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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 man ..."
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Cited by 36 (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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. 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...
GENERIC ABSOLUTENESS AND THE CONTINUUM
 MATHEMATICAL RESEARCH LETTERS 9, 465–471
, 2002
"... Let Hω2 denote the collection of all sets whose transitive closure has size at most ℵ1. Thus, (Hω2, ∈) is a natural model of ZFC minus the powerset axiom which correctly estimates manyof the problems left open by the smaller and better understood structure (Hω1, ∈) of hereditarily countable sets. O ..."
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Cited by 15 (2 self)
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Let Hω2 denote the collection of all sets whose transitive closure has size at most ℵ1. Thus, (Hω2, ∈) is a natural model of ZFC minus the powerset axiom which correctly estimates manyof the problems left open by the smaller and better understood structure (Hω1, ∈) of hereditarily countable sets. One of such problems is, for example, the Continuum Hypothesis. It is largely for this reason that the structure (Hω2, ∈) has recently received a considerable amount of study (see e.g. [15] and [16]). Recall the wellknown LevySchoenfield absoluteness theorem ([10, §2]) which states that for everyΣ0−sentence ϕ(x, a) with one free variable x and parameter a from Hω2, if there is an x such that ϕ(x, a) holds then there is such an x in Hω2, or in other words, (1) (Hω2, ∈) ≺1 (V,∈). Strictly speaking, what is usuallycalled the LevySchoenfield absoluteness theorem is a bit stronger result than this, but this is the form of their absoluteness theorem that allows a variation of interest to us here. The generic absoluteness
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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(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).
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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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.
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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The author’s preparation of this article and his travel to the 2010 meeting of
Canonical structure for the universe of set theory; part 2
 ANN. PURE AND APPL. LOGIC, TO APPEAR
"... 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 concentrat ..."
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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.
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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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.