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47
Squares, Scales and Stationary Reflection
"... Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by ZermeloFraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many are ..."
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Cited by 25 (10 self)
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Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by ZermeloFraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven eective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(}) and square() discovered by Jensen. Simultaneously, attempts have been made to nd suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of c...
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 17 (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...
Partial orderings with the weak FreeseNation property
 Annals of Pure and Applied Logic
, 1996
"... (revised according to some suggestions by the referee) to appear in Annals of Pure and Applied Logic 549 revision:19950825 modified:19950827 A partial ordering P is said to have the weak FreeseNation property (WFN) if there is a mapping f: P → [P] ≤ℵ0 such that, for any a, b ∈ P, if a ≤ b then ..."
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Cited by 9 (5 self)
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(revised according to some suggestions by the referee) to appear in Annals of Pure and Applied Logic 549 revision:19950825 modified:19950827 A partial ordering P is said to have the weak FreeseNation property (WFN) if there is a mapping f: P → [P] ≤ℵ0 such that, for any a, b ∈ P, if a ≤ b then there exists c ∈ f(a) ∩ f(b) such that a ≤ c ≤ b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g., under CH, every ccc complete Boolean algebra has this property while, under b ≥ ℵ2, there exists no complete Boolean algebra with the WFN. (Theorem 6.2).
Forcing axioms and projective sets of reals
 Proceedings of Foundations of the Formal Sciences III, in: “Classical and new paradigms of computation and their complexity hierarchies” (Löwe
"... Abstract. This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper for ..."
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Cited by 8 (0 self)
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Abstract. This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper forcing axiom (BPFA) + “every projective set of reals is Lebesgue measurable ” is equiconsistent with ZFC + “there is a Σ1 reflecting cardinal above a remarkable cardinal.” 1. Introduction. The current paper ∗ is in the tradition of the following result.
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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Cited by 8 (0 self)
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Canonical structure for the universe of set theory; part 2
 Ann. Pure and Appl. Logic, To Appear
"... Abstract. We prove a number of consistency results complementary to the ZFC results from our paper [4]. We produce examples of nontightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not co ..."
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Cited by 6 (2 self)
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Abstract. We prove a number of consistency results complementary to the ZFC results from our paper [4]. We produce examples of nontightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the existence of stationarily many nongood points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness. 1.
The Proper Forcing Axiom and the Singular Cardinal Hypothesis
, 2005
"... We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses ideas of Moore from [11] and the notion of a relativized trace function on pairs of ordinals. ..."
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Cited by 6 (0 self)
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We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses ideas of Moore from [11] and the notion of a relativized trace function on pairs of ordinals.
Generic Absoluteness
 Annals of Pure and Applied Logic, Vol.108
, 2001
"... We explore the consistency strength of \Sigma 1 3 and \Sigma 1 4 absoluteness, for a variety of forcing notions. ..."
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Cited by 4 (2 self)
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We explore the consistency strength of \Sigma 1 3 and \Sigma 1 4 absoluteness, for a variety of forcing notions.