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  1. Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (forthcoming). Multiverse Conceptions in Set Theory. Synthese:1-26.
    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 potentialism with regard to the (...)
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  2. Vincenzo Dimonte & Sy-David Friedman (forthcoming). Rank-Into-Rank Hypotheses and the Failure of GCH. Archive for Mathematical Logic.
  3. Tatiana Arrigoni & Sy-David Friedman (2013). The Hyperuniverse Program. Bulletin of Symbolic Logic 19 (1):77-96.
    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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  4. David Asperó, Sy-David Friedman, Miguel Angel Mota & Marcin Sabok (2013). Baumgartnerʼs Conjecture and Bounded Forcing Axioms. Annals of Pure and Applied Logic 164 (12):1178-1186.
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  5. Vera Fischer, Sy David Friedman & Yurii Khomskii (2013). Co-Analytic Mad Families and Definable Wellorders. Archive for Mathematical Logic 52 (7-8):809-822.
    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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  6. Vera Fischer, Sy David Friedman & Lyubomyr Zdomskyy (2013). Cardinal Characteristics, Projective Wellorders and Large Continuum. Annals of Pure and Applied Logic 164 (7-8):763-770.
    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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  7. Sy-David Friedman, Tapani Hyttinen & Martin Koerwien (2013). The Nonabsoluteness of Model Existence in Uncountable Cardinals for $L{Omega{1},Omega}$. Notre Dame Journal of Formal Logic 54 (2):137-151.
    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 in $\aleph_{\alpha}$ is nonabsolute . Finally, we present a complete (...)
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  8. Sy-David Friedman, Michael Rathjen & Andreas Weiermann (2013). Slow Consistency. Annals of Pure and Applied Logic 164 (3):382-393.
    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 (...)
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  9. Tatiana Arrigoni & Sy-David Friedman (2012). Foundational Implications of the Inner Model Hypothesis. Annals of Pure and Applied Logic 163 (10):1360-1366.
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  10. David Asperó & Sy-David Friedman (2012). Definable Well-Orders of $H(\Omega _2)$ and $GCH$. Journal of Symbolic Logic 77 (4):1101-1121.
    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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  11. Ekaterina B. Fokina, Sy-David Friedman, Valentina Harizanov, Julia F. Knight, Charles McCoy & Antonio Montalbán (2012). Isomorphism Relations on Computable Structures. Journal of Symbolic Logic 77 (1):122-132.
    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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  12. Ekaterina B. Fokina & Sy‐David Friedman (2012). On Σ11 Equivalence Relations Over the Natural Numbers. Mathematical Logic Quarterly 58 (1‐2):113-124.
    We study the structure of Σ11 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly equation imagei.e., Σ11 but not equation image equivalence classes. We also show the existence of incomparable Σ11 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ11 (...)
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  13. Sy David Friedman, Radek Honzik & Lyubomyr Zdomskyy (2012). Fusion and Large Cardinal Preservation. Annals of Pure and Applied Logic 2 (12):1247-1273.
    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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  14. Sy-David Friedman (2012). The Stable Core. Bulletin of Symbolic Logic 18 (2):261-267.
    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 (...)
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  15. Sy-David Friedman & Mohammad Golshani (2012). Independence of Higher Kurepa Hypotheses. Archive for Mathematical Logic 51 (5-6):621-633.
    We study the Generalized Kurepa hypothesis introduced by Chang. We show that relative to the existence of an inaccessible cardinal the Gap-n-Kurepa hypothesis does not follow from the Gap-m-Kurepa hypothesis for m different from n. The use of an inaccessible is necessary for this result.
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  16. Sy-David Friedman & Radek Honzik (2012). Eastonʼs Theorem and Large Cardinals From the Optimal Hypothesis. Annals of Pure and Applied Logic 163 (12):1738-1747.
    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 cardinals (...)
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  17. Sy-David Friedman & Tapani Hyttinen (2012). On Borel Equivalence Relations in Generalized Baire Space. Archive for Mathematical Logic 51 (3-4):299-304.
    We construct two Borel equivalence relations on the generalized Baire space κ κ , κ <κ = 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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  18. Sam Buss, Yijia Chen, Jörg Flum, Sy-David Friedman & Moritz Müller (2011). Strong Isomorphism Reductions in Complexity Theory. Journal of Symbolic Logic 76 (4):1381-1402.
    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 cardinality at most n in both (...)
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  19. Andrés Eduardo Caicedo & Sy-David Friedman (2011). BPFA and Projective Well-Orderings of the Reals. Journal of Symbolic Logic 76 (4):1126-1136.
    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 certified" relations R on $\mathrm{\mathbb{R}}$ . (...)
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  20. Vera Fischer, Sy David Friedman & Lyubomyr Zdomskyy (2011). Projective Wellorders and Mad Families with Large Continuum. Annals of Pure and Applied Logic 162 (11):853-862.
    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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  21. Sy-David Friedman & Ajdin Halilović (2011). The Tree Property at א Ω+2. Journal of Symbolic Logic 76 (2):477 - 490.
    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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  22. Sy-David Friedman, Tapani Hyttinen & Agatha C. Walczak-Typke (2011). Potential Isomorphism of Elementary Substructures of a Strictly Stable Homogeneous Model. Journal of Symbolic Logic 76 (3):987 - 1004.
    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 (...)
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  23. Sy-David Friedman & Luca Motto Ros (2011). Analytic Equivalence Relations and Bi-Embeddability. Journal of Symbolic Logic 76 (1):243-266.
    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 ℒω1ω) is far from complete (see [5, 2]). In this article we strengthen (...)
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  24. Sy-David Friedman & Luca Motto Ros (2011). Analytic Equivalence Relations and Bi-Embeddability. Journal of Symbolic Logic 76 (1):243 - 266.
    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 (...)
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  25. Sy-David Friedman & P. D. Welch (2011). Hypermachines. Journal of Symbolic Logic 76 (2):620 - 636.
    The Infinite Time Turing Machine model [8] of Hamkins and Kidder is, in an essential sense, a "Σ₂-machine" in that it uses a Σ₂ Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σ n rule. Such machines either halt or enter an infinite loop by stage ζ(n) = df μζ(n)[∃Σ(n) > ζ(n) L ζ(n) ≺ Σn L Σ(n) ], again generalising precisely the ITTM case. The collection of (...)
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  26. Vera Fischer & Sy David Friedman (2010). Cardinal Characteristics and Projective Wellorders. Annals of Pure and Applied Logic 161 (7):916-922.
    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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  27. Ekaterina B. Fokina, Sy-David Friedman & Asger Törnquist (2010). The Effective Theory of Borel Equivalence Relations. Annals of Pure and Applied Logic 161 (7):837-850.
    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 (...)
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  28. Sy-David Friedman (2010). Generalizations of Gödel's Universe of Constructible Sets. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic
     
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  29. Sy-David Friedman & Martin Koerwien (2010). On Absoluteness of Categoricity in Abstract Elementary Classes. Notre Dame Journal of Formal Logic 52 (4):395-402.
    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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  30. Sy-David Friedman & Lyubomyr Zdomskyy (2010). Projective Mad Families. Annals of Pure and Applied Logic 161 (12):1581-1587.
    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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  31. David Asperó & Sy-David Friedman (2009). Large Cardinals and Locally Defined Well-Orders of the Universe. Annals of Pure and Applied Logic 157 (1):1-15.
    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 (...)
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  32. Andrew D. Brooke-Taylor & Sy-David Friedman (2009). Large Cardinals and Gap-1 Morasses. Annals of Pure and Applied Logic 159 (1):71-99.
    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 the partial order; we (...)
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  33. Sy-David Friedman & Menachem Magidor (2009). The Number of Normal Measures. Journal of Symbolic Logic 74 (3):1069-1080.
    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 (...)
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  34. Sy-David Friedman & Katherine Thompson (2009). An Inner Model for Global Domination. Journal of Symbolic Logic 74 (1):251-264.
    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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  35. James Cummings & Sy-David Friedman (2008). □ On the Singular Cardinals. Journal of Symbolic Logic 73 (4):1307-1314.
    We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen. "Square on singular cardinals".
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  36. Natasha Dobrinen & Sy-David Friedman (2008). Homogeneous Iteration and Measure One Covering Relative to HOD. Archive for Mathematical Logic 47 (7-8):711-718.
    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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  37. Natasha Dobrinen & Sy-David Friedman (2008). Internal Consistency and Global Co-Stationarity of the Ground Model. Journal of Symbolic Logic 73 (2):512 - 521.
    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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  38. Sy David& Thompson Friedman (2008). Katherine," Perfect Trees and Elementary Embeddings. Journal of Symbolic Logic 73:3.
     
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  39. Sy-David Friedman & Radek Honzik (2008). Easton's Theorem and Large Cardinals. Annals of Pure and Applied Logic 154 (3):191-208.
    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<β→F≤F. The classic example of an Easton function is the continuum function αmaps to2α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V[G]; we say that F is realised in V[G]. However if we also wish (...)
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  40. Sy-David Friedman & Pavel Ondrejovič (2008). The Internal Consistency of Easton's Theorem. Annals of Pure and Applied Logic 156 (2):259-269.
    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 are L-definable (...)
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  41. Sy-David Friedman & Katherine Thompson (2008). Internal Consistency for Embedding Complexity. Journal of Symbolic Logic 73 (3):831-844.
    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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  42. Sy-David Friedman & Katherine Thompson (2008). Perfect Trees and Elementary Embeddings. Journal of Symbolic Logic 73 (3):906-918.
    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 (...)
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  43. Sy-David Friedman, Philip Welch & W. Woodin (2008). On the Consistency Strength of the Inner Model Hypothesis. Journal of Symbolic Logic 73 (2):391-400.
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  44. Sy-David Friedman, Philip Welch & W. Hugh Woodin (2008). On the Consistency Strength of the Inner Model Hypothesis. Journal of Symbolic Logic 73 (2):391 - 400.
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  45. Jeremy Avigad, Sy Friedman, Akihiro Kanamori, Elisabeth Bouscaren, Philip Kremer, Claude Laflamme, Antonio Montalbán, Justin Moore & Helmut Schwichtenberg (2007). Montréal, Québec, Canada May 17–21, 2006. Bulletin of Symbolic Logic 13 (1).
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  46. Natasha Dobrinen & Sy-David Friedman (2006). Co-Stationarity of the Ground Model. Journal of Symbolic Logic 71 (3):1029 - 1043.
    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 (...)
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  47. Sy-David Friedman (2006). Internal Consistency and the Inner Model Hypothesis. Bulletin of Symbolic Logic 12 (4):591-600.
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  48. Sy-David Friedman (2006). Theorem 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).
  49. Sy-David Friedman, Peter Koepke & Boris Piwinger (2006). Hyperfine Structure Theory and Gap 1 Morasses. Journal of Symbolic Logic 71 (2):480 - 490.
    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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  50. Sy D. Friedman (2005). Definability Degrees. Mathematical Logic Quarterly 51 (5):448-449.
    We establish the equiconsistency of a simple statement in definability theory with the failure of the GCH at all infinite cardinals. The latter was shown by Foreman and Woodin to be consistent, relative to the existence of large cardinals.
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