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19Jensen's $Sigma^ast$ Theory and the Combinatorial Content of $V = L$Journal of Symbolic Logic 59 (3): 1096-1104. 1994.
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57Projective wellorders and mad families with large continuumAnnals of Pure and Applied Logic 162 (11): 853-862. 2011.We show that is consistent with the existence of a -definable wellorder of the reals and a -definable ω-mad subfamily of [ω]ω
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30Jensen's Σ* theory and the combinatorial content of V = LJournal of Symbolic Logic 59 (3). 1994.
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60Universally 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 (relative to the consistency of n - 2 strong cardinals) that every $\Sigma_n^1-set$ of reals is universally Baire yet there is a (lightface) 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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42Cardinal-preserving extensionsJournal of Symbolic Logic 68 (4): 1163-1170. 2003.A classic result of Baumgartner-Harrington-Kleinberg [1] implies that assuming CH a stationary subset of ω1 has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω1. Therefore, assuming that $\omega_2^L$ is countable: { $X \in L \mid X \subseteq \omega_1^L$ and X has a CUB subset in a cardinal -preserving extension of L} is constructible, as it equals the set of constructible subsets of $\omega_1^L$ which in L are stationary. Is there a similar such resul…Read more
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32An elementary approach to the fine structure of LBulletin of Symbolic Logic 3 (4): 453-468. 1997.We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the traditional Lα -or Jα-seque…Read more
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31Annual meeting of the association for symbolic logicJournal of Symbolic Logic 58 (1): 370-382. 1993.
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804The Search for New Axioms in the Hyperuniverse ProgrammeIn Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics, Springer International Publishing. pp. 165-188. 2016.The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identifies higher-order statements motivated by the ma…Read more
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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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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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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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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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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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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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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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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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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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67Foundational implications of the inner model hypothesisAnnals of Pure and Applied Logic 163 (10): 1360-1366. 2012.
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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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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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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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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