The Generalized Continuum Hypothesis revisited

We argue that we solved Hilbert's first problem positively (after reformulating it just to avoid the known consistency results) and give some applications. Let lambda to the revised power of kappa, denoted lambda^{[kappa]}, be the minimal cardinality of a family of subsets of lambda each of cardinality kappa such that any other subset of lambda of cardinality kappa is included in the union of <kappa members of the family. The main theorem says that almost always this revised power is equal to lambda. Our main result is The Revised GCH Theorem: Assume we fix an uncountable strong limit cardinal mu (i.e., mu>aleph_0, (for all theta= mu for some kappa<mu we have: (a) kappa lambda^{[theta]}= lambda and (b) there is a family P of lambda subsets of lambda each of cardinality < mu such that every subset of lambda of cardinality mu is equal to the union of < kappa members of P

Multiscale Analysis and Localization of Random Operators

A discussion of the method of multiscale analysis in the study of localization of random operators based on lectures given at \emph{Random Schr\"odinger operators: methods, results, and perspectives}, \'Etats de la recherche, Universit\'e Paris 13, June 200

Deciding the Continuum Hypothesis with the Inverse Powerset

We introduce the concept of inverse powerset by adding three axioms to the Zermelo-Fraenkel set theory. This extends the Zermelo-Fraenkel set theory with a new type of set which is motivated by an intuitive meaning and interesting applications. We present different ways to extend the definition of cardinality and show that one implies the continuum hypothesis while another implies the negation of the continuum hypothesis. We will also explore the idea of empty sets of different cardinalities which could be seen as the empty counterpart of Cantor's theorem for infinite sets.Comment: 37 pages; added and refined a few definition

Localization for Anderson Models on Metric and Discrete Tree Graphs

We establish spectral and dynamical localization for several Anderson models on metric and discrete radial trees. The localization results are obtained on compact intervals contained in the complement of discrete sets of exceptional energies. All results are proved under the minimal hypothesis on the type of disorder: the random variables generating the trees assume at least two distinct values. This level of generality, in particular, allows us to treat radial trees with disordered geometry as well as Schr\"odinger operators with Bernoulli-type singular potentials. Our methods are based on an interplay between graph-theoretical properties of radial trees and spectral analysis of the associated random differential and difference operators on the half-line.Comment: 55 pages; several changes to the exposition in v

Saturation and elementary equivalence of C*-algebras

We study the saturation properties of several classes of $C^*$-algebras. Saturation has been shown by Farah and Hart to unify the proofs of several properties of coronas of $\sigma$-unital $C^*$-algebras; we extend their results by showing that some coronas of non-$\sigma$-unital $C^*$-algebras are countably degree-$1$ saturated. We then relate saturation of the abelian $C^*$-algebra $C(X)$, where $X$ is $0$-dimensional, to topological properties of $X$, particularly the saturation of $CL(X)$.Comment: 36 pages. Version 4 is rewritten for clarity in several place

A Note On Separating Function Sets

We study separating function sets. We find some necessary and sufficient conditions for $C_p(X)$ or $C_p^2(X)$ to have a point-separating subspace that is a metric space with certain nice properties. One of the corollaries to our discussion is that for a zero-dimensional $X$, $C_p(X)$ has a discrete point-separating space if and only if $C_p^2(X)$ does.Comment: Lobachevskii Journal of Mathematics, accepted, 201

In Memoriam: James Earl Baumgartner (1943-2011)

James Earl Baumgartner (March 23, 1943 - December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations.Comment: 51 page

There is no bound on sizes of indecomposable Banach spaces

Assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable Banach spaces. i.e., such that whenever they are decomposed as $X\oplus Y$, then one of the closed subspaces $X$ or $Y$ must be finite dimensional. It requires alternative techniques compared to those which were initiated by Gowers and Maurey or Argyros with the coauthors. This is because hereditarily indecomposable Banach spaces always embed into $\ell_\infty$ and so their density and cardinality is bounded by the continuum and because dual Banach spaces of densities bigger than continuum are decomposable by a result due to Heinrich and Mankiewicz. The obtained Banach spaces are of the form $C(K)$ for some compact connected Hausdorff space and have few operators in the sense that every linear bounded operator $T$ on $C(K)$ for every $f\in C(K)$ satisfies $T(f)=gf+S(f)$ where $g\in C(K)$ and $S$ is weakly compact or equivalently strictly singular. In particular, the spaces carry the structure of a Banach algebra and in the complex case even the structure of a $C^*$-algebra

A Note on Congruences of Infinite Bounded Involution Lattices

We prove that an infinite (bounded) involution lattice and even pseudo--Kleene algebra can have any number of congruences between $2$ and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals as elements or as many ideals as subsets; consequently, the same holds for antiortholattices. Under the Generalized Continuum Hypothesis, this means that an infinite (bounded) involution lattice, pseudo--Kleene algebra or antiortholattice can have any number of congruences between $2$ and its number of subsets, regardless of its number of ideals.Comment: 10 page

Topological detection of Lyapunov instability

Given an arbitrary continuous flow on a manifold M, let CMin be the set of its compact minimal sets, endowed with the Hausdorff metric, and S the subset of those that are Lyapunov stable. A topological characterization of the interior of S, the set of Lyapunov stable compact minimal sets that are away from Lyapunov unstable ones is given, together with a description of the dynamics around it. In particular, int S is locally a Peano continuum (Peano curve) and each of its countably many connected components admits a complete geodesic metric. This result establishes unexpected connections between the local topology of CMin and the dynamics of the flow, providing criteria for the local detection of Lyapunov instability by merely looking at the topology of CMin. For instance, if CMin is not locally connected at some compact minimal set Q (seen as a "point" of CMin), then every neighbourhood of Q in M contains Lyapunov unstable compact minimal sets (hence, if CMin is nowhere locally connected, then every neighbourhood of each compact minimal set contains infinitely many Lyapunov unstable compact minimal sets).Comment: Replaces "Lyapunov stability away from instability"; pages 1-3 and Section 3.1 are new; minor lapses corrected; 29 pages, 7 figures