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 Zermelo-Fraenkel 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...

. Starting with GCH and a P3-hypermeasurable 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 set-theoretic 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 ...

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 ω-inaccessible 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.

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

. This note investigates the class of finite initial segments of the cumulative hierarchy of pure sets. We show that this class is first-order definable over the class of finite directed graphs and that this class admits a first-order 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...

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.

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 J-trivial (resp. R-trivial) semigroups. In particular, we show that the pseudovariety V ∨ W is (completely) κ-tame when V is a subpseudovariety of J and W is (completely) κ-tame. Moreover, if W is a κ-tame pseudovariety which satisfies the pseudoidentity x1 · · · xry ω+1 zt ω = x1 · · · xryzt ω, then we prove that R ∨ W is also κ-tame. In particular the joins R ∨ Ab, R ∨ G, R ∨ OCR, and R ∨ CR are decidable.

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 Branch-and-Bound (sBB) algorithms.

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 extremely-close-to-polynomial 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 Non-Uniform Circuit complexity models. e-mail: shai@cs.technion.ac.il 0 1

. We construct a model in which there are no @n-Aronszajn trees for any finite n 2, starting from a model with infinitely many supercompact cardinals. We also construct a model in which there is no ++ -Aronszajn tree for a strong limit cardinal of cofinality !, starting from a model with a supercompact cardinal and a weakly compact cardinal above it. 1. Introduction We will prove the following theorems. Theorem 1. If "ZFC + there exist infinitely many supercompact cardinals" is consistent, then "ZFC + there are no @n-Aronszajn trees for 2 n ! !" is also consistent. Theorem 2. If "ZFC + there exists a supercompact cardinal with a weakly compact cardinal above it" is consistent then "ZFC + there exists a strong limit cardinal of cofinality ! such that there are no ++ -Aronszajn trees" is also consistent. We start by recalling the definition of "-Aronszajn tree" and some related concepts. Definition 1.1. Let be regular. 1. A -tree is a tree of height whose every lev...