Results for 'countable compactness'

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  1.  1
    Measurement of Countable Compactness and Lindelöf Property in RL -Fuzzy Topological Spaces.Xiongwei Zhang, Ibtesam Alshammari & A. Ghareeb - 2021 - Complexity 2021:1-7.
    Based on the concepts of pseudocomplement of L -subsets and the implication operator where L is a completely distributive lattice with order-reversing involution, the definition of countable RL -fuzzy compactness degree and the Lindelöf property degree of an L -subset in RL -fuzzy topology are introduced and characterized. Since L -fuzzy topology in the sense of Kubiak and Šostak is a special case of RL -fuzzy topology, the degrees of RL -fuzzy compactness and the Lindelöf property are (...)
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  2.  3
    On Ordering of the Family of Logics with Skolem-Löwenheim Property and Countable Compactness Property.Marek Wacławek - 1995 - In M. Krynicki, M. Mostowski & L. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Kluwer Academic Publishers. pp. 229--236.
  3.  7
    The Anti-Specker Property, Uniform Sequential Continuity, and a Countable Compactness Property.Douglas Bridges - 2011 - Logic Journal of the IGPL 19 (1):174-182.
    It is shown constructively that, on a metric space that is dense in itself, if every pointwise continuous, real-valued function is uniformly sequentially continuous, then the space has the anti-Specker property. The converse is also discussed. Finally, we show that the anti-Specker property implies a restricted form of countable compactness.
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  4.  17
    Compactness in Countable Tychonoff Products and Choice.Paul Howard, K. Keremedis & J. E. Rubin - 2000 - Mathematical Logic Quarterly 46 (1):3-16.
    We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.
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  5.  7
    Compact Spaces, Elementary Submodels, and the Countable Chain Condition.Lúcia R. Junqueira, Paul Larson & Franklin D. Tall - 2006 - Annals of Pure and Applied Logic 144 (1-3):107-116.
    Given a space in an elementary submodel M of H, define XM to be X∩M with the topology generated by . It is established, using anti-large-cardinals assumptions, that if XM is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X=XM. Assuming in addition, the result holds for any compact XM satisfying the countable chain condition.
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  6. On Compact Hausdorff Spaces of Countable Tightness.Piotr Koszmider, Z. Szentmiklossy, A. Csaszar & Zoltan Balogh - 2002 - Bulletin of Symbolic Logic 8 (2):306.
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  7.  8
    Proof-theoretic uniform boundedness and bounded collection principles and countable Heine–Borel compactness.Ulrich Kohlenbach - 2021 - Archive for Mathematical Logic 60 (7):995-1003.
    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.
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  8.  28
    S-Spaces and L-Spaces Under Martin's AxiomOn Compact Hausdorff Spaces of Countable Tightness.Piotr Koszmider, Z. Szentmiklossy, A. Csaszar & Zoltan Balogh - 2002 - Bulletin of Symbolic Logic 8 (2):306.
  9.  16
    Murray G. Bell. Spaces of Ideals of Partial Functions. Set Theory and its Applications, Proceedings of a Conference Held at York University, Ontario, Canada, Aug. 10–21,1987, Edited by J. Streprāns and S. Watson, Lecture Notes in Mathematics, Vol. 1401, Springer-Verlag, Berlin Etc. 1989, Pp. 1–4. - Alan Dow. Compact Spaces of Countable Tightness in the Cohen Model. Set Theory and its Applications, Proceedings of a Conference Held at York University, Ontario, Canada, Aug. 10–21,1987, Edited by J. Streprāns and S. Watson, Lecture Notes in Mathematics, Vol. 1401, Springer-Verlag, Berlin Etc. 1989, Pp. 55–67. - Peter J. Nyikos. Classes of Compact Sequential Spaces. Set Theory and its Applications, Proceedings of a Conference Held at York University, Ontario, Canada, Aug. 10–21,1987, Edited by J. Streprāns and S. Watson, Lecture Notes in Mathematics, Vol. 1401, Springer-Verlag, Berlin Etc. 1989, Pp. 135–159. - Franklin D. Tall. Topological Problems for Set-Theorists. Set Theory and its Appl. [REVIEW]Judith Roitman - 1991 - Journal of Symbolic Logic 56 (2):753-755.
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  10.  22
    Szentmiklóssy Z.. S-Spaces and L-Spaces Under Martin's Axiom. Topology, Volume II, Edited by Császár A., Colloquia Mathematica Societatis János Bolyai, No. 23, János Bolyai Mathematical Society, Budapest, and North-Holland Publishing Company, Amsterdam, Oxford, and New York, 1980, Pp. 1139–1145. Balogh Zoltán. On Compact Hausdorff Spaces of Countable Tightness. Proceedings of the American Mathematical Society, Vol. 105 (1989), Pp. 755–764. [REVIEW]Piotr Koszmider - 2002 - Bulletin of Symbolic Logic 8 (2):306-307.
  11.  9
    Compactness Under Constructive Scrutiny.Hajime Ishihara & Peter Schuster - 2004 - Mathematical Logic Quarterly 50 (6):540-550.
    How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop-style constructive mathematics as the basic system – that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected constructive implication that every sequentially compact separable metric space is totally bounded. As (...)
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  12.  10
    Compactness and Normality in Abstract Logics.Xavier Caicedo - 1993 - Annals of Pure and Applied Logic 59 (1):33-43.
    We generalize a theorem of Mundici relating compactness of a regular logic L to a strong form of normality of the associated spaces of models. Moreover, it is shown that compactness is in fact equivalent to ordinary normality of the model spaces when L has uniform reduction for infinite disjoint sums of structures. Some applications follow. For example, a countably generated logic is countably compact if and only if every clopen class in the model spaces is elementary. The (...)
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  13.  32
    Compact Metric Spaces and Weak Forms of the Axiom of Choice.E. Tachtsis & K. Keremedis - 2001 - Mathematical Logic Quarterly 47 (1):117-128.
    It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, (...)
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  14.  33
    On Compactness of Logics That Can Express Properties of Symmetry or Connectivity.Vera Koponen & Tapani Hyttinen - 2015 - Studia Logica 103 (1):1-20.
    A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. (...)
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  15.  4
    Tychonoff Products of Compact Spaces in ZF and Closed Ultrafilters.Kyriakos Keremedis - 2010 - Mathematical Logic Quarterly 56 (5):474-487.
    Let {: i ∈I } be a family of compact spaces and let X be their Tychonoff product. [MATHEMATICAL SCRIPT CAPITAL C] denotes the family of all basic non-trivial closed subsets of X and [MATHEMATICAL SCRIPT CAPITAL C]R denotes the family of all closed subsets H = V × Πmath imageXi of X, where V is a non-trivial closed subset of Πmath imageXi and QH is a finite non-empty subset of I. We show: Every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL (...)
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  16.  18
    Countable Sums and Products of Metrizable Spaces in ZF.Kyriakos Keremedis & Eleftherios Tachtsis - 2005 - Mathematical Logic Quarterly 51 (1):95-103.
    We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, Lindelöf metric spaces, and in disjoint topological unions of countably many such spaces.
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  17.  14
    Proofs of the Compactness Theorem.Alexander Paseau - 2011 - History and Philosophy of Logic 32 (4):407-407.
    In this study, the author compares several proofs of the compactness theorem for propositional logic with countably many atomic sentences. He thereby takes some steps towards a systematic philosophical study of the compactness theorem. He also presents some data and morals for the theory of mathematical explanation. [The author is not responsible for the horrific mathematical typo in the second sentence.].
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  18.  7
    On Countable Products of Finite Hausdorff Spaces.Horst Herrlich & Kyriakos Keremedis - 2000 - Mathematical Logic Quarterly 46 (4):537-542.
    We investigate in ZF conditions that are necessary and sufficient for countable products ∏m∈ℕXm of finite Hausdorff spaces Xm resp. Hausdorff spaces Xm with at most n points to be compact resp. Baire. Typica results: Countable products of finite Hausdorff spaces are compact if and only if countable products of non-empty finite sets are non-empty. Countable products of discrete spaces with at most n + 1 points are compact if and only if countable products of (...)
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  19.  65
    Notions of Compactness for Special Subsets of ℝ I and Some Weak Forms of the Axiom of Choice.Marianne Morillon - 2010 - Journal of Symbolic Logic 75 (1):255-268.
    We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results where AC N (resp. the (...)
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  20.  44
    Products of Compact Spaces and the Axiom of Choice II.Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin - 2003 - Mathematical Logic Quarterly 49 (1):57-71.
    This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.
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  21.  16
    Countably Decomposable Admissible Sets.Menachem Magidor, Saharon Shelah & Jonathan Stavi - 1984 - Annals of Pure and Applied Logic 26 (3):287-361.
    The known results about Σ 1 -completeness, Σ 1 -compactness, ordinal omitting etc. are given a unified treatment, which yields many new examples. It is shown that the unifying theorem is best possible in several ways, assuming V = L.
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  22. Proofs of the Compactness Theorem.Alexander Paseau - 2010 - History and Philosophy of Logic 31 (1):73-98.
    In this study, several proofs of the compactness theorem for propositional logic with countably many atomic sentences are compared. Thereby some steps are taken towards a systematic philosophical study of the compactness theorem. In addition, some related data and morals for the theory of mathematical explanation are presented.
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  23.  10
    Universal Theories and Compactly Expandable Models.Enrique Casanovas & Saharon Shelah - 2019 - Journal of Symbolic Logic 84 (3):1215-1223.
    Our aim is to solve a quite old question on the difference between expandability and compact expandability. Toward this, we further investigate the logic of countable cofinality.
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  24.  20
    Products of Some Special Compact Spaces and Restricted Forms of AC.Kyriakos Keremedis & Eleftherios Tachtsis - 2010 - Journal of Symbolic Logic 75 (3):996-1006.
    We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1] (...)
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  25.  29
    Actions of Non-Compact and Non-Locally Compact Polish Groups.Sławomir Solecki - 2000 - Journal of Symbolic Logic 65 (4):1881-1894.
    We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a (...)
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  26.  21
    Universally Measurable Subgroups of Countable Index.Christian Rosendal - 2010 - Journal of Symbolic Logic 75 (3):1081-1086.
    It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group S ∞ is continuous. It is also shown that a universally measurable homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous.
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  27.  25
    On the Elementary Equivalence of Automorphism Groups of Boolean Algebras; Downward Skolem Löwenheim Theorems and Compactness of Related Quantifiers.Matatyahu Rubin & Saharon Shelah - 1980 - Journal of Symbolic Logic 45 (2):265-283.
    THEOREM 1. (⋄ ℵ 1 ) If B is an infinite Boolean algebra (BA), then there is B 1 such that $|\operatorname{Aut} (B_1)| \leq B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut} (B_1)\rangle \equiv \langle B, \operatorname{Aut}(B)\rangle$ . THEOREM 2. (⋄ ℵ 1 ) There is a countably compact logic stronger than first-order logic even on finite models. This partially answers a question of H. Friedman. These theorems appear in §§ 1 and 2. THEOREM 3. (a) (⋄ ℵ 1 ) If (...)
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  28.  12
    Codings of Separable Compact Subsets of the First Baire Class.Pandelis Dodos - 2006 - Annals of Pure and Applied Logic 142 (1):425-441.
    Let X be a Polish space and a separable compact subset of the first Baire class on X. For every sequence dense in , the descriptive set-theoretic properties of the set are analyzed. It is shown that if is not first countable, then is -complete. This can also happen even if is a pre-metric compactum of degree at most two, in the sense of S. Todorčević. However, if is of degree exactly two, then is always Borel. A deep result (...)
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  29.  18
    Inverse Topological Systems and Compactness in Abstract Model Theory.Daniele Mundici - 1986 - Journal of Symbolic Logic 51 (3):785-794.
    Given an abstract logic L = L(Q i ) i ∈ I generated by a set of quantifiers Q i , one can construct for each type τ a topological space S τ exactly as one constructs the Stone space for τ in first-order logic. Letting T be an arbitrary directed set of types, the set $S_T = \{(S_\tau, \pi^\tau_\sigma)\mid\sigma, \tau \in T, \sigma \subset \tau\}$ is an inverse topological system whose bonding mappings π τ σ are naturally determined by (...)
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  30. On the Axiomatisability of the Dual of Compact Ordered Spaces.Marco Abbadini - 2021 - Bulletin of Symbolic Logic 27 (4):526-526.
    We prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to any (...)
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  31.  9
    Discrete Subspaces of Countably Tight Compacta.I. Juhász & Z. Szentmiklóssy - 2006 - Annals of Pure and Applied Logic 140 (1):72-74.
    Our main result is that the following cardinal arithmetic assumption, which is a slight weakening of GCH, “2κ is a finite successor of κ for every cardinal κ”, implies that in any countably tight compactum X there is a discrete subspace D with . This yields a confirmation of Alan Dow’s Conjecture 2 from [A. Dow, Closures of discrete sets in compact spaces, Studia Math. Sci. Hung. 42 227–234].
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  32.  11
    Embeddings of Countable Closed Sets and Reverse Mathematics.Jeffry L. Hirst - 1993 - Archive for Mathematical Logic 32 (6):443-449.
    If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR 0. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of ℝ.
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  33.  51
    Generic Embeddings Associated to an Indestructibly Weakly Compact Cardinal.Gunter Fuchs - 2010 - Annals of Pure and Applied Logic 162 (1):89-105.
    I use generic embeddings induced by generic normal measures on that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be applied in order to obtain the forcing axioms in forcing extensions. This has consequences in : The Singular Cardinal Hypothesis holds above κ, and κ has a useful Jónsson-like property. This in turn implies that the countable tower works much like it does when κ is a Woodin limit of Woodin cardinals. (...)
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  34.  12
    Gδ Sets in Σ-Ideals Generated by Compact Sets.Maya Saran - 2019 - Journal of Symbolic Logic 84 (2):781-797.
    Given a compact Polish space E and the hyperspace of its compact subsets , we consider Gδσ-ideals of compact subsets of E. Solecki has shown that any σ-ideal in a broad natural class of Gδ ideals can be represented via a compact subset of ; in this article we examine the behaviour of Gδ subsets of E with respect to the representing set. Given an ideal I in this class, we construct a representing set that recognises a compact subset of (...)
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  35.  16
    On Uniformly Continuous Functions Between Pseudometric Spaces and the Axiom of Countable Choice.Samuel G. da Silva - 2019 - Archive for Mathematical Logic 58 (3-4):353-358.
    In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between metric spaces, and the second declares that sequentially compact pseudometric spaces are \—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.
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  36.  14
    The Model N = ∪ {L[A]: A Countable Set of Ordinals}.Claude Sureson - 1987 - Annals of Pure and Applied Logic 36:289-313.
    This paper continues the study of covering properties of models closed under countable sequences. In a previous article we focused on C. Chang's Model . Our purpose is now to deal with the model N = ∪ { L [A]: A countable ⊂ Ord}. We study here relations between covering properties, satisfaction of ZF by N , and cardinality of power sets. Under large cardinal assumptions N is strictly included in Chang's Model C , it may thus be (...)
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  37.  34
    An Algebraic Treatment of the Barwise Compactness Theory.Isidore Fleischer & Philip Scott - 1991 - Studia Logica 50 (2):217 - 223.
    A theorem on the extendability of certain subsets of a Boolean algebra to ultrafilters which preserve countably many infinite meets (generalizing Rasiowa-Sikorski) is used to pinpoint the mechanism of the Barwise proof in a way which bypasses the set theoretical elaborations.
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  38.  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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  39.  7
    Some Weak Forms of the Baire Category Theorem.Kyriakos Kermedis - 2003 - Mathematical Logic Quarterly 49 (4):369.
    We show that the statement “separable, countably compact, regular spaces are Baire” is deducible from a strictly weaker form than AC, namely, CAC . We also find some characterizations of the axiom of dependent choices.
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  40.  15
    On the Consistency of Some Partition Theorems for Continuous Colorings, and the Structure of ℵ1-Dense Real Order Types.Uri Abraham, Matatyahu Rubin & Saharon Shelah - 1985 - Annals of Pure and Applied Logic 29 (2):123-206.
    We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ℵ 1 -dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ℵ 1 -dense subsets of R are isomorphic, is consistent. We e.g. prove Con). Let K H, be the set of order types of ℵ 1 -dense homogeneous subsets of R (...)
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  41.  13
    Two Applications of Topology to Model Theory.Christopher J. Eagle, Clovis Hamel & Franklin D. Tall - 2021 - Annals of Pure and Applied Logic 172 (5):102907.
    By utilizing the topological concept of pseudocompactness, we simplify and improve a proof of Caicedo, Dueñez, and Iovino concerning Terence Tao's metastability. We also pinpoint the exact relationship between the Omitting Types Theorem and the Baire Category Theorem by developing a machine that turns topological spaces into abstract logics.
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  42.  20
    On the Consistency of Some Partition Theorems for Continuous Colorings, and the Structure of ℵ 1 -Dense Real Order Types.J. Steprans, Uri Abraham, Matatyahu Rubin & Saharon Shelah - 2002 - Bulletin of Symbolic Logic 8 (2):303.
    We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ℵ 1 -dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ℵ 1 -dense subsets of R are isomorphic, is consistent. We e.g. prove Con). Let K H , be the set of order types of ℵ 1 -dense homogeneous subsets of (...)
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  43.  10
    Filter Logics on $Omega$.Matt Kaufmann - 1984 - Journal of Symbolic Logic 49 (1):241-256.
    Logics $L^F(M)$ are considered, in which $M$ ("most") is a new first-order quantifier whose interpretation depends on a given filter $F$ of subsets of $\omega$. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that $F$ is not of the form $\{(\bigcap F) \cup X: |\omega - X| < \omega\}$ with $|\omega - \bigcap F| = \omega$. Moreover the set of validities of $L^F(M)$ and even of $L^F_{\omega_1\omega}(M)$ depends only on a few basic (...)
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  44.  5
    Special ultrafilters and cofinal subsets of $$({}^omega omega, <^*)$$.Peter Nyikos - 2020 - Archive for Mathematical Logic 59 (7-8):1009-1026.
    The interplay between ultrafilters and unbounded subsets of \ with the order \ of strict eventual domination is studied. Among the tools are special kinds of non-principal ultrafilters on \. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \ of almost inclusion. It is shown that the cofinality of such a base must be either \, the least cardinality of \-unbounded set, or \, the least cardinality of (...)
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  45.  8
    EM Constructions for a Class of Generalized Quantifiers.Martin Otto - 1992 - Archive for Mathematical Logic 31 (5):355-371.
    We consider a class of Lindström extensions of first-order logic which are susceptible to a natural Skolemization procedure. In these logics Ehrenfeucht Mostowski (EM) functors for theories with arbitrarily large models can be obtained under suitable restrictions. Characteristic dependencies between algebraic properties of the quantifiers and the maximal domains of EM functors are investigated.Results are applied to Magidor Malitz logic,L(Q <ω), showing e.g. its Hanf number to be equal to ℶω(ℵ1) in the countably compact case. Using results of Baumgartner, the (...)
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  46.  25
    Definability Properties and the Congruence Closure.Xavier Caicedo - 1990 - Archive for Mathematical Logic 30 (4):231-240.
    We introduce a natural class of quantifiersTh containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic ofL ωω (Th) or countably compact regular sublogic ofL ∞ω (Th), properly extendingL ωω , satisfies the uniform reduction property for quotients. As a consequence, none of these logics satisfies eitherΔ-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic ofL (...)
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  47.  16
    Quantifier Elimination for Neocompact Sets.H. Jerome Keisler - 1998 - Journal of Symbolic Logic 63 (4):1442-1472.
    We shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide an easy alternative to nonstandard methods of proving existence theorems in probability theory, where they behave like compact sets. The quantifier elimination theorems in this paper can be applied in a general setting to show that the family of neocompact sets is countably (...)
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  48.  30
    Filter Logics on Ω.Matt Kaufmann - 1984 - Journal of Symbolic Logic 49 (1):241-256.
    Logics L F (M) are considered, in which M ("most") is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form $\{(\bigcap F) \cup X:|\omega - X| with $|\omega - \bigcap F| = \omega$ . Moreover the set of validities of L F (M) and even of L F ω 1 ω (...)
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  49.  45
    Disasters in Topology Without the Axiom of Choice.Kyriakos Keremedis - 2001 - Archive for Mathematical Logic 40 (8):569-580.
    We show that some well known theorems in topology may not be true without the axiom of choice.
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  50.  26
    Unique Solutions.Peter Schuster - 2006 - Mathematical Logic Quarterly 52 (6):534-539.
    It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness (...)
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