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29
Set mapping reflection
, 2003
"... Abstract. In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2ω = ω2 and that L(P(ω1)) satisfies the Axiom of Choice. It will also be demonstra ..."
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Cited by 38 (7 self)
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Abstract. In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2ω = ω2 and that L(P(ω1)) satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that (κ) fails for all regular κ> ω1. 1.
Deconstructing inner model theory
 J. Symbolic Logic
"... In this paper we shall repair some errors and fill some gaps in the inner model theory of [2]. The problems we shall address affect some quite basic definitions and proofs. We shall be concerned with condensation properties of canonical inner ..."
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Cited by 27 (5 self)
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In this paper we shall repair some errors and fill some gaps in the inner model theory of [2]. The problems we shall address affect some quite basic definitions and proofs. We shall be concerned with condensation properties of canonical inner
Indexed Squares
 Journal of Symbolic Logic
"... . We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1 ..."
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Cited by 24 (8 self)
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. We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1extendibility is incompatible with square; we prove this is close to optimal by showing that 1extendibility is compatible with square. 1. Introduction In this paper we study some variations on Jensen's celebrated combinatorial principle (variously pronounced as \square kappa" or \box kappa"). is a principle which is helpful in constructing objects of cardinality + ; for example Jensen showed that if holds then there is a special + Aronszajn tree, and every stationary subset of + contains a nonreecting stationary subset. Jensen proved [Je1] that if V = L then holds for every uncountable cardinal ( ! is a trivial theorem in ZFC). In combination with Jen...
PFA implies AD^L(R)
, 2007
"... In this paper we shall prove Theorem 0.1 Suppose there is a singular strong limit cardinal κ such that κ fails; then AD holds in L(R). See [11] for a discussion of the background to this problem. We suspect that more work will produce a proof of the theorem with its hypothesis that κ is a strong lim ..."
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Cited by 18 (4 self)
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In this paper we shall prove Theorem 0.1 Suppose there is a singular strong limit cardinal κ such that κ fails; then AD holds in L(R). See [11] for a discussion of the background to this problem. We suspect that more work will produce a proof of the theorem with its hypothesis that κ is a strong limit weakened to ∀α < κ(αω < κ), and significantly more work will enable one to drop the hypothesis that κ is a strong limit entirely. At present, we do not see how to carry out even the less ambitious project. Todorcevic ([23]) has shown that if the Proper Forcing Axiom (PFA) holds, then κ fails for all uncountable cardinals κ. Thus we get immediately: Corollary 0.2 PFA implies ADL(R). It has been known since the early 90’s that PFA implies PD, that PFA plus the existence of a strongly inaccessible cardinal implies ADL(R), and that PFA plus a measurable yields an inner model of ADR containing all reals and ordinals. 1 As we do here, these arguments made use of Tororcevic’s work, so that logical strength is ultimately coming from a failure of covering for some appropriate core models.
CHARACTERIZATION OF �κ IN CORE MODELS
, 2001
"... We present a general construction of a �κsequence in Jensen’s fine structural extender models. This construction yields a local definition of a canonical �κsequence as well as a characterization of those cardinals κ, for which the principle �κ fails. Such cardinals are called subcompact and can be ..."
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Cited by 18 (6 self)
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We present a general construction of a �κsequence in Jensen’s fine structural extender models. This construction yields a local definition of a canonical �κsequence as well as a characterization of those cardinals κ, for which the principle �κ fails. Such cardinals are called subcompact and can be described in terms of elementary embeddings. Our construction is carried out abstractly, making use only of a few fine structural properties of levels of the model, such as solidity and condensation.
Stacking mice
 JOURNAL OF SYMB. LOGIC
"... We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ ℵ3 such that □κ and □(κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the ..."
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Cited by 11 (5 self)
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We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ ℵ3 such that □κ and □(κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ +). Of special interest is 1) with κ = ℵ3 since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over K c ‖κ.
The proper forcing axiom
 Proceedings of the ICM 2010
"... The author’s preparation of this article and his travel to the 2010 meeting of ..."
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Cited by 9 (1 self)
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The author’s preparation of this article and his travel to the 2010 meeting of
The core model for almost linear iterations
 Annals of Pure and Appl. Logic 116 (2002
"... We introduce 0 • (“zero handgrenade”) as a sharp for an inner model with a proper class of strong cardinals. We prove the existence of the core model K in the theory “ZFC + 0 • doesn’t exist. ” Combined with work of Woodin, Steel, and earlier work of the author, this provides the last step for de ..."
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Cited by 8 (4 self)
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We introduce 0 • (“zero handgrenade”) as a sharp for an inner model with a proper class of strong cardinals. We prove the existence of the core model K in the theory “ZFC + 0 • doesn’t exist. ” Combined with work of Woodin, Steel, and earlier work of the author, this provides the last step for determining the exact consistency strength of the assumption in the statement of the 12th Delfino problem (cf. [12]). 0 Introduction. Core models were constructed in the papers [2], [13], [7], [15] and [16], [8] (see also [23]), [27], and [28]. We refer the reader to [6], [17], and [14] for less painful introductions into core model theory. A core model is intended to be an inner model of set theory (that is, a transitive
Woodin’s axiom (∗), bounded forcing axioms, and precipitous ideals on ω1
"... If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at ℵ2 with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC–model. This yields that if Woodin’s Pmax axiom (∗) holds, then BPFA implies that V is closed under the “WoodininthenextZFC–mod ..."
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Cited by 5 (3 self)
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If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at ℵ2 with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC–model. This yields that if Woodin’s Pmax axiom (∗) holds, then BPFA implies that V is closed under the “WoodininthenextZFC–model ” operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus “NSω1 is precipitous” and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor, [2, Question 6.11].