinal \, the set of elements whose orbit is of complexity less than \ is a Borel set.
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  18. Representability and compactness for pseudopowers.Todd Eisworth - 2022 - Archive for Mathematical Logic 61 (1):55-80.
    We prove a compactness theorem for pseudopower operations of the form \}\) where \\le {{\,\mathrm{cf}\,}}\). 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, in particular, implying the existence of a progressive set A of regular cardinals for which \\) has an inaccessible accumulation point.
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  19.  6
    Small $$\mathfrak {u}(\kappa )$$ u ( κ ) at singular $$\kappa $$ κ with compactness at $$\kappa ^{++}$$ κ + +.Radek Honzik & Šárka Stejskalová - 2022 - Archive for Mathematical Logic 61 (1):33-54.
    We show that the tree property, stationary reflection and the failure of approachability at \ are consistent with \= \kappa ^+ < 2^\kappa \), where \ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if \ is a regular cardinal, then stationary reflection at \ is indestructible under all \-cc forcings.
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  20.  1
    First-order theories of bounded trees.Ruaan Kellerman - 2022 - Archive for Mathematical Logic 61 (1):263-297.
    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 of the first-order theory of the class of trees whose paths are all isomorphic to some ordinal (...)
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  21. Normalisation and Subformula Property for a System of Classical Logic with Tarski’s Rule.Nils Kürbis - 2022 - Archive for Mathematical Logic 61 (1):105-129.
    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 segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule for implication. (...)
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  22.  2
    Sprague–Grundy theory in bounded arithmetic.Satoru Kuroda - 2022 - Archive for Mathematical Logic 61 (1):233-262.
    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 Kayles position is provably equivalent to some NIM heap.
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  23.  1
    Does weak quasi-o-minimality behave better than weak o-minimality?Slavko Moconja & Predrag Tanović - 2022 - Archive for Mathematical Logic 61 (1):81-103.
    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.
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  24.  1
    Rosenthal families, filters, and semifilters.Miroslav Repický - 2022 - Archive for Mathematical Logic 61 (1):131-153.
    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 \-filter, properly between a selective filter and a \-filter. We prove that every \-ultrafilter is a Rosenthal family. We prove that it is consistent with ZFC to have uncountably many \-ultrafilters such that any intersection of (...)
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