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31Co-stationarity of the Ground ModelJournal of Symbolic Logic 71 (3). 2006.This paper investigates when it is possible for a partial ordering P to force Pκ(λ) \ V to be stationary in VP. It follows from a result of Gitik that whenever P adds a new real, then Pκ(λ) \ V is stationary in VP for each regular uncountable cardinal κ in VP and all cardinals λ > κ in VP [4]. However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω₁-Erdős cardinals: If …Read more
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26Baumgartnerʼs conjecture and bounded forcing axiomsAnnals of Pure and Applied Logic 164 (12): 1178-1186. 2013.
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1Theorem 1 (Easton's Theorem). There is a forcing extension L [G] of L in which GCH fails at every regular cardinal. Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L [G] and V? In particular, are these models compatible (review)Bulletin of Symbolic Logic 12 (4). 2006.
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28Large cardinals and lightface definable well-orders, without the gchJournal of Symbolic Logic 80 (1): 251-284. 2015.
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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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65The effective theory of Borel equivalence relationsAnnals of Pure and Applied Logic 161 (7): 837-850. 2010.The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver [20] and Harrington, Kechris and Louveau [6] show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on , the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on . In this article we examine the effective …Read more
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25The stable coreBulletin of Symbolic Logic 18 (2): 261-267. 2012.Vopenka [2] proved long ago that every set of ordinals is set-generic over HOD, Gödel's inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over, and indeed over the even smaller inner model $\mathbb{S}=$, where S is the Stability predicate. We refer to the inner model $\mathbb{S}$ as the Stable Core of V. The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible to add reals w…Read more
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56Projective mad familiesAnnals of Pure and Applied Logic 161 (12): 1581-1587. 2010.Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω together with
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6Internal consistency for embedding complexityJournal of Symbolic Logic 73 (3): 831-844. 2008.In a previous paper with M. Džamonja, class forcings were given which fixed the complexity (a universality covering number) for certain types of structures of size λ together with the value of 2λ for every regular λ. As part of a programme for examining when such global results can be true in an inner model, we build generics for these class forcings
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38Eastonʼs theorem and large cardinals from the optimal hypothesisAnnals of Pure and Applied Logic 163 (12): 1738-1747. 2012.The equiconsistency of a measurable cardinal with Mitchell order o=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ .In Friedman and Honzik [5], we formulated and proved Eastonʼs theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik , for a suitable μ, instead of the cardinal…Read more
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34Rank-into-rank hypotheses and the failure of GCHArchive for Mathematical Logic 53 (3-4): 351-366. 2014.In this paper we are concerned about the ways GCH can fail in relation to rank-into-rank hypotheses, i.e., very large cardinals usually denoted by I3, I2, I1 and I0. The main results are a satisfactory analysis of the way the power function can vary on regular cardinals in the presence of rank-into-rank hypotheses and the consistency under I0 of the existence of j:Vλ+1≺Vλ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepacka…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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432Definable well-orders of $H(\omega _2)$ and $GCH$Journal of Symbolic Logic 77 (4): 1101-1121. 2012.Assuming ${2^{{N_0}}}$ = N₁ and ${2^{{N_1}}}$ = N₂, we build a partial order that forces the existence of a well-order of H(ω₂) lightface definable over ⟨H(ω₂), Є⟩ and that preserves cardinal exponentiation and cofinalities.
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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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25On Absoluteness of Categoricity in Abstract Elementary ClassesNotre Dame Journal of Formal Logic 52 (4): 395-402. 2011.Shelah has shown that $\aleph_1$-categoricity for Abstract Elementary Classes (AECs) is not absolute in the following sense: There is an example $K$ of an AEC (which is actually axiomatizable in the logic $L(Q)$) such that if $2^{\aleph_0}
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Generalizations of Gödel's universe of constructible setsIn Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial, Association For Symbolic Logic. 2010.
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33Analytic equivalence relations and bi-embeddabilityJournal of Symbolic Logic 76 (1). 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 (or on any class of countable structures consisting of the models of a sentence of L ω ₁ ω ) is far from complete (see [5, 2]). In this article we strength…Read more
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29Large cardinals and gap-1 morassesAnnals of Pure and Applied Logic 159 (1-2): 71-99. 2009.We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong , hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of …Read more
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19The tree property at א ω+2Journal of Symbolic Logic 76 (2). 2011.Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension א ω is a strong limit cardinal and א ω+2 has the tree property. This improves a result of Matthew Foreman (see [2])
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67Foundational implications of the inner model hypothesisAnnals of Pure and Applied Logic 163 (10): 1360-1366. 2012.
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46Perfect trees and elementary embeddingsJournal of Symbolic Logic 73 (3): 906-918. 2008.An important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j*: M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the generic G*…Read more
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22Easton’s theorem and large cardinalsAnnals of Pure and Applied Logic 154 (3): 191-208. 2008.The continuum function αmaps to2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, image and α
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33Homogeneous iteration and measure one covering relative to HODArchive for Mathematical Logic 47 (7-8): 711-718. 2008.Relative to a hyperstrong cardinal, it is consistent that measure one covering fails relative to HOD. In fact it is consistent that there is a superstrong cardinal and for every regular cardinal κ, κ + is greater than κ + of HOD. The proof uses a very general lemma showing that homogeneity is preserved through certain reverse Easton iterations
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50Hyperfine Structure Theory and Gap 1 MorassesJournal of Symbolic Logic 71 (2). 2006.Using the Friedman-Koepke Hyperfine Structure Theory of [2], we provide a short construction of a gap 1 morass in the constructible universe
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47Large cardinals and locally defined well-orders of the universeAnnals of Pure and Applied Logic 157 (1): 1-15. 2009.By forcing over a model of with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H is a well-order of H definable over the structure H, by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in add…Read more
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33The number of normal measuresJournal of Symbolic Logic 74 (3): 1069-1080. 2009.There have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH, where a is a cardinal at most κ⁺⁺. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = κ⁺⁺, the maximum possible) and [1] (for α = κ⁺, after collapsing κ⁺⁺) . In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of Mitchell order α ) , [2] (as in [12], but where κ…Read more