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806The 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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662It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that the continu…Read more
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590Universism and extensions of VReview of Symbolic Logic 14 (1): 112-154. 2021.A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret t…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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363Set Theory and StructuresIn Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics, Springer Verlag. pp. 223-253. 2019.Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological p…Read more
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358Maximality and ontology: how axiom content varies across philosophical frameworksSynthese 197 (2): 623-649. 2017.Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in partic…Read more
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182Multiverse Conceptions in Set TheorySynthese 192 (8): 2463-2488. 2015.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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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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86On the Consistency Strength of the Inner Model HypothesisJournal of Symbolic Logic 73 (2). 2008.
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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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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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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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63A guide to "coding the universe" by Beller, Jensen, WelchJournal of Symbolic Logic 50 (4): 1002-1019. 1985.
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62Co-analytic mad families and definable wellordersArchive for Mathematical Logic 52 (7-8): 809-822. 2013.We show that the existence of a ${\Pi^1_1}$ -definable mad family is consistent with the existence of a ${\Delta^{1}_{3}}$ -definable well-order of the reals and ${\mathfrak{b}=\mathfrak{c}=\aleph_3}$
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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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60Cardinal characteristics and projective wellordersAnnals of Pure and Applied Logic 161 (7): 916-922. 2010.Using countable support iterations of S-proper posets, we show that the existence of a definable wellorder of the reals is consistent with each of the following: , and
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58Projective 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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57Slow 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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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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55On Borel equivalence relations in generalized Baire spaceArchive for Mathematical Logic 51 (3-4): 299-304. 2012.We construct two Borel equivalence relations on the generalized Baire space κκ, κ ω, with the property that neither of them is Borel reducible to the other. A small modification of the construction shows that the straightforward generalization of the Glimm-Effros dichotomy fails.
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52Cardinal characteristics, projective wellorders and large continuumAnnals of Pure and Applied Logic 164 (7-8): 763-770. 2013.We extend the work of Fischer et al. [6] by presenting a method for controlling cardinal characteristics in the presence of a projective wellorder and 2ℵ0>ℵ2. This also answers a question of Harrington [9] by showing that the existence of a Δ31 wellorder of the reals is consistent with Martinʼs axiom and 2ℵ0=ℵ3
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51Large cardinals need not be large in HODAnnals of Pure and Applied Logic 166 (11): 1186-1198. 2015.
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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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49Fusion 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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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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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