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Saharon Shelah [375]S. Shelah [51]
  1. Haim Judah & Saharon Shelah (1993). ▵13-Sets of Reals. Journal of Symbolic Logic 58 (1):72 - 80.
    We build models where all $\underset{\sim}{\triangle}^1_3$ -sets of reals are measurable and (or) have the property of Baire and (or) are Ramsey. We will show that there is no implication between any of these properties for $\underset{\sim}{\triangle}^1_3$ -sets of reals.
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  2. Mirna Džamonja & Saharon Shelah (2003). Universal Graphs at the Successor of a Singular Cardinal. Journal of Symbolic Logic 68 (2):366-388.
    The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are $\mu^{++}$ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2^{\mu^+} \gg \mu^{++}$ . The paper also addresses the general problem of obtaining a framework for consistency results at the (...)
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  3. Tapani Hyttinen & Saharon Shelah (1994). Constructing Strongly Equivalent Nonisomorphic Models for Unsuperstable Theories, Part A. Journal of Symbolic Logic 59 (3):984-996.
    In this paper we prove a strong nonstructure theorem for κ(T)-saturated models of a stable theory T with dop. This paper continues the work started in [1].
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  4. Saharon Shelah (1979). Hanf Number of Omitting Type for Simple First-Order Theories. Journal of Symbolic Logic 44 (3):319-324.
    Let T be a complete countable first-order theory such that every ultrapower of a model of T is saturated. If T has a model omitting a type p in every cardinality $ then T has a model omitting p in every cardinality. There is also a related theorem, and an example showing the $\beth_\omega$ cannot be improved.
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  5. Menachem Kojman & Saharon Shelah (1992). Nonexistence of Universal Orders in Many Cardinals. Journal of Symbolic Logic 57 (3):875-891.
    Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1 without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove--again in ZFC--that for a large class of cardinals there is no universal linear order (e.g. in every regular $\aleph_1 < (...)
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  6. Tomek Bartoszynski, Jaime I. Ihoda & Saharon Shelah (1989). The Cofinality of Cardinal Invariants Related to Measure and Category. Journal of Symbolic Logic 54 (3):719-726.
    We prove that the following are consistent with ZFC. 1. 2 ω = ℵ ω 1 + K C = ℵ ω 1 + K B = K U = ω 2 (for measure and category simultaneously). 2. 2 ω = ℵ ω 1 = K C (L) + K C (M) = ω 2 . This concludes the discussion about the cofinality of K C.
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  7. G. Sageev & S. Shelah (1985). On the Structure of $\Operatorname{Ext}(a, \Mathbf{Z})$ in ZFC+. Journal of Symbolic Logic 50 (2):302 - 315.
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  8. Saharon Shelah & Juris Steprāns (2001). The Covering Numbers of Mycielski Ideals Are All Equal. Journal of Symbolic Logic 66 (2):707-718.
    The Mycielski ideal M k is defined to consist of all sets $A \subseteq ^{\mathbb{N}}k$ such that $\{f \upharpoonright X: f \in A\} \neq ^Xk$ for all X ∈ [N] ℵ 0 . It will be shown that the covering numbers for these ideals are all equal. However, the covering numbers of the closely associated Roslanowski ideals will be shown to be consistently different.
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  9. Renling Jin & Saharon Shelah (1994). The Strength of the Isomorphism Property. Journal of Symbolic Logic 59 (1):292-301.
    In § 1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second-order types in model theory. In § 2, several applications are given. One of the applications answers a question of D. Ross in [this Journal, vol. 55 (1990), pp. 1233-1242] about infinite Loeb measure spaces.
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  10. Alan H. Mekler & Saharon Shelah (1989). Uniformization Principles. Journal of Symbolic Logic 54 (2):441-459.
    It is consistent that for many cardinals λ there is a family of at least λ + unbounded subsets of λ which have uniformization properties. In particular if it is consistent that a supercompact cardinal exists, then it is consistent that ℵ ω has such a family. We have applications to point set topology, Whitehead groups and reconstructing separable abelian p-groups from their socles.
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  11. Jaime I. Ihoda & Saharon Shelah (1989). Martin's Axioms, Measurability and Equiconsistency Results. Journal of Symbolic Logic 54 (1):78-94.
    We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (MA(σ-centered), (MA(σ-linked)), MA(Γ + ℵ 0 ), MA(K)), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists r ∈ (...)
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  12. Saharon Shelah (1984). Diamonds, Uniformization. Journal of Symbolic Logic 49 (4):1022-1033.
    Assume G.C.H. We prove that for singular λ, □ λ implies the diamonds hold for many $S \subseteq \lambda^+$ (including $S \subseteq \{\delta:\delta \in \lambda^+, \mathrm{cf}\delta = \mathrm{cf}\delta = \mathrm{cf}\lambda\}$ . We also have complementary consistency results.
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  13. Saharon Shelah (2000). Applications of PCF Theory. Journal of Symbolic Logic 65 (4):1624-1674.
    We deal with several pcf problems: we characterize another version of exponentiation: maximal number of κ-branches in a tree with λ nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for each λ with a pcf restriction and investigate further T D (f). The sections can be read independently, although there are some minor dependencies.
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  14. Andreas Blass, Yuri Gurevich & Saharon Shelah (2002). On Polynomial Time Computation Over Unordered Structures. Journal of Symbolic Logic 67 (3):1093-1125.
    This paper is motivated by the question whether there exists a logic capturing polynomial time computation over unordered structures. We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time. We show that fixpoint logic plus counting is stronger than might be expected, in that it can express the existence of a complete matching in a bipartite graph. We revisit the known examples that separate polynomial time from fixpoint plus counting. We show (...)
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  15. Thomas Jech & Saharon Shelah (1990). Full Reflection of Stationary Sets Below ℵω. Journal of Symbolic Logic 55 (2):822 - 830.
    It is consistent that, for every n ≥ 2, every stationary subset of ω n consisting of ordinals of cofinality ω k , where k = 0 or k ≤ n - 3, reflects fully in the set of ordinals of cofinality ω n - 1 . We also show that this result is best possible.
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  16.  6
    Adi Jarden & Saharon Shelah (2013). Non-Forking Frames in Abstract Elementary Classes. Annals of Pure and Applied Logic 164 (3):135-191.
    The stability theory of first order theories was initiated by Saharon Shelah in 1969. The classification of abstract elementary classes was initiated by Shelah, too. In several papers, he introduced non-forking relations. Later, Shelah [17, II] introduced the good non-forking frame, an axiomatization of the non-forking notion.We improve results of Shelah on good non-forking frames, mainly by weakening the stability hypothesis in several important theorems, replacing it by the almost λ-stability hypothesis: The number of types over a model of cardinality (...)
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  17. Saharon Shelah & Simon Thomas (1997). The Cofinality Spectrum of the Infinite Symmetric Group. Journal of Symbolic Logic 62 (3):902-916.
    Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem is the main result of this paper. Theorem. Suppose that $V \models GCH$ . Let C be (...)
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  18. Rami Grossberg & Saharon Shelah (1986). On the Number of Nonisomorphic Models of an Infinitary Theory Which has the Infinitary Order Property. Part A. Journal of Symbolic Logic 51 (2):302-322.
    Let κ and λ be infinite cardinals such that κ ≤ λ (we have new information for the case when $\kappa ). Let T be a theory in L κ +, ω of cardinality at most κ, let φ(x̄, ȳ) ∈ L λ +, ω . Now define $\mu^\ast_\varphi (\lambda, T) = \operatorname{Min} \{\mu^\ast:$ If T satisfies $(\forall\mu \kappa)(\exists M_\chi \models T)(\exists \{a_i: i Our main concept in this paper is $\mu^\ast_\varphi (\lambda, \kappa) = \operatorname{Sup}\{\mu^\ast(\lambda, T): T$ is a theory (...)
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  19. Yuri Gurevich & Saharon Shelah (1996). On Finite Rigid Structures. Journal of Symbolic Logic 61 (2):549-562.
    The main result of this paper is a probabilistic construction of finite rigid structures. It yields a finitely axiomatizable class of finite rigid structures where no L ω ∞,ω formula with counting quantifiers defines a linear order.
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  20.  7
    Saharon Shelah (1991). Reflecting Stationary Sets and Successors of Singular Cardinals. Archive for Mathematical Logic 31 (1):25-53.
    REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure (...)
    No categories
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  21. Saharon Shelah (1980). Simple Unstable Theories. Annals of Mathematical Logic 19 (3):177-203.
    No categories
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  22.  6
    Moti Gitik & Saharon Shelah (1989). On Certain Indestructibility of Strong Cardinals and a Question of Hajnal. Archive for Mathematical Logic 28 (1):35-42.
    A model in which strongness ofκ is indestructible under κ+ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of {λ δ |2 δ <λ}.
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  23.  3
    Saharon Shelah (1999). Categoricity for Abstract Classes with Amalgamation. Annals of Pure and Applied Logic 98 (1-3):261-294.
    Let be an abstract elementary class with amalgamation, and Lowenheim Skolem number LS. We prove that for a suitable Hanf number gc0 if χ0 < λ0 λ1, and is categorical inλ1+ then it is categorical in λ0.
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  24.  35
    Vladimir Kanovei & Saharon Shelah (2004). A Definable Nonstandard Model of the Reals. Journal of Symbolic Logic 69 (1):159-164.
    We prove, in ZFC,the existence of a definable, countably saturated elementary extension of the reals.
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  25. Saharon Shelah (1972). Uniqueness and Characterization of Prime Models Over Sets for Totally Transcendental First-Order Theories. Journal of Symbolic Logic 37 (1):107-113.
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  26. Saharon Shelah (1987). Semiproper Forcing Axiom Implies Martin Maximum but Not |mathrmPFA+. Journal of Symbolic Logic 52 (2):360 - 367.
    We prove that MM (Martin maximum) is equivalent (in ZFC) to the older axiom SPFA (semiproper forcing axiom). We also prove that SPFA does not imply SPFA + or even PFA + (using the consistency of a large cardinal).
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  27. Saharon Shelah (1996). Toward Classifying Unstable Theories. Annals of Pure and Applied Logic 80 (3):229-255.
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  28.  2
    Andreas Blass & Saharon Shelah (1987). There May Be Simple Pℵ1 and Pℵ2-Points and the Rudin-Keisler Ordering May Be Downward Directed. Annals of Pure and Applied Logic 33 (3):213-243.
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  29.  2
    Mirna Džamonja & Saharon Shelah (2004). On ◁∗-Maximality. Annals of Pure and Applied Logic 125 (1-3):119-158.
    This paper investigates a connection between the semantic notion provided by the ordering * among theories in model theory and the syntactic SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelah's article 229) it was shown that SOP3 implies *-maximality and we prove here that *-maximality in a model of GCH implies a property called SOP2″. It has been (...)
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  30. Saharon Shelah, Heikki Tuuri & Jouko Väänänen (1993). On the Number of Automorphisms of Uncountable Models. Journal of Symbolic Logic 58 (4):1402-1418.
    Let σ(U) denote the number of automorphisms of a model U of power ω1. We derive a necessary and sufficient condition in terms of trees for the existence of an U with $\omega_1 < \sigma(\mathfrak{U}) < 2^{\omega_1}$. We study the sufficiency of some conditions for σ(U) = 2ω1 . These conditions are analogous to conditions studied by D. Kueker in connection with countable models.
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  31.  23
    Todd Eisworth & Saharon Shelah (2005). Successors of Singular Cardinals and Coloring Theorems I. Archive for Mathematical Logic 44 (5):597-618.
  32.  16
    Heike Mildenberger & Saharon Shelah (2003). Specialising Aronszajn Trees by Countable Approximations. Archive for Mathematical Logic 42 (7):627-647.
    We show that there are proper forcings based upon countable trees of creatures that specialise a given Aronszajn tree.
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  33. Menachem Magidor, Saharon Shelah & Jonathan Stavi (1983). On the Standard Part of Nonstandard Models of Set Theory. Journal of Symbolic Logic 48 (1):33-38.
    We characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).
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  34.  37
    Tomek Bartoszyński, Andrzej Roslanowski & Saharon Shelah (2000). After All, There Are Some Inequalities Which Are Provable in ZFC. 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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  35.  55
    Shmuel Lifsches & Saharon Shelah (1996). Uniformization, Choice Functions and Well Orders in the Class of Trees. Journal of Symbolic Logic 61 (4):1206-1227.
    The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have a definable choice function (by a monadic formula with parameters)? A natural dichotomy arises where the trees that fall in the first class don't have a definable choice function and the trees in the second class have even a definable well ordering of their elements. This has a close connection to (...)
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  36. Saharon Shelah & Alexander Usvyatsov (2008). More on SOP 1 and SOP 2. Annals of Pure and Applied Logic 155 (1):16-31.
    This paper continues the work in [S. Shelah, Towards classifying unstable theories, Annals of Pure and Applied Logic 80 229–255] and [M. Džamonja, S. Shelah, On left triangle, open*-maximality, Annals of Pure and Applied Logic 125 119–158]. We present a rank function for NSOP1 theories and give an example of a theory which is NSOP1 but not simple. We also investigate the connection between maximality in the ordering left triangle, open* among complete first order theories and the SOP2 property. We (...)
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  37.  3
    Andreas Blass & Saharon Shelah (1987). There May Be Simple Pℵ1 and Pℵ2-Points and the Rudin-Keisler Ordering May Be Downward Directed. Annals of Pure and Applied Logic 33 (3):213-243.
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  38.  1
    Saharon Shelah (1971). Stability, the F.C.P., and Superstability; Model Theoretic Properties of Formulas in First Order Theory. Annals of Mathematical Logic 3 (3):271-362.
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  39.  1
    Leo Harrington & Saharon Shelah (1985). Some Exact Equiconsistency Results in Set Theory. Notre Dame Journal of Formal Logic 26 (2):178-188.
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  40.  2
    Saharon Shelah (1970). Finite Diagrams Stable in Power. Annals of Mathematical Logic 2 (1):69-118.
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  41.  9
    Haim Judah, Arnold W. Miller & Saharon Shelah (1992). Sacks Forcing, Laver Forcing, and Martin's Axiom. Archive for Mathematical Logic 31 (3):145-161.
    In this paper we study the question assuming MA+⌝CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalent to the question of what is the additivity of Marczewski's ideals 0. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals.
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  42.  8
    Menachem Magidor & Saharon Shelah (1996). The Tree Property at Successors of Singular Cardinals. Archive for Mathematical Logic 35 (5-6):385-404.
    Assuming some large cardinals, a model of ZFC is obtained in which $\aleph_{\omega+1}$ carries no Aronszajn trees. It is also shown that if $\lambda$ is a singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees.
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  43.  28
    Uri Abraham & Saharon Shelah (1996). Martin's Axiom and Well-Ordering of the Reals. Archive for Mathematical Logic 35 (5):287-298.
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  44.  4
    Yuri Gurevich & Saharon Shelah (1986). Fixed-Point Extensions of First-Order Logic. Annals of Pure and Applied Logic 32 (3):265-280.
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  45.  1
    Tapani Hyttinen & Saharon Shelah (2000). Strong Splitting in Stable Homogeneous Models. Annals of Pure and Applied Logic 103 (1-3):201-228.
    In this paper we study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in Hyttinen 167–182). We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure, see Hyttinen.
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  46.  15
    Tomek Bartoszynski & Saharon Shelah (1992). Intersection of Ultrafilters May Have Measure Zero. Archive for Mathematical Logic 31 (4):221-226.
    We show that it is consistent with ZFC that the intersection of some family of less than ultrafilters have measure zero. This answers a question of D. Fremlin.
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  47.  7
    Uri Abraham & Saharon Shelah (1993). A Δ22 Well-Order of the Reals and Incompactness of L. Annals of Pure and Applied Logic 59 (1):1-32.
    A forcing poset of size 221 which adds no new reals is described and shown to provide a Δ22 definable well-order of the reals . The encoding of this well-order is obtained by playing with products of Aronszajn trees: some products are special while other are Suslin trees. The paper also deals with the Magidor–Malitz logic: it is consistent that this logic is highly noncompact.
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  48.  6
    Saharon Shelah & Michael Makkai (1990). Categoricity of Theories in Lϰω, with Κ a Compact Cardinal. Annals of Pure and Applied Logic 47 (1):41-97.
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  49.  5
    Saharon Shelah & Andrés Villaveces (1999). Toward Categoricity for Classes with No Maximal Models. Annals of Pure and Applied Logic 97 (1-3):1-25.
    We provide here the first steps toward a Classification Theory ofElementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some λ greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non μ-splitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the o Conjecture for these classes. Further (...)
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  50.  11
    Martin Goldstern & Saharon Shelah (1995). The Bounded Proper Forcing Axiom. Journal of Symbolic Logic 60 (1):58-73.
    The bounded proper forcing axiom BPFA is the statement that for any family of ℵ 1 many maximal antichains of a proper forcing notion, each of size ℵ 1 , there is a directed set meeting all these antichains. A regular cardinal κ is called Σ 1 -reflecting, if for any regular cardinal χ, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies " $\exists\delta . We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded (...)
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