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Parameterfree Comprehension Does Not Imply Full Comprehension in Second Order Peano ArithmeticStudia Logica 1-16. forthcoming.The parameter-free part $$\textbf{PA}_2^*$$ of $$\textbf{PA}_2$$, second order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an $$\omega $$ -model of $$\textbf{PA}_2^*+ \textbf{CA}(\Sigma ^1_2)$$, in which an example of the full Comprehension schema $$\textbf{CA}$$ fails. Using Cohen’s forcing, we also define an $$\omega $$ -model of $$\textbf{PA}_2^*$$, in which not every set has its complement, and hence the full $$\textbf{CA}$$ fails in a rather el…Read more
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9A good lightface Δ n 1 well-ordering of the reals does not imply the existence of boldface Δ n − 1 1 well-orderingsAnnals of Pure and Applied Logic 175 (6): 103426. 2024.
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24A Groszek‐Laver pair of undistinguishable ‐classesMathematical Logic Quarterly 63 (1-2): 19-31. 2017.A generic extension of the constructible universe by reals is defined, in which the union of ‐classes of x and y is a lightface set, but neither of these two ‐classes is separately ordinal‐definable.
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35A model of second-order arithmetic satisfying AC but not DCJournal of Mathematical Logic 19 (1): 1850013. 2019.We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. G. Kanovei, On…Read more
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29On a Spector Ultrapower for the Solovay ModelMathematical Logic Quarterly 43 (3): 389-395. 1997.We prove that a Spector‐like ultrapower extension ???? of a countable Solovay model ???? (where all sets of reals are Lebesgue measurable) is equal to the set of all sets constructible from reals in a generic extension ????[a], where a is a random real over ????. The proof involves the Solovay almost everywhere uniformization technique.
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1318Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in MathematicsFoundations of Science 18 (2): 259-296. 2013.We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclus…Read more
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596Is Leibnizian calculus embeddable in first order logic?Foundations of Science 22 (4). 2017.To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found…Read more
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44Cauchy’s Infinitesimals, His Sum Theorem, and Foundational ParadigmsFoundations of Science 23 (2): 267-296. 2018.Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.
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57A Non-Standard Analysis of a Cultural Icon: The Case of Paul HalmosLogica Universalis 10 (4): 393-405. 2016.We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematic…Read more
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131Interpreting the Infinitesimal Mathematics of Leibniz and EulerJournal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 48 (2): 195-238. 2017.We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a …Read more
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15On coding uncountable sets by realsMathematical Logic Quarterly 56 (4): 409-424. 2010.If A ⊆ ω1, then there exists a cardinal preserving generic extension [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ][x ] of [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ] by a real x such that1) A ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ] and A is Δ1HC in [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ];2) x is minimal over [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ], that is, if a set Y belongs to [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ], then either x ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A, Y ] or Y …Read more
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19An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisitedJournal of Mathematical Logic 21 (3): 2150014. 2020.A definable pair of disjoint non-OD sets of reals exists in the Sacks and ????0-large generic extensions of the constructible universe L. More specifically, if a∈2ω is eith...
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17The full basis theorem does not imply analytic wellorderingAnnals of Pure and Applied Logic 172 (4): 102929. 2021.
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8Canonization of Smooth Equivalence Relations on Infinite-Dimensional E0-Large ProductsNotre Dame Journal of Formal Logic 61 (1): 117-128. 2020.We propose a canonization scheme for smooth equivalence relations on Rω modulo restriction to E0-large infinite products. It shows that, given a pair of Borel smooth equivalence relations E, F on Rω, there is an infinite E0-large perfect product P⊆Rω such that either F⊆E on P, or, for some ℓ
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12Definable minimal collapse functions at arbitrary projective levelsJournal of Symbolic Logic 84 (1): 266-289. 2019.
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31Minimal axiomatic frameworks for definable hyperreals with transferJournal of Symbolic Logic 83 (1): 385-391. 2018.
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20Definable E 0 classes at arbitrary projective levelsAnnals of Pure and Applied Logic 169 (9): 851-871. 2018.
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14Countable OD sets of reals belong to the ground modelArchive for Mathematical Logic 57 (3-4): 285-298. 2018.It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \ elements.
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50Gregory’s Sixth OperationFoundations of Science 23 (1): 133-144. 2018.In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here Gregory…Read more
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591What Makes a Theory of Infinitesimals Useful? A View by Klein and FraenkelJournal of Humanistic Mathematics 8 (1). 2018.Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.
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49Uniqueness, 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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41Isomorphism 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.
Areas of Interest
17th/18th Century Philosophy |