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19Failures of the silver dichotomy in the generalized baire spaceJournal of Symbolic Logic 80 (2): 661-670. 2015.We prove results that falsify Silver’s dichotomy for Borel equivalence relations on the generalized Baire space under the assumptionV=L.
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18Universally Baire sets and definable well-orderings of the realsJournal of Symbolic Logic 68 (4): 1065-1081. 2003.Let n ≥ 3 be an integer. We show that it is consistent that every σ1n-set of reals is universally Baire yet there is a projective well-ordering of the reals. The proof uses “David’s trick” in the presence of inner models with strong cardinals.
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18The Hyperuniverse Project and Maximality (edited book)Birkhäuser. 2018.This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperu…Read more
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18Handbook of mathematical logic, edited by Barwise Jon with the cooperation of Keisler H. J., Kunen K., Moschovakis Y. N., and Troelstra A. S., Studies in logic and the foundations of mathematics, vol. 90, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1978 , xi + 1165 pp (review)Journal of Symbolic Logic 49 (3): 975-980. 1984.
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18Δ1-DefinabilityAnnals of Pure and Applied Logic 89 (1): 93-99. 1997.We isolate a condition on a class A of ordinals sufficient to Δ1-code it by a real in a class-generic extension of L. We then apply this condition to show that the class of ordinals of L-cofinality ω is Δ1 in a real of L-degree strictly below O#
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18The tree property at the ℵ 2 n 's and the failure of SCH at ℵ ωAnnals of Pure and Applied Logic 166 (4): 526-552. 2015.
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18Review: James E. Baumgartner, On the Size of Closed Unbounded Sets (review)Bulletin of Symbolic Logic 7 (4): 538-539. 2001.
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17Analytic equivalence relations and bi-embeddabilityJournal of Symbolic Logic 76 (1): 243-266. 2011.Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs is far from complete.In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence rela…Read more
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17Internal Consistency and Global Co-stationarity of the Ground ModelJournal of Symbolic Logic 73 (2). 2008.Global co-stationarity of the ground model from an N₂-c.c, forcing which adds a new subset of N₁ is internally consistent relative to an ω₁-Erdös hyperstrong cardinal and a sufficiently large measurable above
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17Multiverse Conceptions in Set TheoryIn Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 47-73. 2018.We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and pot…Read more
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17On Strong Forms of Reflection in Set TheoryIn Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 125-134. 2018.In this paper we review the most common forms of reflection and introduce a new form which we call sharp-generated reflection. We argue that sharp-generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp-maximality with the corresponding hypothesis IMH#. IMH# is an analogue of the IMH :591–600, 2006)) which is compatible with the existence of large cardinals.
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17Downey, R., Gasarch, W. and Moses, M., The structureAnnals of Pure and Applied Logic 70 (1): 287. 1994.
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161996–97 Annual Meeting of the Association for Symbolic LogicBulletin of Symbolic Logic 3 (3): 378-396. 1997.
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15Annual meeting of the association for symbolic logic: Boston 1983Journal of Symbolic Logic 49 (4): 1441-1449. 1984.
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14Fragments of Kripke–Platek set theory and the metamathematics of $$\alpha $$ α -recursion theoryArchive for Mathematical Logic 55 (7-8): 899-924. 2016.The foundation scheme in set theory asserts that every nonempty class has an ∈\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\in $$\end{document}-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and α\documentclass[12pt]{minimal} \usepackage{amsmath}…Read more
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13Embeddings Into Outer ModelsJournal of Symbolic Logic 87 (4): 1301-1321. 2022.We explore the possibilities for elementary embeddings $j : M \to N$, where M and N are models of ZFC with the same ordinals, $M \subseteq N$, and N has access to large pieces of j. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line ${\mathbb R}$.
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13Hyperclass Forcing in Morse-Kelley Class TheoryIn Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 17-46. 2018.In this article we introduce and study hyperclass-forcing in the context of an extension of Morse-Kelley class theory, called MK∗∗. We define this forcing by using a symmetry between MK∗∗ models and models of ZFC− plus there exists a strongly inaccessible cardinal. We develop a coding between β-models ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsid…Read more
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13On the Set-Generic MultiverseIn Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 109-124. 2018.The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver’s theorem and Bukovský’s theorem assert that set-generic extensions of a given ground model constitute a quite reasonable and sufficiently general class of standard models of set-theory.In Sects. 2 and 3 of this note, we give a proof of Bukovsky’s theorem in a modern setting ). In Sect. 4 we check that the multiverse of set-generic extensions c…Read more
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12The internal consistency of Easton’s theoremAnnals of Pure and Applied Logic 156 (2): 259-269. 2008.An Easton function is a monotone function C from infinite regular cardinals to cardinals such that C has cofinality greater than α for each infinite regular cardinal α. Easton showed that assuming GCH, if C is a definable Easton function then in some cofinality-preserving extension, C=2α for all infinite regular cardinals α. Using “generic modification”, we show that over the ground model L, models witnessing Easton’s theorem can be obtained as inner models of L[0#], for Easton functions which a…Read more
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12A wellorder of the reals with saturatedJournal of Symbolic Logic 84 (4): 1466-1483. 2019.We show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $, there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.
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12Definability degreesMathematical Logic Quarterly 51 (5): 448-449. 2005.We establish the equiconsistency of a simple statement in definability theory with the failure of the GCH at all infinite cardinals. The latter was shown by Foreman and Woodin to be consistent, relative to the existence of large cardinals