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72
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...
A Model In Which GCH Holds At Successors But Fails At Limits
 Transactions of the American Mathematical Society
, 1992
"... . Starting with GCH and a P3hypermeasurable cardinal, a model is produced in which 2 = + if is a successor cardinal and 2 = ++ if is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing. 1. INTRODUCTION 1. Historical remarks and statement o ..."
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Cited by 18 (5 self)
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. Starting with GCH and a P3hypermeasurable cardinal, a model is produced in which 2 = + if is a successor cardinal and 2 = ++ if is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing. 1. INTRODUCTION 1. Historical remarks and statement of the main result. The continuum problem is an old one, dating back to Cantor and his statement of the Continuum Hypothesis in [Ca]. Put in a modern form which might have puzzled Cantor, the problem is to determine which behaviours of the continuum function 7\Gamma! 2 are consistent with ZFC. Throughout this paper ZFC will be the base set theory, though as we see below strong settheoretic hypotheses will play an essential role in the result. Before Godel progress on the continuum problem was made by the descriptive set theorists, who showed that certain easily definable sets of reals could not be counterexamples to CH. Godel [G] took the major step forward of showing that in a certain ...
Un Calcul De Constructions Infinies Et Son Application A La Verification De Systemes Communicants
, 1996
"... m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to ..."
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Cited by 16 (0 self)
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m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to Luc Boug'e, who accepted to be my oficial supervisor, and to the chair of the jury, Michel Cosnard, who opened to me the doors of the LIP. During these last three years in Lyon I met many wonderful people, who then become wonderful friends. Miguel, Nuria, Veronique, Patricia, Philippe, Pia, Rodrigo, Salvador, Sophie : : : with you I have shared the happiness and sadness of everyday life, those little things which make us to remember someone forever. I also would like to thank the people from "Tango de Soie", for all those funny nights at the Caf'e Moulin Joly. Thanks too to the Uruguayan research community in Computer Science (specially to Cristina Cornes and Alberto Pardo) w
Forcing positive partition relations
 Trans. Amer. Math. Soc
, 1983
"... Abstract. We show how to force two strong positive partition relations on u, and use them in considering several wellknown open problems. In [32] Sierpiñski proved that the wellknown Ramsey Theorem [27] does not generalize to the first uncountable cardinal by constructing a partition [ío,]2 = KQ U ..."
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Cited by 13 (2 self)
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Abstract. We show how to force two strong positive partition relations on u, and use them in considering several wellknown open problems. In [32] Sierpiñski proved that the wellknown Ramsey Theorem [27] does not generalize to the first uncountable cardinal by constructing a partition [ío,]2 = KQ U Kx with no uncountable homogeneous sets. Sierpinski's partition has been analyzed in several directions. One direction was to improve this relation so as to get much stronger negative partition relations on ux. The direction taken in this paper is to prove stronger and stronger positive relations on <*>, which do not appear to be refutable by Sierpinski's partition. The first result of this kind is due to Dushnik and Miller [9] who proved W,»(«p w). This was later improved by Erdös and Rado [11] to to,> (<«J, « + 1). In [17] Hajnal proved the following result which shows that the ErdösRado theorem is, in a sense, a best possible result of this sort in ZFC: CH implies uxr * (u>x,u + 2). Problem 8 of Erdös and Hajnal [12, 13] asks whether w,r * (w „ w + 2)2 can be proved without the continuum hypothesis, i.e., whether w, » (ax, u + 2)2 is consistent with ZFC. The first result on this problem is due to Laver [24] who proved that MA8  implies to, ^(w,,("'jj. This result was improved by Hajnal (see [24]) to MAN  implies to,> I to,, I ')) for all a <wx. Clearly these results leave open the problem whether w, » (to,, w + 2)2 is consistent. In this paper we shall prove the consistency of ux>iux,ct) for all a <w,. Received by the editors November 19, 1982.
Combinatorial principles from adding Cohen reals
 in Logic Colloquium 95, Proceedings of the Annual European Summer Meeting of the Association of Symbolic Logic
, 1998
"... Abstract. We first formulate several “combinatorial principles” concerning κ × ω matrices of subsets of ω and prove that they are valid in the generic extension obtained by adding any number of Cohen reals to any ground model V, provided that the parameter κ is an ωinaccessible regular cardinal in ..."
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Cited by 12 (4 self)
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Abstract. We first formulate several “combinatorial principles” concerning κ × ω matrices of subsets of ω and prove that they are valid in the generic extension obtained by adding any number of Cohen reals to any ground model V, provided that the parameter κ is an ωinaccessible regular cardinal in V. Then in section 4 we present a large number of applications of these principles, mainly to topology. Some of these consequences had been established earlier in generic extensions obtained by adding ω2 Cohen reals to ground models satisfying CH, mostly for the case κ = ω2. 1.
Reformulation and Convex Relaxation Techniques for Global Optimization
 4OR
, 2004
"... Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested i ..."
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Cited by 9 (7 self)
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Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested in determining the globally optimal point. This thesis is concerned with techniques for establishing such global optima using spatial BranchandBound (sBB) algorithms.
Elementary Properties Of The Finite Ranks
 Mathematical Logic Quarterly
, 1998
"... . This note investigates the class of finite initial segments of the cumulative hierarchy of pure sets. We show that this class is firstorder definable over the class of finite directed graphs and that this class admits a firstorder definable global linear order. We apply this last result to show ..."
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Cited by 8 (0 self)
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. This note investigates the class of finite initial segments of the cumulative hierarchy of pure sets. We show that this class is firstorder definable over the class of finite directed graphs and that this class admits a firstorder definable global linear order. We apply this last result to show that FO(!; BIT) = FO(BIT): This note establishes some elementary properties of the finite initial segments of the cumulative hierarchy of pure sets. We define the sets Vn ; n 2 ! by induction as follows: V 0 = ;; Vn+1 = P(Vn ): (Here, P(X) is the power set of X; that is, the set of all subsets of X:) The collection of nonempty finite ranks, FR, is fVn : n ? 0g: We will often use Vn to denote the directed graph whose set of nodes is Vn and whose edge relation is 2, the set membership relation; similarly, we will use FR to denote the collection of such graphs Vn for n ? 0: More generally, if M is a collection of sets, we will also use M to denote the directed graph whose set of nodes is M and...
Locally compact perfectly normal spaces may all be paracompact
, 2010
"... We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, th ..."
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Cited by 7 (6 self)
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We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space with a hereditarily normal square is metrizable. We also solve a problem raised by the second author, proving it consistent with ZFC that every first countable hereditarily normal countable chain condition space is hereditarily separable.
Tameness of pseudovariety joins involving R
, 2004
"... In this paper, we establish several decidability results for pseudovariety joins of the form V ∨ W, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all Jtrivial (resp. Rtrivial) semigroups. In particular, we show that the pseudovariety V ∨ ..."
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Cited by 7 (6 self)
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In this paper, we establish several decidability results for pseudovariety joins of the form V ∨ W, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all Jtrivial (resp. Rtrivial) semigroups. In particular, we show that the pseudovariety V ∨ W is (completely) κtame when V is a subpseudovariety of J and W is (completely) κtame. Moreover, if W is a κtame pseudovariety which satisfies the pseudoidentity x1 · · · xry ω+1 zt ω = x1 · · · xryzt ω, then we prove that R ∨ W is also κtame. In particular the joins R ∨ Ab, R ∨ G, R ∨ OCR, and R ∨ CR are decidable.
On the Independence of P Versus NP
, 1991
"... We investigate the possibility that the current failure to resolve basic complexity theoretic questions stems from the independence of these questions with respect to the formal theories underlying our mathematical reasoning. We show that, any question in the field of computational complexity tha ..."
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Cited by 6 (1 self)
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We investigate the possibility that the current failure to resolve basic complexity theoretic questions stems from the independence of these questions with respect to the formal theories underlying our mathematical reasoning. We show that, any question in the field of computational complexity that is independent of a certain extension of the axioms of Peano Arithmetic, using currntly available techniques, is `practically insignificant'. This implies that if P 6= NP can be shown to be independent of Peano Arithmetic, using any currently known mathematical paradigm, then NP has extremelyclosetopolynomial deterministic time upper bounds. In particular, in such a case, there is a DTIME(n log (n) ) algorithm that computes SAT correctly on infinitely many huge intervals of input lengths. We provide a complete characterization of the worst case behavior of languages whose location in the complexity hierarchy is independent (with respect to sufficiently strong proof systems, including Peano Arithmetic). Such languages are, on one hand easily computable for long stretches of inputs, and, on the other hand, they are complex infinitely often. (We also construct an explicit example of such a language). Our results hold for both the Turing Machine and the NonUniform Circuit complexity models. email: shai@cs.technion.ac.il 0 1