• ##### Canonization of Smooth Equivalence Relations on Infinite-Dimensional $mathsf{E}_{0}$-Large Products with Vassily Lyubetsky Notre 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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##### Countable OD sets of reals belong to the ground model with Vassily Lyubetsky Archive 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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##### Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms with Tiziana Bascelli, Piotr Błaszczyk, Alexandre Borovik, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Thomas McGaffey, David M. Schaps, and David Sherry Foundations 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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##### Gregory’s Sixth Operation with Tiziana Bascelli, Piotr Błaszczyk, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Tahl Nowik, David M. Schaps, and David Sherry Foundations 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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##### What Makes a Theory of Infinitesimals Useful? A View by Klein and Fraenkel with K. Katz, M. Katz, and Thomas Mormann Journal 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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##### A Definable Nonstandard Model Of The Reals with Saharon Shelah Journal 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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##### A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos with Piotr Błaszczyk, Alexandre Borovik, Mikhail G. Katz, Taras Kudryk, Semen S. Kutateladze, and David Sherry Logica 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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##### Uniqueness, collection, and external collapse of cardinals in ist and models of peano arithmetic Journal 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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##### Internal Approach to External Sets and Universes: Part 1. Bounded Set Theory with Michael Reeken Studia 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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##### Special Model Axiom in Nonstandard Set Theory with Michael Reeken Mathematical Logic Quarterly 45 (3): 371-384. 1999.
We demonstrate that the special model axiom SMA of Ross admits a natural formalization in Kawai's nonstandard set theory KST but is independent of KST. As an application of our methods to classical model theory, we present a short proof of the consistency of the existence of a k+ like k-saturated model of PA for a given cardinal k
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##### Linearization of definable order relations Annals of Pure and Applied Logic 102 (1-2): 69-100. 2000.
We prove that if ≼ is an analytic partial order then either ≼ can be extended to a Δ 2 1 linear order similar to an antichain in 2 ω 1 , ordered lexicographically, or a certain Borel partial order ⩽ 0 embeds in ≼. Similar linearization results are presented, for κ -bi-Souslin partial orders and real-ordinal definable orders in the Solovay model. A corollary for analytic equivalence relations says that any Σ 1 1 equivalence relation E , such that E 0 does not embed in E , is fully determined by i…Read more
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##### Extending standard models of ZFC to models of nonstandard set theories with Michael Reeken Studia Logica 64 (1): 37-59. 2000.
We study those models of ZFCwhich are embeddable, as the class of all standard sets, in a model of internal set theory &gt;ISTor models of some other nonstandard set theories.