61 found
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  1.  56
    Model Theory for Infinitary Logic: Logic with Countable Conjunctions and Finite Quantifiers.H. Jerome Keisler - 1971 - Amsterdam: North-Holland Pub. Co..
    Provability, Computability and Reflection.
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  2.  25
    Logic with the Quantifier “There Exist Uncountably Many”.H. Jerome Keisler - 1970 - Annals of Mathematical Logic 1 (1):1-93.
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  3. Model Theory.C. C. Chang & H. Jerome Keisler - 1992 - Studia Logica 51 (1):154-155.
     
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  4.  11
    [Omnibus Review].H. Jerome Keisler - 1970 - Journal of Symbolic Logic 35 (2):342-344.
  5.  12
    Frege Structures and the Notions of Proposition, Truth and Set.Peter Aczel, Jon Barwise, H. Jerome Keisler & Kenneth Kunen - 1986 - Journal of Symbolic Logic 51 (1):244-246.
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  6.  50
    Nonstandard Arithmetic and Reverse Mathematics.H. Jerome Keisler - 2006 - Bulletin of Symbolic Logic 12 (1):100-125.
    We show that each of the five basic theories of second order arithmetic that play a central role in reverse mathematics has a natural counterpart in the language of nonstandard arithmetic. In the earlier paper [3] we introduced saturation principles in nonstandard arithmetic which are equivalent in strength to strong choice axioms in second order arithmetic. This paper studies principles which are equivalent in strength to weaker theories in second order arithmetic.
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  7.  37
    Ultraproducts Which Are Not Saturated.H. Jerome Keisler - 1967 - Journal of Symbolic Logic 32 (1):23-46.
    In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. IfDis an ultrafilter over a setI, andis a structure, the ultrapower ofmoduloDis denoted byD-prod. The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure. Our ultimate aim is to find out what kinds of structure are ultrapowers of. We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis, for (...)
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  8.  43
    Theory of Models with Generalized Atomic Formulas.H. Jerome Keisler - 1960 - Journal of Symbolic Logic 25 (1):1-26.
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  9.  53
    On the Strength of Nonstandard Analysis.C. Ward Henson & H. Jerome Keisler - 1986 - Journal of Symbolic Logic 51 (2):377-386.
  10.  50
    An Impossibility Theorem on Beliefs in Games.Adam Brandenburger & H. Jerome Keisler - 2006 - Studia Logica 84 (2):211-240.
    A paradox of self-reference in beliefs in games is identified, which yields a game-theoretic impossibility theorem akin to Russell’s Paradox. An informal version of the paradox is that the following configuration of beliefs is impossible:Ann believes that Bob assumes that.
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  11.  18
    Measures and Forking.H. Jerome Keisler - 1987 - Annals of Pure and Applied Logic 34 (2):119-169.
    Shelah's theory of forking is generalized in a way which deals with measures instead of complete types. This allows us to extend the method of forking from the class of stable theories to the larger class of theories which do not have the independence property. When restricted to the special case of stable theories, this paper reduces to a reformulation of the classical approach. However, it goes beyond the classical approach in the case of unstable theories. Methods from ordinary forking (...)
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  12. The Strength of Nonstandard Methods in Arithmetic.C. Ward Henson, Matt Kaufmann & H. Jerome Keisler - 1984 - Journal of Symbolic Logic 49 (4):1039-1058.
    We consider extensions of Peano arithmetic suitable for doing some of nonstandard analysis, in which there is a predicate N(x) for an elementary initial segment, along with axiom schemes approximating ω 1 -saturation. We prove that such systems have the same proof-theoretic strength as their natural analogues in second order arithmetic. We close by presenting an even stronger extension of Peano arithmetic, which is equivalent to ZF for arithmetic statements.
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  13.  33
    Some Applications of Infinitely Long Formulas.H. Jerome Keisler - 1965 - Journal of Symbolic Logic 30 (3):339-349.
    Introduction. This paper is a sequel to our paper [3]. In that paper we introduced the notion of a finite approximation to an infinitely long formula, in a language L with infinitely long expressions of the type considered by Henkin in [2]. The results of the paper [3] show relationships between the models of an infinitely long sentence and the models of its finite approximations. In the present paper we shall apply the main result of [3] to prove a number (...)
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  14.  11
    Separable Models of Randomizations.Uri Andrews & H. Jerome Keisler - 2015 - Journal of Symbolic Logic 80 (4):1149-1181.
  15.  12
    Finite Approximations of Infinitely Long Formulas.H. Jerome Keisler, J. W. Addison, Leon Henkin & Alfred Tarski - 1969 - Journal of Symbolic Logic 34 (1):129-130.
  16.  10
    Definable Closure in Randomizations.Uri Andrews, Isaac Goldbring & H. Jerome Keisler - 2015 - Annals of Pure and Applied Logic 166 (3):325-341.
  17.  86
    Barwise: Infinitary Logic and Admissible Sets.H. Jerome Keisler & Julia F. Knight - 2004 - Bulletin of Symbolic Logic 10 (1):4-36.
  18.  23
    Limit Ultraproducts.H. Jerome Keisler - 1965 - Journal of Symbolic Logic 30 (2):212-234.
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  19.  32
    Making the Hyperreal Line Both Saturated and Complete.H. Jerome Keisler & James H. Schmerl - 1991 - Journal of Symbolic Logic 56 (3):1016-1025.
    In a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a κ-saturated nonstandard (...)
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  20.  60
    The Diversity of Quantifier Prefixes.H. Jerome Keisler & Wilbur Walkoe - 1973 - Journal of Symbolic Logic 38 (1):79-85.
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  21.  12
    Making the Hyperreal Line Both Saturated and Complete.H. Jerome Keisler & James H. Schmerl - 1991 - Journal of Symbolic Logic 56 (3):1016-1025.
    In a nonstandard universe, the $\kappa$-saturation property states that any family of fewer than $\kappa$ internal sets with the finite intersection property has a nonempty intersection. An ordered field $F$ is said to have the $\lambda$-Bolzano-Weierstrass property iff $F$ has cofinality $\lambda$ and every bounded $\lambda$-sequence in $F$ has a convergent $\lambda$-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a $\kappa$-saturated nonstandard (...)
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  22.  22
    Descriptive Set Theory Over Hyperfinite Sets.H. Jerome Keisler, Kenneth Kunen, Arnold Miller & Steven Leth - 1989 - Journal of Symbolic Logic 54 (4):1167-1180.
    The separation, uniformization, and other properties of the Borel and projective hierarchies over hyperfinite sets are investigated and compared to the corresponding properties in classical descriptive set theory. The techniques used in this investigation also provide some results about countably determined sets and functions, as well as an improvement of an earlier theorem of Kunen and Miller.
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  23.  36
    Ultraproducts of Finite Sets.H. Jerome Keisler - 1967 - Journal of Symbolic Logic 32 (1):47-57.
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  24.  18
    Hyperfinite Models of Adapted Probability Logic.H. Jerome Keisler - 1986 - Annals of Pure and Applied Logic 31:71-86.
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  25.  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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  26. First Order Quantifiers in Monadic Second Order Logic.H. Jerome Keisler & Wafik Boulos Lotfallah - 2004 - Journal of Symbolic Logic 69 (1):118-136.
    This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn on properties S that says "there are n components having S". We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if (...)
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  27.  10
    Game Sentences and Ultrapowers.Renling Jin & H. Jerome Keisler - 1993 - Annals of Pure and Applied Logic 60 (3):261-274.
    We prove that if is a model of size at most [kappa], λ[kappa] = λ, and a game sentence of length 2λ is true in a 2λ-saturated model ≡ , then player has a winning strategy for a related game in some ultrapower ΠD of . The moves in the new game are taken in the cartesian power λA, and the ultrafilter D over λ must be chosen after the game is played. By taking advantage of the expressive power of (...)
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  28. The Kleene Symposium: Proceedings of the Symposium Held June 18-24, 1978 at Madison, Wisconsin, U.S.A.Stephen Cole Kleene, Jon Barwise, H. Jerome Keisler & Kenneth Kunen (eds.) - 1980 - Amsterdam, Netherlands: Sole Distributors for the U.S.A. And Canada, Elsevier North-Holland.
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  29.  11
    From Discrete to Continuous Time.H. Jerome Keisler - 1991 - Annals of Pure and Applied Logic 52 (1-2):99-141.
    A general metatheorem is proved which reduces a wide class of statements about continuous time stochastic processes to statements about discrete time processes. We introduce a strong language for stochastic processes, and a concept of forcing for sequences of discrete time processes. The main theorem states that a sentence in the language is true if and only if it is forced. Although the stochastic process case is emphasized in order to motivate the results, they apply to a wider class of (...)
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  30.  20
    Abraham Robinson. Forcing in Model Theory. Symposia Mathematica, Vol. 5, Istituto Nazionale di Alta Matematica, Academic Press, London and New York 1971, Pp. 69–82. - Jon Barwise and Abraham Robinson. Completing Theories by Forcing. Annals of Mathematical Logic, Vol. 2 No. 2 , Pp. 119–142. - Abraham Robinson. Infinite Forcing in Model Theory. Proceedings of the Second Scandinavian Logic Symposium, Edited by J. E. Fenstad, Studies in Logic and the Foundations of Mathematics, Vol. 63, North-Holland Publishing Company, Amsterdam and London 1971, Pp. 317–340. - Abraham Robinson. Forcing in Model Theory. Actes du Congrès International des Mathematiciens 1970, Gauthier-Villars, Paris 1971, Vol. 1, Pp. 245–250. [REVIEW]H. Jerome Keisler - 1975 - Journal of Symbolic Logic 40 (4):633-634.
  31.  13
    Meager Sets on the Hyperfinite Time Line.H. Jerome Keisler & Steven C. Leth - 1991 - Journal of Symbolic Logic 56 (1):71-102.
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  32.  35
    M. Makkai. On the Model Theory of Denumerably Long Formulas with Finite Strings of Quantifiers. The Journal of Symbolic Logic, Vol. 34 , Pp. 437–459. [REVIEW]H. Jerome Keisler - 1973 - Journal of Symbolic Logic 38 (2):337-337.
  33.  38
    Almost Everywhere Elimination of Probability Quantifiers.H. Jerome Keisler & Wafik Boulos Lotfallah - 2009 - Journal of Symbolic Logic 74 (4):1121 - 1142.
    We obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like $\exists ^{ \ge 3/4} y$ which says that "for at least 3/4 of all y". These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are: 1. We deal with the quantifier $\exists ^{ \ge r} y$ , where y is a tuple (...)
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  34.  18
    Miodrag Rašković and Radosav ĐorĐević. Probability Quantifiers and Operators. Series in Pure and Applied Mathematics. Vesta, Belgrade 1996, Also Distributed by Bid International Co., Sherman Oaks, Calif., Iv + 121 Pp. [REVIEW]H. Jerome Keisler - 1998 - Journal of Symbolic Logic 63 (3):1191-1193.
  35.  8
    A Completeness Proof for Adapted Probability Logic.H. Jerome Keisler - 1986 - Annals of Pure and Applied Logic 31:61-70.
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  36.  39
    A Result Concerning Cardinalities of Ultraproducts.H. Jerome Keisler & Karel Prikry - 1974 - Journal of Symbolic Logic 39 (1):43-48.
  37. Applications of Ultraproducts of Pairs of Cardinals to the Theory of Models.C. C. Chang & H. Jerome Keisler - 1971 - Journal of Symbolic Logic 36 (2):338-339.
  38. Continuous Sentences Preserved Under Reduced Products.Isaac Goldbring & H. Jerome Keisler - 2022 - Journal of Symbolic Logic 87 (2):649-681.
    Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.
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  39. Review: K. I. Appel, Horn Sentences in Identity Theory. [REVIEW]H. Jerome Keisler - 1966 - Journal of Symbolic Logic 31 (1):131-132.
  40.  18
    A Canonical Hidden-Variable Space.Adam Brandenburger & H. Jerome Keisler - 2018 - Annals of Pure and Applied Logic 169 (12):1295-1302.
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  41.  7
    Continuous Sentences Preserved Under Reduced Products.Isaac Goldbring & H. Jerome Keisler - 2020 - Journal of Symbolic Logic:1-33.
    Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.
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  42.  95
    Maharam Spectra of Loeb Spaces.Renling Jin & H. Jerome Keisler - 2000 - Journal of Symbolic Logic 65 (2):550-566.
    We characterize Maharam spectra of Loeb probability spaces and give some applications of the results.
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  43.  13
    Independence in Randomizations.Uri Andrews, Isaac Goldbring & H. Jerome Keisler - 2019 - Journal of Mathematical Logic 19 (1):1950005.
    The randomization of a complete first-order theory [Formula: see text] is the complete continuous theory [Formula: see text] with two sorts, a sort for random elements of models of [Formula: see text] and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of [Formula: see text]. We show that if [Formula: see text] has the exchange property and [Formula: see text], then [Formula: see text] has a strict independence (...)
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  44.  20
    Łos J.. On the Extending of Models . Fundamenta Mathematicae, Vol. 42 , Pp. 38–54.Łos J. And Suszko R.. On the Extending of Models . Common Extensions. Fundamenta Mathematicae, Vol. 42 , Pp. 343–347.Słomiński J.. On the Extending of Models . Extensions in Equationally Definable Classes of Algebras. Fundamenta Mathematicae, Vol. 43 , Pp. 69–76.Łos J. And Suszko R.. On the Extending of Models . Infinite Sums of Models. Fundamenta Mathematicae, Vol. 44 , Pp. 52–60. [REVIEW]H. Jerome Keisler - 1962 - Journal of Symbolic Logic 27 (1):93-95.
  45.  2
    Madison 1970 Meeting of the Association for Symbolic Logic.H. Jerome Keisler & Kenneth Kunen - 1971 - Journal of Symbolic Logic 36 (2):368-378.
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  46.  37
    Meeting of the Association for Symbolic Logic: Madison 1982.H. Jerome Keisler - 1983 - Journal of Symbolic Logic 48 (4):1233-1239.
  47.  26
    The Stability Function of a Theory.H. Jerome Keisler - 1978 - Journal of Symbolic Logic 43 (3):481-486.
    Let T be a complete theory with infinite models in a countable language. The stability function g T (κ) is defined as the supremum of the number of types over models of T of power κ. It is proved that there are only six possible stability functions, namely $\kappa, \kappa + 2^\omega, \kappa^\omega, \operatorname{ded} \kappa, (\operatorname{ded} \kappa)^\omega, 2^\kappa$.
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  48.  18
    Definability with a Predicate for a Semi-Linear Set.Michael Benedikt & H. Jerome Keisler - 2003 - Journal of Symbolic Logic 68 (1):319-351.
    We settle a number of questions concerning definability in first order logic with an extra predicate symbol ranging over semi-linear sets. We give new results both on the positive and negative side: we show that in first-order logic one cannot query a semi-linear set as to whether or not it contains a line, or whether or not it contains the line segment between two given points. However, we show that some of these queries become definable if one makes small restrictions (...)
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  49.  14
    1995–1996 Annual Meeting of the Association for Symbolic Logic.H. Jerome Keisler - 1996 - Bulletin of Symbolic Logic 2 (4):448-472.
  50.  14
    Review: Miodrag Raskovic, Radosav Dordevic, Probability Quantifiers and Operators. [REVIEW]H. Jerome Keisler - 1998 - Journal of Symbolic Logic 63 (3):1191-1193.
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