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9A theorem on ROD-hypersmooth equivalence relations in the Solovay modelMathematical Logic Quarterly 49 (3): 299. 2003.It is known that every Borel hypersmooth but non-smooth equivalence relation is Borel bi-reducible to E1. We prove a ROD version of this result in the Solovay model
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60On non-wellfounded iterations of the perfect set forcingJournal of Symbolic Logic 64 (2): 551-574. 1999.We prove that if I is a partially ordered set in a countable transitive model M of ZFC then M can be extended by a generic sequence of reals a i , i ∈ I, such that ℵ M 1 is preserved and every a i is Sacks generic over $\mathfrak{M}[\langle \mathbf{a}_j: j . The structure of the degrees of M-constructibility of reals in the extension is investigated. As applications of the methods involved, we define a cardinal invariant to distinguish product and iterated Sacks extensions, and give a short proo…Read more
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8Internal Approach to External Sets and Universes: Part 2. External Universes over the Universe of Bounded Set TheoryStudia Logica 55 (3): 347-376. 1995.In this article we show how the universe of BST, bounded set theory can be enlarged by definable subclasses of sets so that Separation and Replacement are true in the enlargement for all formulas, including those in which the standardness predicate may occur. Thus BST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop topics in nonstandard analysis inaccessible in the framework of a purely internal approach, such as Loeb measures.
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92A version of the Jensen–Johnsbråten coding at arbitrary level n≥ 3Archive for Mathematical Logic 40 (8): 615-628. 2001.We generalize, on higher projective levels, a construction of “incompatible” generic Δ1 3 real singletons given by Jensen and Johnsbråten
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67A definable nonstandard model of the realsJournal of Symbolic Logic 69 (1): 159-164. 2004.We prove, in ZFC,the existence of a definable, countably saturated elementary extension of the reals
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47Uniqueness, collection, and external collapse of cardinals in ist and models of peano arithmeticJournal of Symbolic Logic 60 (1): 318-324. 1995.We prove that in IST, Nelson's internal set theory, the Uniqueness and Collection principles, hold for all (including external) formulas. A corollary of the Collection theorem shows that in IST there are no definable mappings of a set X onto a set Y of greater (not equal) cardinality unless both sets are finite and #(Y) ≤ n #(X) for some standard n. Proofs are based on a rather general technique which may be applied to other nonstandard structures. In particular we prove that in a nonstandard mo…Read more
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22On Baire Measurable Homomorphisms of Quotients of the Additive Group of the RealsMathematical Logic Quarterly 46 (3): 377-384. 2000.The quotient ℝ/G of the additive group of the reals modulo a countable subgroup G does not admit nontrivial Baire measurable automorphisms
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150Internal approach to external sets and universesStudia Logica 55 (2). 1995.In this article we show how the universe of BST, bounded set theory can be enlarged by definable subclasses of sets so that Separation and Replacement are true in the enlargement for all formulas, including those in which the standardness predicate may occur. Thus BST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop topics in nonstandard analysis inaccessible in the framework of a purely internal approach, such as Loeb measures
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17A nonstandard set theory in the [mathematical formula]-languageArchive for Mathematical Logic 39 (6): 403-416. 2000.
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32Leibniz versus Ishiguro: Closing a Quarter Century of SyncategoremaniaHopos: The Journal of the International Society for the History of Philosophy of Science 6 (1): 117-147. 2016.Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well a…Read more
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5On Non-Wellfounded Iterations of the Perfect Set ForcingJournal of Symbolic Logic 64 (2): 551-574. 1999.We prove that if I is a partially ordered set in a countable transitive model $\mathfrak{M}$ of $\mathbf{ZFC}$ then $\mathfrak{M}$ can be extended by a generic sequence of reals $\mathbf{a}_i$, i $\in$ I, such that $\aleph^{\mathfrak{M}}_1$ is preserved and every $\mathbf{a}_i$ is Sacks generic over $\mathfrak{M}[\langle \mathbf{a}_j : j < i\rangle]$. The structure of the degrees of $\mathfrak{M}$-constructibility of reals in the extension is investigated. As applications of the methods involved…Read more
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40Isomorphism property in nonstandard extensions of theZFC universeAnnals of Pure and Applied Logic 88 (1): 1-25. 1997.We study models of HST . This theory admits an adequate formulation of the isomorphism propertyIP, which postulates that any two elementarily equivalent internally presented structures of a well-orderable language are isomorphic. We prove that IP is independent of HST and consistent with HST
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20Counterexamples to countable-section Π 2 1 uniformization and Π 3 1 separationAnnals of Pure and Applied Logic 167 (3): 262-283. 2016.
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7A Definable Nonstandard Model Of The RealsJournal of Symbolic Logic 69 (1): 159-164. 2004.We prove, in ZFC, the existence of a definable, countably saturated elementary extension of the reals.
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25Ulm Classification of Analytic Equivalence Relations in Generic UniversesMathematical Logic Quarterly 44 (3): 287-303. 1998.
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5Internal Approach to External Sets and Universes: Part 1. Bounded Set TheoryStudia Logica 55 (2): 229-257. 1995.A problem which enthusiasts of IST, Nelson's internal set theory, usually face is how to treat external sets in the internal universe which does not contain them directly. To solve this problem, we consider BST, bounded set theory, a modification of IST which is, briefly, a theory for the family of those IST sets which are members of standard sets. We show that BST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop the most advanced applications…Read more
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57An Ulm-type classification theorem for equivalence relations in Solovay modelJournal of Symbolic Logic 62 (4): 1333-1351. 1997.We prove that in the Solovay model, every OD equivalence relation, E, over the reals, either admits an OD reduction to the equality relation on the set of all countable (of length $ ) binary sequences, or continuously embeds E 0 , the Vitali equivalence. If E is a Σ 1 1 (resp. Σ 1 2 ) relation then the reduction above can be chosen in the class of all ▵ 1 (resp. ▵ 2 ) functions. The proofs are based on a topology generated by OD sets
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23Toward a History of Mathematics Focused on ProceduresFoundations of Science 22 (4): 763-783. 2017.Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the ela…Read more
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43Proofs and Retributions, Or: Why Sarah Can’t Take LimitsFoundations of Science 20 (1): 1-25. 2015.The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal a…Read more
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17Loeb Measure from the Point of View of a Coin Flipping GameMathematical Logic Quarterly 42 (1): 19-26. 1996.A hyperfinitely long coin flipping game between the Gambler and the Casino, associated with a given set A, is considered. It turns out that the Gambler has a winning strategy if and only if A has Loeb measure 0. The Casino has a winning strategy if and only if A contains an internal subset of positive Loeb measure
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53Elementary extensions of external classes in a nonstandard universeStudia Logica 60 (2): 253-273. 1998.In continuation of our study of HST, Hrbaek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st--language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe.We show that, given a standard cardinal , any set R * generates an "internal" class S(R) of all sets standard relatively to elements of R, and an "e…Read more
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103A nonstandard set theory in the $\displaystyle\in$ -languageArchive for Mathematical Logic 39 (6): 403-416. 2000.. We demonstrate that a comprehensive nonstandard set theory can be developed in the standard $\displaystyle{\in}$ -language. As an illustration, a nonstandard ${\sf Law of Large Numbers}$ is obtained
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43On external Scott algebras in nonstandard models of peano arithmeticJournal of Symbolic Logic 61 (2): 586-607. 1996.We prove that a necessary and sufficient condition for a countable set L of sets of integers to be equal to the algebra of all sets of integers definable in a nonstandard elementary extension of ω by a formula of the PA language which may include the standardness predicate but does not contain nonstandard parameters, is as follows: L is closed under arithmetical definability and contains 0 (ω) , the set of all (Gödel numbers of) true arithmetical sentences. Some results related to definability o…Read more
Areas of Interest
17th/18th Century Philosophy |