24 found
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  1.  11
    Independence Results Around Constructive ZF.Robert S. Lubarsky - 2005 - Annals of Pure and Applied Logic 132 (2-3):209-225.
    CZF is an intuitionistic set theory that does not contain Power Set, substituting instead a weaker version, Subset Collection. In this paper a Kripke model of CZF is presented in which Power Set is false. In addition, another Kripke model is presented of CZF with Subset Collection replaced by Exponentiation, in which Subset Collection fails.
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  2.  44
    On the Constructive Dedekind Reals.Robert S. Lubarsky & Michael Rathjen - 2008 - Logic and Analysis 1 (2):131-152.
    In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy (...)
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  3.  7
    CZF and Second Order Arithmetic.Robert S. Lubarsky - 2006 - Annals of Pure and Applied Logic 141 (1):29-34.
    Constructive ZF + full separation is shown to be equiconsistent with Second Order Arithmetic.
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  4.  3
    Separating the Fan Theorem and its Weakenings.Robert S. Lubarsky & Hannes Diener - 2014 - Journal of Symbolic Logic 79 (3):792-813.
    Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.
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  5.  19
    On the Regular Extension Axiom and its Variants.Robert S. Lubarsky & Michael Rathjen - 2003 - Mathematical Logic Quarterly 49 (5):511.
    The regular extension axiom, REA, was first considered by Peter Aczel in the context of Constructive Zermelo-Fraenkel Set Theory as an axiom that ensures the existence of many inductively defined sets. REA has several natural variants. In this note we gather together metamathematical results about these variants from the point of view of both classical and constructive set theory.
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  6.  14
    On the Cauchy Completeness of the Constructive Cauchy Reals.Robert S. Lubarsky - 2007 - Mathematical Logic Quarterly 53 (4‐5):396-414.
    It is consistent with constructive set theory (without Countable Choice, clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of rationals) are not Cauchy complete. Related results are also shown, such as that a Cauchy sequence of rationals may not have a modulus of convergence, and that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy sequence, among others.
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  7.  16
    IKP and Friends.Robert S. Lubarsky - 2002 - Journal of Symbolic Logic 67 (4):1295-1322.
  8.  54
    Topological Forcing Semantics with Settling.Robert S. Lubarsky - 2012 - Annals of Pure and Applied Logic 163 (7):820-830.
  9.  10
    Μ-Definable Sets of Integers.Robert S. Lubarsky - 1993 - Journal of Symbolic Logic 58 (1):291-313.
  10.  4
    On Extensions of Supercompactness.Robert S. Lubarsky & Norman Lewis Perlmutter - 2015 - Mathematical Logic Quarterly 61 (3):217-223.
  11.  38
    A. Louveau. Some Results in the Wadge Hierarchy of Borel Sets. Cabal Seminar 79–81, Proceedings, Caltech-UCLA Logic Seminar 1979–81, Edited by A. S. Kechris, D. A. Martin, and Y. N. Moschovakis, Lecture Notes in Mathematics, Vol. 1019, Springer-Verlag, Berlin Etc. 1983, Pp. 28–55. - A. Louveau and J. Saint Raymond. Borel Classes and Closed Games: Wadge-Type and Hurewicz-Type Results. Transactions of the American Mathematical Society, Vol. 304 , Pp. 431–467. - Alain Louveau and Jean Saint Raymond. The Strength of Borel Wadge Determinacy. Cabal Seminar 81–85, Proceedings, Caltech-UCLA Logic Seminar 1981–85, Edited by A. S. Kechris, D. A. Martin, and J. R. Steel, Lecture Notes in Mathematics, Vol. 1333, Springer-Verlag, Berlin Etc. 1988, Pp. 1–30. [REVIEW]Robert S. Lubarsky - 1992 - Journal of Symbolic Logic 57 (1):264-266.
  12.  3
    Lattices of C-Degrees.Robert S. Lubarsky - 1987 - Annals of Pure and Applied Logic 36:115-118.
  13.  28
    Definability and Initial Segments of C-Degrees.Robert S. Lubarsky - 1988 - Journal of Symbolic Logic 53 (4):1070-1081.
    We combine two techniques of set theory relating to minimal degrees of constructibility. Jensen constructed a minimal real which is additionally a Π 1 2 singleton. Groszek built an initial segment of order type 1 + α * , for any ordinal α. This paper shows how to force a Π 1 2 singleton such that the c-degrees beneath it, all represented by reals, are of type 1 + α * , for many ordinals α. We also examine the definability (...)
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  14.  7
    Separating the Fan Theorem and its Weakenings II.Robert S. Lubarsky - 2019 - Journal of Symbolic Logic 84 (4):1484-1509.
    Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Earlier work showed all of these implications to be strict. Here we reprove one of the strictness results, using very different arguments. The technique used is a mixture of realizability, forcing in the guise of Heyting-valued models, and Kripke models.
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  15.  16
    Simple R. E. Degree Structures.Robert S. Lubarsky - 1987 - Journal of Symbolic Logic 52 (1):208-213.
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  16.  19
    On the Failure of BD-ࡃ and BD, and an Application to the Anti-Specker Property.Robert S. Lubarsky - 2013 - Journal of Symbolic Logic 78 (1):39-56.
    We give the natural topological model for $\neg$BD-${\mathbb N}$, and use it to show that the closure of spaces with the anti-Specker property under product does not imply BD-${\mathbb N}$. Also, the natural topological model for $\neg$BD is presented. Finally, for some of the realizability models known indirectly to falsify BD-$\mathbb{N}$, it is brought out in detail how BD-$\mathbb N$ fails.
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  17.  9
    Sacks Forcing Sometimes Needs Help to Produce a Minimal Upper Bound.Robert S. Lubarsky - 1989 - Journal of Symbolic Logic 54 (2):490-498.
  18.  6
    Correction to “Simple R. E. Degree Structures”.Robert S. Lubarsky - 1988 - Journal of Symbolic Logic 53 (1):103-104.
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  19.  5
    Correction to "Simple R. E. Degree Structures".Robert S. Lubarsky - 1988 - Journal of Symbolic Logic 53 (1):103-104.
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  20.  8
    Uncountable Master Codes and the Jump Hierarchy.Robert S. Lubarsky - 1987 - Journal of Symbolic Logic 52 (4):952-958.
  21.  4
    Principles Weaker Than BD-N.Robert S. Lubarsky & Hannes Diener - 2013 - Journal of Symbolic Logic 78 (3):873-885.
  22.  7
    An Introduction to Γ-Recursion Theory (or What to Do in KP - Foundation).Robert S. Lubarsky - 1990 - Journal of Symbolic Logic 55 (1):194-206.
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  23.  3
    Another Extension of Van de Wiele's Theorem.Robert S. Lubarsky - 1988 - Annals of Pure and Applied Logic 38 (3):301-306.
  24.  2
    Admissibility Spectra and Minimality.Robert S. Lubarsky - 1988 - Annals of Pure and Applied Logic 40 (3):257-276.