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  1. Vincenzo Dimonte & Sy-David Friedman (forthcoming). Rank-Into-Rank Hypotheses and the Failure of GCH. Archive for Mathematical Logic.
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  2. Sy-David Friedman, Philip Welch & W. Hugh Woodin (forthcoming). On the Consistency Strength of the Inner Model Hypothesis. Journal of Symbolic Logic.
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  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.
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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 (GCH) 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 (relative to large cardinal hypotheses). (...)
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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.
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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.
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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.
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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 (HOD,S), and indeed over the even smaller inner model $\mathbb{S}=(L[S],S)$, 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 (...)
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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.
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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.
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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 & 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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  25. Vera Fischer & Sy David Friedman (2010). Cardinal Characteristics and Projective Wellorders. Annals of Pure and Applied Logic 161 (7):916-922.
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  26. 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.
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  27. 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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  28. 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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  29. Sy-David Friedman & Lyubomyr Zdomskyy (2010). Projective Mad Families. Annals of Pure and Applied Logic 161 (12):1581-1587.
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  30. 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.
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  31. Andrew D. Brooke-Taylor & Sy-David Friedman (2009). Large Cardinals and Gap-1 Morasses. Annals of Pure and Applied Logic 159 (1):71-99.
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  32. 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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  33. 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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  34. 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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  35. 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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  36. 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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  37. Sy David& Thompson Friedman (2008). Katherine," Perfect Trees and Elementary Embeddings. Journal of Symbolic Logic 73:3.
     
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  38. Sy-David Friedman & Radek Honzik (2008). Easton's Theorem and Large Cardinals. Annals of Pure and Applied Logic 154 (3):191-208.
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  39. Sy-David Friedman & Pavel Ondrejovič (2008). The Internal Consistency of Easton's Theorem. Annals of Pure and Applied Logic 156 (2):259-269.
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  40. 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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  41. 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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  42. 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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  43. 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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  44. 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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  45. Sy-David Friedman (2006). Internal Consistency and the Inner Model Hypothesis. Bulletin of Symbolic Logic 12 (4):591-600.
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  46. 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).
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  47. 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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  48. Sy D. Friedman (2005). Definability Degrees. Mathematical Logic Quarterly 51 (5):448-449.
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  49. Sy D. Friedman (2005). Genericity and Large Cardinals. Journal of Mathematical Logic 5 (02):149-166.
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  50. Sy D. Friedman (2004). Generic Σ₃¹ Absoluteness. Journal of Symbolic Logic 69 (1):73 - 80.
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