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16Jensen's $Sigma^ast$ Theory and the Combinatorial Content of $V = L$Journal of Symbolic Logic 59 (3): 1096-1104. 1994.
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56Projective 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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59Universally 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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31Annual meeting of the association for symbolic logicJournal of Symbolic Logic 58 (1): 370-382. 1993.
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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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797The 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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11An inner model for global dominationJournal of Symbolic Logic 74 (1): 251-264. 2009.In this paper it is shown that the global statement that the dominating number for k is less than $2^k $ for all regular k, is internally consistent, given the existence of $0^\# $ . The possible range of values for the dominating number for k and $2^k $ which may be simultaneously true in an inner model is also explored
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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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40Strong isomorphism reductions in complexity theoryJournal of Symbolic Logic 76 (4): 1381-1402. 2011.We give the first systematic study of strong isomorphism reductions, a notion of reduction more appropriate than polynomial time reduction when, for example, comparing the computational complexity of the isomorphim problem for different classes of structures. We show that the partial ordering of its degrees is quite rich. We analyze its relationship to a further type of reduction between classes of structures based on purely comparing for every n the number of nonisomorphic structures of cardina…Read more
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56Slow consistencyAnnals of Pure and Applied Logic 164 (3): 382-393. 2013.The fact that “natural” theories, i.e. theories which have something like an “idea” to them, are almost always linearly ordered with regard to logical strength has been called one of the great mysteries of the foundation of mathematics. However, one easily establishes the existence of theories with incomparable logical strengths using self-reference . As a result, PA+Con is not the least theory whose strength is greater than that of PA. But still we can ask: is there a sense in which PA+Con is t…Read more
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139The hyperuniverse programBulletin of Symbolic Logic 19 (1): 77-96. 2013.The Hyperuniverse Program is a new approach to set-theoretic truth which is based on justifiable principles and leads to the resolution of many questions independent from ZFC. The purpose of this paper is to present this program, to illustrate its mathematical content and implications, and to discuss its philosophical assumptions.
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21Large cardinals and definable well-orders, without the GCHAnnals of Pure and Applied Logic 166 (3): 306-324. 2015.
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48Fusion and large cardinal preservationAnnals of Pure and Applied Logic 164 (12): 1247-1273. 2013.In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ⩽κ not only does not collapse κ+ but also preserves the strength of κ. This provides a general theory covering the known cases of tree iterations which preserve large cardinals [3], Friedman and Halilović [5], Friedman and Honzik [6], Friedman and Magidor [8], Friedman and Zdomskyy [10], Honzik [12]).
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84On the Consistency Strength of the Inner Model HypothesisJournal of Symbolic Logic 73 (2). 2008.
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67Isomorphism relations on computable structuresJournal of Symbolic Logic 77 (1): 122-132. 2012.We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all ${\mathrm{\Sigma }}_{1}^{1}$ equivalence relations on hyperarithmetical subsets of ω
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22The Nonabsoluteness of Model Existence in Uncountable Cardinals for $L{omega{1},omega}$Notre Dame Journal of Formal Logic 54 (2): 137-151. 2013.For sentences $\phi$ of $L_{\omega_{1},\omega}$, we investigate the question of absoluteness of $\phi$ having models in uncountable cardinalities. We first observe that having a model in $\aleph_{1}$ is an absolute property, but having a model in $\aleph_{2}$ is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis context and provide sentences for any $\alpha\in\omega_{1}\setminus\{0,1,\omega\}$ for which the existence of a model…Read more
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22Potential isomorphism of elementary substructures of a strictly stable homogeneous modelJournal of Symbolic Logic 76 (3). 2011.The results herein form part of a larger project to characterize the classification properties of the class of submodels of a homogeneous stable diagram in terms of the solvability (in the sense of [1]) of the potential isomorphism problem for this class of submodels. We restrict ourselves to locally saturated submodels of the monster model m of some power π. We assume that in Gödel's constructible universe , π is a regular cardinal at least the successor of the first cardinal in which is stabl…Read more
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48Internal consistency and the inner model hypothesisBulletin of Symbolic Logic 12 (4): 591-600. 2006.There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this by examining Easton's strengthening of Cohen's result:Theorem 1. The…Read more
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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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18BPFA and projective well-orderings of the realsJournal of Symbolic Logic 76 (4): 1126-1136. 2011.If the bounded proper forcing axiom BPFA holds and ω 1 = ${\mathrm{\omega }}_{1}^{\mathrm{L}}$ , then there is a lightface ${\mathrm{\Sigma }}_{3}^{1}$ well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of "David's trick." We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface ${\mathrm{\Sigma }}_{4}^{1}$ , for many "consistently locally c…Read more
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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.