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Set Theory

Journal of Symbolic Logic 46 (4):876-877 (1981)

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  1. On the Consistency of the Definable Tree Property on $\aleph_1$.Amir Leshem - 2000 - Journal of Symbolic Logic 65 (3):1204-1214.
    In this paper we prove the equiconsistency of "Every $\omega_1$-tree which is first order definable over has a cofinal branch" with the existence of a $\Pi^1_1$ reflecting cardinal. We also prove that the addition of MA to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.
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  • Two Applications Of Inner Model Theory To The Study Of \sigma^1_2 Sets.Greg Hjorth - 1996 - Bulletin of Symbolic Logic 2 (1):94-107.
    §0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may (...)
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  • Precipitous Towers of Normal Filters.Douglas R. Burke - 1997 - Journal of Symbolic Logic 62 (3):741-754.
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  • After All, There are Some Inequalities which are Provable in ZFC.Tomek Bartoszyński, Andrzej Rosłanowski & Saharon Shelah - 2000 - Journal of Symbolic Logic 65 (2):803-816.
    We address ZFC inequalities between some cardinal invariants of the continuum, which turned out to be true in spite of strong expectations given by [11].
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  • Laver Indestructibility and the Class of Compact Cardinals.Arthur W. Apter - 1998 - Journal of Symbolic Logic 63 (1):149-157.
    Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal $\kappa$ indestructible under $\kappa$-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly (...)
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  • Variations on a game of Gale (III): Remainder strategies.Marion Scheepers & William Weiss - 1997 - Journal of Symbolic Logic 62 (4):1253-1264.
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  • Proper forcing and remarkable cardinals II.Ralf-Dieter Schindler - 2001 - Journal of Symbolic Logic 66 (3):1481-1492.
    The current paper proves the results announced in [5]. We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and ω-Erdos cardinals. They are characterized by the existence of "O # -like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness for proper forcings. In particular, said absoluteness does not imply Π 1 1 determinacy.
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  • A many permutation group result for unstable theories.Mark D. Schlatter - 1998 - Journal of Symbolic Logic 63 (2):694-708.
    We extend Shelah's first many model result to show that an unstable theory has 2 κ many non-permutation group isomorphic models of size κ, where κ is an uncountable regular cardinal.
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  • Simple forcing notions and forcing axioms.Andrzej Rosłanowski & Saharon Shelah - 1997 - Journal of Symbolic Logic 62 (4):1297-1314.
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  • Proper forcing and l(ℝ).Itay Neeman & Jindřich Zapletal - 2001 - Journal of Symbolic Logic 66 (2):801-810.
    We present two ways in which the model L(R) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing; we show further that a set of ordinals in V cannot be added to L(R) by small forcing. The large cardinal needed corresponds to the consistency strength of AD L (R); roughly ω Woodin cardinals.
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  • Changing cardinal invariants of the reals without changing cardinals or the reals.Heike Mildenberger - 1998 - Journal of Symbolic Logic 63 (2):593-599.
    We show: The procedure mentioned in the title is often impossible. It requires at least an inner model with a measurable cardinal. The consistency strength of changing b and d from a regular κ to some regular δ < κ is a measurable of Mitchell order δ. There is an application to Cichon's diagram.
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  • The complexity of the modal predicate logic of "true in every transitive model of ZF".Vann McGee - 1997 - Journal of Symbolic Logic 62 (4):1371-1378.
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  • Nowhere precipitousness of some ideals.Yo Matsubara & Masahiro Shioya - 1998 - Journal of Symbolic Logic 63 (3):1003-1006.
    In this paper we will present a simple condition for an ideal to be nowhere precipitous. Through this condition we show nowhere precipitousness of fundamental ideals onPkλ, in particular the non-stationary idealNSkλunder cardinal arithmetic assumptions.In this sectionIdenotes a non-principal ideal on an infinite setA. LetI+=PA/I(ordered by inclusion as a forcing notion) andI∣X= {Y⊂A:Y⋂X∈I}, which is also an ideal onAforX∈I+. We refer the reader to [8, Section 35] for the general theory of generic ultrapowers associated with an ideal. We recallIis said (...)
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  • Singular σ-dense trees.Avner Landver - 1992 - Journal of Symbolic Logic 57 (4):1403-1416.
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  • Cofinitary groups, almost disjoint and dominating families.Michael Hrušák, Juris Steprans & Yi Zhang - 2001 - Journal of Symbolic Logic 66 (3):1259-1276.
    In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(ω) is strictly greater than the cardinal numbers o and a.
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  • Strong axioms of infinity in NFU.M. Randall Holmes - 2001 - Journal of Symbolic Logic 66 (1):87-116.
    This paper discusses a sequence of extensions ofNFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].The original theoryNFof Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] thatNFdisproves Choice (and so proves Infinity). Even if one assumes the consistency ofNF, one is hampered by the lack of powerful methods for proofs of consistency and (...)
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  • Some applications of coarse inner model theory.Greg Hjorth - 1997 - Journal of Symbolic Logic 62 (2):337-365.
    The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. $\underset{\sim}{\Pi}$ determinacy implies that for every thin Σ 1 2 equivalence relation there is a Δ 1 3 real, N, over which every equivalence class is generic--and hence there is a good Δ 1 2 (N ♯ ) wellordering of the equivalence classes. Analogous results are obtained for Π 1 2 and Δ 1 2 quasilinear orderings and $\underset{\sim}{\Pi}^1_2$ determinacy is shown to imply that (...)
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  • Small forcing makes any cardinal superdestructible.Joel David Hamkins - 1998 - Journal of Symbolic Logic 63 (1):51-58.
    Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal κ becomes superdestructible--any further <κ--closed forcing which adds a subset to κ will destroy the measurability, even the weak compactness, of κ. Nevertheless, after small forcing indestructible cardinals remain resurrectible, but never strongly resurrectible.
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  • Canonical seeds and Prikry trees.Joel David Hamkins - 1997 - Journal of Symbolic Logic 62 (2):373-396.
    Applying the seed concept to Prikry tree forcing P μ , I investigate how well P μ preserves the maximality property of ordinary Prikry forcing and prove that P μ Prikry sequences are maximal exactly when μ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if μ is a strongly normal supercompactness measure, then P μ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. Hugh Woodin's.
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  • The consistency strength of successive cardinals with the tree property.Matthew Foreman, Menachem Magidor & Ralf-Dieter Schindler - 2001 - Journal of Symbolic Logic 66 (4):1837-1847.
    If ω n has the tree property for all $2 \leq n and $2^{ , then for all X ∈ H ℵ ω and $n exists.
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  • On the relationship between ATR 0 and.Jeremy Avigad - 1996 - Journal of Symbolic Logic 61 (3):768-779.
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  • Pseudo-Superstructures as Nonstandard Universes.Mauro Di Nasso - 1998 - Journal of Symbolic Logic 63 (1):222 - 236.
    A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed. Though necessarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values. Nonstandard universes are also constructed which model the whole ZFC theory without regularity and satisfy the κ-saturation property.
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  • The fine structure of real mice.Daniel W. Cunningham - 1998 - Journal of Symbolic Logic 63 (3):937-994.
    Before one can construct scales of minimal complexity in the Real Core Model, K(R), one needs to develop the fine-structure theory of K(R). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice (...)
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  • Ideals and combinatorial principles.Douglas Burke & Yo Matsubara - 1997 - Journal of Symbolic Logic 62 (1):117-122.
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  • Supplements of bounded permutation groups.Stephen Bigelow - 1998 - Journal of Symbolic Logic 63 (1):89-102.
    Let λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation group B λ (Ω), or simply B λ , is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation group G acting on Ω is a supplement of B λ if B λ G is the full symmetric group on Ω. In [7], Macpherson and Neumann claimed to have classified (...)
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  • The ⊲-ordering on normal ultrafilters.Stewart Baldwin - 1985 - Journal of Symbolic Logic 50 (4):936-952.
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  • A characterization of Martin's axiom in terms of absoluteness.Joan Bagaria - 1997 - Journal of Symbolic Logic 62 (2):366-372.
    Martin's axiom is equivalent to the statement that the universe is absolute under ccc forcing extensions for Σ 1 sentences with a subset of $\kappa, \kappa , as a parameter.
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  • The least measurable can be strongly compact and indestructible.Arthur W. Apter & Moti Gitik - 1998 - Journal of Symbolic Logic 63 (4):1404-1412.
    We show the consistency, relative to a supercompact cardinal, of the least measurable cardinal being both strongly compact and fully Laver indestructible. We also show the consistency, relative to a supercompact cardinal, of the least strongly compact cardinal being somewhat supercompact yet not completely supercompact and having both its strong compactness and degree of supercompactness fully Laver indestructible.
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  • Identity crises and strong compactness.Arthur W. Apter & James Cummings - 2000 - Journal of Symbolic Logic 65 (4):1895-1910.
    Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals κ 1 ,..., κ n are so that κ i for i = 1,..., n is both the i th measurable cardinal and κ + i supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.
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  • The independence of.Amir Leshem & Menachem Magidor - 1999 - Journal of Symbolic Logic 64 (1):350-362.
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  • Small forcings and Cohen reals.Jindřich Zapletal - 1997 - Journal of Symbolic Logic 62 (1):280-284.
    We show that all posets of uniform density ℵ 1 may have to add a Cohen real and develop some forcing machinery for obtaining this sort of result.
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  • ◇ at Mahlo cardinals.Martin Zeman - 2000 - Journal of Symbolic Logic 65 (4):1813-1822.
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  • Splitting number at uncountable cardinals.Jindřich Zapletal - 1997 - Journal of Symbolic Logic 62 (1):35-42.
    We study a generalization of the splitting number s to uncountable cardinals. We prove that $\mathfrak{s}(\kappa) > \kappa^+$ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption $\mathfrak{s}(\aleph_\omega) > \aleph_{\omega + 1}$ has a considerable large cardinal strength as well.
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  • Killing ideals and adding reals.Jindřich Zapletal - 2000 - Journal of Symbolic Logic 65 (2):747-755.
    The relationship between killing ideals and adding reals by forcings is analysed.
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  • Preserving σ-ideals.Jindřich Zapletal - 1998 - Journal of Symbolic Logic 63 (4):1437-1441.
    It is proved consistent that there be a proper σ-ideal ℑ on ω 1 and an ℵ 1 -preserving poset P such that $\mathbb{P} \Vdash$ the σ-ideal generated by ℑ̌ is not proper.
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  • On the formalization of semantic conventions.James G. Williams - 1990 - Journal of Symbolic Logic 55 (1):220-243.
    This paper discusses six formalization techniques, of varying strengths, for extending a formal system based on traditional mathematical logic. The purpose of these formalization techniques is to simulate the introduction of new syntactic constructs, along with associated semantics for them. We show that certain techniques (among the six) subsume others. To illustrate sharpness, we also consider a selection of constructs and show which techniques can and cannot be used to introduce them. The six studied techniques were selected on the basis (...)
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  • Satisfaction relations for proper classes: Applications in logic and set theory.Robert A. Van Wesep - 2013 - Journal of Symbolic Logic 78 (2):345-368.
    We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate ($\models^*$) appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension $\Theta$ of ZF there is a finitely axiomatizable extension $\Theta'$ of GB that is a conservative extension of $\Theta$. We also (...)
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  • On Gupta-Belnap revision theories of truth, Kripkean fixed points, and the next stable set.P. D. Welch - 2001 - Bulletin of Symbolic Logic 7 (3):345-360.
    We consider various concepts associated with the revision theory of truth of Gupta and Belnap. We categorize the notions definable using their theory of circular definitions as those notions universally definable over the next stable set. We give a simplified account of varied revision sequences-as a generalised algorithmic theory of truth. This enables something of a unification with the Kripkean theory of truth using supervaluation schemes.
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  • Nonabsoluteness of elementary embeddings.Friedrich Wehrung - 1989 - Journal of Symbolic Logic 54 (3):774-778.
    Ifκis a measurable cardinal, let us say that a measure onκis aκ-complete nonprincipal ultrafilter onκ. IfUis a measure onκ, letjUbe the canonical elementary embedding ofVinto its Ultrapower UltU. Ifxis a set, say thatUmovesxwhenjU≠x; say thatκmovesxwhen some measure onκmovesx. Recall Kunen's lemma : “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof and Fleissner's proof are essentially nonconstructive.The following proposition can be proved by using elementary facts about iterated ultrapowers.Proposition.Let ‹Un: n ∈ ω› be a sequence of measures (...)
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  • Abstract logic and set theory. II. large cardinals.Jouko Väänänen - 1982 - Journal of Symbolic Logic 47 (2):335-346.
    The following problem is studied: How large and how small can the Löwenheim and Hanf numbers of unbounded logics be in relation to the most common large cardinals? The main result is that the Löwenheim number of the logic with the Härtig-quantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly Mahlo, the first weakly compact, the first Ramsey, the first measurable and the first supercompact cardinals.
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  • Second order logic or set theory?Jouko Väänänen - 2012 - Bulletin of Symbolic Logic 18 (1):91-121.
    We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each (...)
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  • Jensen's □ principles and the Novak number of partially ordered sets.Boban Veličković - 1986 - Journal of Symbolic Logic 51 (1):47-58.
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  • Toward a stability theory of tame abstract elementary classes.Sebastien Vasey - 2018 - Journal of Mathematical Logic 18 (2):1850009.
    We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness, and are stable in some cardinal. Assuming the singular cardinal hypothesis, we prove a full characterization of the stability cardinals, and connect the stability spectrum with the behavior of saturated models.We deduce that if a class is stable on a tail of cardinals, then it has no long splitting chains. This indicates that there is a clear notion of superstability in this framework.We also present an (...)
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  • Consistency of strictly impredicative NF and a little more ….Sergei Tupailo - 2010 - Journal of Symbolic Logic 75 (4):1326-1338.
    An instance of Stratified Comprehension ∀x₁ … ∀x n ∃y∀x (x ∈ y ↔ φ(x, x₁, …, x n )) is called strictly impredicative iff, under minimal stratification, the type of x is 0. Using the technology of forcing, we prove that the fragment of NF based on strictly impredicative Stratified Comprehension is consistent. A crucial part in this proof, namely showing genericity of a certain symmetric filter, is due to Robert Solovay. As a bonus, our interpretation also satisfies some (...)
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  • Combinatorial dichotomies in set theory.Stevo Todorcevic - 2011 - Bulletin of Symbolic Logic 17 (1):1-72.
    We give an overview of a research line concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered on some other perhaps not so large cardinal. While this is of great interest to set theorists, one of the main motivations behind this line of research is in its applicability to other areas of mathematics. We give some details about this and we expose some possible directions for further (...)
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  • Partition properties of m-ultrafilters and ideals.Joji Takahashi - 1987 - Journal of Symbolic Logic 52 (4):897-907.
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  • A saturation property of ideals and weakly compact cardinals.Joji Takahashi - 1986 - Journal of Symbolic Logic 51 (3):513-525.
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  • Finitist set theory in ontological modeling.Avril Styrman & Aapo Halko - 2018 - Applied ontology 13 (2):107-133.
    This article introduces finitist set theory (FST) and shows how it can be applied in modeling finite nested structures. Mereology is a straightforward foundation for transitive chains of part-whole relations between individuals but is incapable of modeling antitransitive chains. Traditional set theories are capable of modeling transitive and antitransitive chains of relations, but due to their function as foundations of mathematics they come with features that make them unnecessarily difficult in modeling finite structures. FST has been designed to function as (...)
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  • HOD L(ℝ) is a Core Model Below Θ.John R. Steel - 1995 - Bulletin of Symbolic Logic 1 (1):75-84.
    In this paper we shall answer some questions in the set theory of L, the universe of all sets constructible from the reals. In order to do so, we shall assume ADL, the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L, and for reasons we cannot discuss here, ZFC + ADL yields the most (...)
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  • Backwards Easton forcing and 0#. [REVIEW]M. C. Stanley - 1988 - Journal of Symbolic Logic 53 (3):809 - 833.
    It is shown that if κ is an uncountable successor cardinal in L[ 0 ♯ ], then there is a normal tree T ∈ L [ 0 ♯ ] of height κ such that $0^\sharp \not\in L\lbrack\mathbf{T}\rbrack$ . Yet T is $ -distributive in L[ 0 ♯ ]. A proper class version of this theorem yields an analogous L[ 0 ♯ ]-definable tree such that distinct branches in the presence of 0 ♯ collapse the universe. A heretofore unutilized method for (...)
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