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  1. H. Jerome Keisler & Jeffrey M. Keisler (2014). Observing, Reporting, and Deciding in Networks of Sentences. Annals of Pure and Applied Logic 165 (3):812-836.
    In prior work [7] we considered networks of agents who have knowledge bases in first order logic, and report facts to their neighbors that are in their common languages and are provable from their knowledge bases, in order to help a decider verify a single sentence. In report complete networks, the signatures of the agents and the links between agents are rich enough to verify any deciderʼs sentence that can be proved from the combined knowledge base. This paper introduces a (...)
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  2. H. Jerome Keisler & Wafik Boulos Lotfallah (2009). Almost Everywhere Elimination of Probability Quantifiers. 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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  3. Adam Brandenburger & H. Jerome Keisler (2006). An Impossibility Theorem on Beliefs in Games. 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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  4. H. Jerome Keisler (2006). Nonstandard Arithmetic and Reverse Mathematics. 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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  5. H. Jerome Keisler & Julia F. Knight (2004). Barwise: Infinitary Logic and Admissible Sets. Bulletin of Symbolic Logic 10 (1):4-36.
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  6. H. Jerome Keisler & Wafik Boulos Lotfallah (2004). First Order Quantifiers in Monadic Second Order Logic. 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 $esists_{n}(S)$ 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, (...)
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  7. H. Jerome Keisler & Wafik Boulos Lotfallah (2004). Shrinking Games and Local Formulas. Annals of Pure and Applied Logic 128 (1-3):215-225.
    Gaifman's normal form theorem showed that every first-order sentence of quantifier rank n is equivalent to a Boolean combination of “scattered local sentences”, where the local neighborhoods have radius at most 7n−1. This bound was improved by Lifsches and Shelah to 3×4n−1. We use Ehrenfeucht–Fraïssé type games with a “shrinking horizon” to get a spectrum of normal form theorems of the Gaifman type, depending on the rate of shrinking. This spectrum includes the result of Lifsches and Shelah, with a more (...)
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  8. Michael Benedikt & H. Jerome Keisler (2003). Definability with a Predicate for a Semi-Linear Set. 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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  9. Renling Jin & H. Jerome Keisler (2000). Maharam Spectra of Loeb Spaces. 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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  10. H. Jerome Keisler (1998). Quantifier Elimination for Neocompact Sets. 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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  11. H. Jerome Keisler (1998). Rašković Miodrag and ĐorĐević Radosav. 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] Journal of Symbolic Logic 63 (3):1191-1193.
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  12. H. Jerome Keisler (1998). Review: Miodrag Raskovic, Radosav Dordevic, Probability Quantifiers and Operators. [REVIEW] Journal of Symbolic Logic 63 (3):1191-1193.
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  13. H. Jerome Keisler (1996). 1995-1996 Annual Meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic 2 (4):448-472.
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  14. Renling Jin & H. Jerome Keisler (1993). Game Sentences and Ultrapowers. 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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  15. H. Jerome Keisler (1991). From Discrete to Continuous Time. 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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  16. H. Jerome Keisler & Steven C. Leth (1991). Meager Sets on the Hyperfinite Time Line. Journal of Symbolic Logic 56 (1):71-102.
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  17. H. Jerome Keisler & James H. Schmerl (1991). Making the Hyperreal Line Both Saturated and Complete. 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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  18. H. Jerome Keisler, Kenneth Kunen, Arnold Miller & Steven Leth (1989). Descriptive Set Theory Over Hyperfinite Sets. 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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  19. H. Jerome Keisler (1987). Measures and Forking. 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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  20. C. Ward Henson & H. Jerome Keisler (1986). On the Strength of Nonstandard Analysis. Journal of Symbolic Logic 51 (2):377-386.
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  21. H. Jerome Keisler (1986). A Completeness Proof for Adapted Probability Logic. Annals of Pure and Applied Logic 31 (1):61-70.
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  22. H. Jerome Keisler (1986). Hyperfinite Models of Adapted Probability Logic. Annals of Pure and Applied Logic 31 (1):71-86.
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  23. C. Ward Henson, Matt Kaufmann & H. Jerome Keisler (1984). The Strength of Nonstandard Methods in Arithmetic. 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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  24. H. Jerome Keisler (1983). Meeting of the Association for Symbolic Logic: Madison 1982. Journal of Symbolic Logic 48 (4):1233-1239.
  25. Stephen Cole Kleene, Jon Barwise, H. Jerome Keisler & Kenneth Kunen (eds.) (1980). The Kleene Symposium: Proceedings of the Symposium Held June 18-24, 1978 at Madison, Wisconsin, U.S.A. Sole Distributors for the U.S.A. And Canada, Elsevier North-Holland.
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  26. John Addison, Jon Barwise, H. Jerome Keisler, Kenneth Kunen & Yiannis N. Moschovakis (1979). The Kleene Symposium and the Summer Meeting of the Association for Symbolic Logic. Journal of Symbolic Logic 44 (3):469-480.
  27. H. Jerome Keisler (1978). The Stability Function of a Theory. 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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  28. H. Jerome Keisler & Karel Prikry (1974). A Result Concerning Cardinalities of Ultraproducts. Journal of Symbolic Logic 39 (1):43-48.
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  29. H. Jerome Keisler (1973). Review: M. Makkai, On the Model Theory of Denumerably Long Formulas with Finite Strings of Quantifiers. [REVIEW] Journal of Symbolic Logic 38 (2):337-337.
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  30. H. Jerome Keisler & Wilbur Walkoe Jr (1973). The Diversity of Quantifier Prefixes. Journal of Symbolic Logic 38 (1):79-85.
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  31. H. Jerome Keisler (1971). Model Theory for Infinitary Logic. Amsterdam,North-Holland Pub. Co..
    Provability, Computability and Reflection.
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  32. H. Jerome Keisler (1971). On Theories Categorical in Their Own Power. Journal of Symbolic Logic 36 (2):240-244.
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  33. H. Jerome Keisler (1971). Review: Carol R. Karp, Finite-Quantifier Equivalence. [REVIEW] Journal of Symbolic Logic 36 (1):158-158.
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  34. H. Jerome Keisler (1971). Review: J. P. Ressayre, Sur les Theories du Premier Ordre Categoriques En Un Cardinal. [REVIEW] Journal of Symbolic Logic 36 (4):684-685.
  35. H. Jerome Keisler (1970). Logic with the Quantifier “There Exist Uncountably Many. Annals of Mathematical Logic 1 (1):1-93.
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  36. H. Jerome Keisler (1970). [Omnibus Review]. Journal of Symbolic Logic 35 (2):342-344.
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  37. H. Jerome Keisler (1970). Review: R. L. Vaught, Models of Complete Theories. [REVIEW] Journal of Symbolic Logic 35 (2):344-344.
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  38. H. Jerome Keisler (1969). Review: Simon Kochen, J. W. Addison, Leon Henkin, Alfred Tarski, Topics in the Theory of Definition; Walter Felscher, On Criteria of Definability. [REVIEW] Journal of Symbolic Logic 34 (2):300-301.
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  39. H. Jerome Keisler (1968). Review: Carol Karp, Nonaxiomatizability Results for Infinitary Systems. [REVIEW] Journal of Symbolic Logic 33 (3):478-479.
  40. H. Jerome Keisler (1967). Ultraproducts of Finite Sets. Journal of Symbolic Logic 32 (1):47-57.
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  41. H. Jerome Keisler (1967). Ultraproducts Which Are Not Saturated. Journal of Symbolic Logic 32 (1):23-46.
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  42. H. Jerome Keisler (1966). Review: K. I. Appel, Horn Sentences in Identity Theory. [REVIEW] Journal of Symbolic Logic 31 (1):131-132.
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  43. H. Jerome Keisler (1965). Limit Ultraproducts. Journal of Symbolic Logic 30 (2):212-234.
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  44. H. Jerome Keisler (1965). Some Applications of Infinitely Long Formulas. Journal of Symbolic Logic 30 (3):339-349.
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  45. C. C. Chang & H. Jerome Keisler (1962). An Improved Prenex Normal Form. Journal of Symbolic Logic 27 (3):317-326.
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  46. H. Jerome Keisler (1962). Review: C. C. Chang, Anne C. Morel, On Closure Under Direct Product. [REVIEW] Journal of Symbolic Logic 27 (2):234-235.
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  47. H. Jerome Keisler (1960). Theory of Models with Generalized Atomic Formulas. Journal of Symbolic Logic 25 (1):1-26.
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