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56Regular ultrafilters and finite square principlesJournal of Symbolic Logic 73 (3): 817-823. 2008.We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle $\square _{\lambda ,D}^{\mathit{fin}}$ introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which Mλ / D is not λ⁺⁺-universal and elementarily equivalent models M and N of size λ for which Mλ / D and Nλ / D are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [1]
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48Gödel’s ModernismGraduate Faculty Philosophy Journal 25 (2): 289-349. 2004.On Friday, November 15, 1940, Kurt Gödel gave a talk on set theory at Brown University. The topic was his recent proof of the consistency of Cantor’s Continuum Hypothesis with the axiomatic system ZFC for set theory. His friend from their days in Vienna, Rudolf Carnap, was in the audience, and afterward wrote a note to himself in which he raised a number of questions on incompleteness
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32On regular reduced productsJournal of Symbolic Logic 67 (3): 1169-1177. 2002.Assume $\langle \aleph_0, \aleph_1 \rangle \rightarrow \langle \lambda, \lambda^+ \rangle$ . Assume M is a model of a first order theory T of cardinality at most λ+ in a language L(T) of cardinality $\leq \lambda$ . Let N be a model with the same language. Let Δ be a set of first order formulas in L(T) and let D be a regular filter on λ. Then M is $\Delta-embeddable$ into the reduced power $N^\lambda/D$ , provided that every $\Delta-existential$ formula true in M is true also in N. We obtain the…Read more
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31Gödel's LogicIn Dov Gabbay (ed.), The Handbook of the History of Logic, Elsevier. pp. 449-509. 2009.
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28More on Regular Reduced ProductsJournal of Symbolic Logic 69 (4). 2004.The authors show. by means of a finitary version $\square_{\lambda D}^{fin}$ of the combinatorial principle $\square_\lambda^{h*}$ of [7]. the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal A. if Mi and Ni are elementarily equivalent models of a language of size $\leq \lambda$ , then the second player has a winning strategy in the Ehrenfeucht- $Fra\uml{i}ss\acute{e}$ game of length $\lambda^{+}$ on $\pi…Read more
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23Gödel's Thesis--An AppreciationIn Baaz Mathias, Christos Papadimitriou, Hilary Putnam, Dana Scott & Charles Harper (eds.), Horizons of Truth, Cambridge University Press. pp. 95. 2011.
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20Regular Ultrapowers at Regular CardinalsNotre Dame Journal of Formal Logic 56 (3): 417-428. 2015.In earlier work by the first and second authors, the equivalence of a finite square principle $\square^{\mathrm{fin}}_{\lambda,D}$ with various model-theoretic properties of structures of size $\lambda $ and regular ultrafilters was established. In this paper we investigate the principle $\square^{\mathrm{fin}}_{\lambda,D}$—and thereby the above model-theoretic properties—at a regular cardinal. By Chang’s two-cardinal theorem, $\square^{\mathrm{fin}}_{\lambda,D}$ holds at regular cardinals for a…Read more
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13On the “Logic without Borders” Point of View: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of MathematicsIn Villaveces Kossak Kontinen Hirvonen Andres Roman Juha Asa (ed.), Logic without Borders, De Gruyter. pp. 1-14. 2015.
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On embedding models of arithmetic of cardinality aleph_1 into reduced powersFundamenta Mathematicae 176 (1). 2003.
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On the Philosophical Development of Kurt GödelIn Robert Tragesser, Mark van Atten & Mark Atten (eds.), Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer, Springer Verlag. 2015.
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On Applications of Transfer Principles in Model TheoryIn Alessandro Andretta (ed.), On Applications of Transfer Principles in Model Theory, Quaderni Di Matematica. 2007.
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Gödel's Modernism: On Set Theoretic Incompleteness, RevisitedIn Sten Lindström, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.), Logicism, Intuitionism and Formalism: What has become of them?, Springer. 2009.
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On embedding models of arithmetic into reduced powersMatematica Contemporanea 24 (1): 91--115. 2003.
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Logic and Philosophy of Logic |
Philosophy of Mathematics |