23 found
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  1.  41
    An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals.Alexandre Borovik, Renling Jin & Mikhail G. Katz - 2012 - Notre Dame Journal of Formal Logic 53 (4):557-570.
    A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On -saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently (...)
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  2.  65
    The Isomorphism Property Versus the Special Model Axiom.Renling Jin - 1992 - Journal of Symbolic Logic 57 (3):975-987.
    This paper answers some questions of D. Ross in [R]. In § 1, we show that some consequences of the ℵ0- or ℵ1-special model axiom in [R] cannot be proved by the κ-isomorphism property for any cardinal κ. In § 2, we show that with one exception, the ℵ0-isomorphism property does imply the remaining consequences of the special model axiom in [R]. In § 3, we improve a result in [R] by showing that the κ-special model axiom is equivalent to (...)
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  3.  67
    A Theorem on the Isomorphism Property.Renling Jin - 1992 - Journal of Symbolic Logic 57 (3):1011-1017.
    An L-structure is called internally presented in a nonstandard universe if its base set and interpretation of every symbol in L are internal. A nonstandard universe is said to satisfy the κ-isomorphism property if for any two internally presented L-structures U and B, where L has less than κ many symbols, U is elementarily equivalent to B implies that U is isomorphic to B. In this paper we prove that the ℵ1-isomorphism property is equivalent to the ℵ0-isomorphism property plus ℵ1-saturation.
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  4. The Strength of the Isomorphism Property.Renling Jin & Saharon Shelah - 1994 - Journal of Symbolic Logic 59 (1):292-301.
    In § 1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second-order types in model theory. In § 2, several applications are given. One of the applications answers a question of D. Ross in [this Journal, vol. 55 (1990), pp. 1233-1242] about infinite Loeb measure spaces.
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  5.  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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  6.  12
    A Model in Which Every Kurepa Tree is Thick.Renling Jin - 1991 - Notre Dame Journal of Formal Logic 33 (1):120-125.
  7.  45
    Possible Size of an Ultrapower of $\Omega$.Renling Jin & Saharon Shelah - 1999 - Archive for Mathematical Logic 38 (1):61-77.
    Let $\omega$ be the first infinite ordinal (or the set of all natural numbers) with the usual order $<$ . In § 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of $\omega$ , whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In § 2 we construct several $\lambda$ -Archimedean ultrapowers of $\omega$ under some large cardinal (...)
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  8.  36
    The Differences Between Kurepa Trees and Jech-Kunen Trees.Renling Jin - 1993 - Archive for Mathematical Logic 32 (5):369-379.
    By an ω1 we mean a tree of power ω1 and height ω1. An ω1-tree is called a Kurepa tree if all its levels are countable and it has more than ω1 branches. An ω1-tree is called a Jech-Kunen tree if it has κ branches for some κ strictly between ω1 and $2^{\omega _1 }$ . In Sect. 1, we construct a model ofCH plus $2^{\omega _1 } > \omega _2$ , in which there exists a Kurepa tree with not (...)
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  9.  22
    Distinguishing Three Strong Saturation Properties in Nonstandard Analysis.Renling Jin - 1999 - Annals of Pure and Applied Logic 98 (1-3):157-171.
    Three results in [14] and one in [8] are analyzed in Sections 3–6 in order to supply examples on Loeb probability spaces, which distinguish the different strength among three generalizations of k-saturation, as well to answer some questions in Section 7 of [15]. In Section 3 we show that not every automorphism of a Loeb algebra is induced by an internal permutation, in Section 4 we show that if the 1-special model axiom is true, then every automorphism of a Loeb (...)
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  10.  14
    Some Independence Results Related to the Kurepa Tree.Renling Jin - 1991 - Notre Dame Journal of Formal Logic 32 (3):448-457.
  11.  13
    Type Two Cuts, Bad Cuts and Very Bad Cuts.Renling Jin - 1997 - Journal of Symbolic Logic 62 (4):1241-1252.
    Type two cuts, bad cuts and very bad cuts are introduced in [10] for studying the relationship between Loeb measure and U-topology of a hyperfinite time line in an ω 1 -saturated nonstandard universe. The questions concerning the existence of those cuts are asked there. In this paper we answer, fully or partially, some of those questions by showing that: (1) type two cuts exist, (2) the ℵ 1 -isomorphism property implies that bad cuts exist, but no bad cuts are (...)
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  12.  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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  13.  23
    Cuts in Hyperfinite Time Lines.Renling Jin - 1992 - Journal of Symbolic Logic 57 (2):522-527.
    In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time (...)
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  14.  23
    Can a Small Forcing Create Kurepa Trees.Renling Jin & Saharon Shelah - 1997 - Annals of Pure and Applied Logic 85 (1):47-68.
    In this paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees by an ω1-preserving forcing notion of size at most ω1. In Section 1 we show that in the Lévy model obtained by collapsing all cardinals between ω1 and a strongly inaccessible cardinal by forcing with a countable support Lévy collapsing order, many ω1-preserving forcing notions of size at most ω1 including all ω-proper forcing notions (...)
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  15.  10
    Essential Kurepa Trees Versus Essential Jech–Kunen Trees.Renling Jin & Saharon Shelah - 1994 - Annals of Pure and Applied Logic 69 (1):107-131.
    By an ω1-tree we mean a tree of cardinality ω1 and height ω1. An ω1-tree is called a Kurepa tree if all its levels are countable and it has more than ω1 branches. An ω1-tree is called a Jech–Kunen tree if it has κ branches for some κ strictly between ω1 and 2ω1. A Kurepa tree is called an essential Kurepa tree if it contains no Jech–Kunen subtrees. A Jech–Kunen tree is called an essential Jech–Kunen tree if it is no (...)
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  16.  5
    High Density Piecewise Syndeticity of Product Sets in Amenable Groups.Mauro di Nasso, Isaac Goldbring, Renling Jin, Steven Leth, Martino Lupini & Karl Mahlburg - 2016 - Journal of Symbolic Logic 81 (4):1555-1562.
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  17.  27
    Applications of Nonstandard Analysis in Additive Number Theory.Renling Jin - 2000 - Bulletin of Symbolic Logic 6 (3):331-341.
    This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.
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  18.  20
    Compactness of Loeb Spaces.Renling Jin & Saharon Shelah - 1998 - Journal of Symbolic Logic 63 (4):1371-1392.
    In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In $\S1$ we prove that Loeb spaces are compact under various assumptions, and in $\S2$ we prove that Loeb spaces are not compact under various other assumptions. The results in $\S1$ and $\S2$ give a quite complete answer to a question of D. Ross in [9], [11] and [12].
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  19.  23
    Existence of Some Sparse Sets of Nonstandard Natural Numbers.Renling Jin - 2001 - Journal of Symbolic Logic 66 (2):959-973.
    Answers are given to two questions concerning the existence of some sparse subsets of $\mathscr{H} = \{0, 1,..., H - 1\} \subseteq * \mathbb{N}$ , where H is a hyperfinite integer. In § 1, we answer a question of Kanovei by showing that for a given cut U in H, there exists a countably determined set $X \subseteq \mathscr{H}$ which contains exactly one element in each U-monad, if and only if U = a · N for some $a \in \mathscr{H} (...)
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  20.  3
    Slow P-Point Ultrafilters.Renling Jin - 2020 - Journal of Symbolic Logic 85 (1):26-36.
    We answer a question of Blass, Di Nasso, and Forti [2, 7] by proving, assuming Continuum Hypothesis or Martin’s Axiom, that there exists a P-point which is not interval-to-one and there exists an interval-to-one P-point which is neither quasi-selective nor weakly Ramsey.
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  21.  27
    U-Lusin Sets in Hyperfinite Time Lines.Renling Jin - 1992 - Journal of Symbolic Logic 57 (2):528-533.
    In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time (...)
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  22.  28
    U-Monad Topologies of Hyperfinite Time Lines.Renling Jin - 1992 - Journal of Symbolic Logic 57 (2):534-539.
    In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time (...)
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  23.  24
    Inverse Problem for Cuts.Renling Jin - 2007 - Logic and Analysis 1 (1):61-89.
    Let U be an initial segment of $^*{\mathbb N}$ closed under addition (such U is called a cut) with uncountable cofinality and A be a subset of U, which is the intersection of U and an internal subset of $^*{\mathbb N}$ . Suppose A has lower U-density α strictly between 0 and 3/5. We show that either there exists a standard real $\epsilon$ > 0 and there are sufficiently large x in A such that | (A+A) ∩ [0, 2x]| > (...)
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