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

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

You Can Enter Cantor's Paradise

This paper is based on the talk given by the author after he received the International Bolyai Prize in Mathematics (on November 4, 2000 in Budapest, Hungary)

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

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

Generalized measurement on size of set

We generalize the measurement using an expanded concept of cover, in order to provide a new approach to size of set other than cardinality. The generalized measurement has application backgrounds such as a generalized problem in dimension reduction, and has reasons from the existence of the minimum of both the positive size and the positive graduation, i.e., both the minimum is the size of the set ${0}$. The minimum of positive graduation in actual measurement provides the possibility that an object cannot be partitioned arbitrarily, e.g., an interval $[0, 1]$ cannot be partitioned by arbitrarily infinite times to keep compatible with the minimum of positive size. For the measurement on size of set, it can be assumed that this minimum is the size of ${0}$, in symbols $|{0}|$ or graduation 1. For a set $S$, we generalize any graduation as the size of a set $C_i$ where $\exists x \in S (x \in C_i)$, and $|S|$ is represented by a pair, in symbols $(C, N(C))$, where ${C} = \cup {C_i}$ and $N(C)$ is a set function on $C_i$, with $C_i$ independent of the order $i$ and $N(C)$ reflecting the quantity of $C_i$. This pair is a generalized form of box-counting dimension. The yielded size satisfies the properties of outer measure in general cases, and satisfies the properties of measure in the case of graduation 1; while in the reverse view, measure is a size using the graduation of size of an interval. As for cardinality, the yielded size is a one-to-one correspondence where only addition is allowable, a weak form of cardinality, and rewrites Continuum Hypothesis using dimension as $\omega \dot |{0,1}| = 1$. In the reverse view, cardinality of a set is a size in the graduation of the set. The generalized measurement provides a unified approach to dimension, measure, cardinality and hence infinity.Comment: 10 pages, 0 figures, using amsar

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

Physical Objects of Dark Systems

The hypothesis of existence of off-site continuums is investigated. Principles of the physical description are formulated. The structure of off-site continuums and opportunities of observation of off-site physical objects from the continuum of the observer is investigated. There are found conformities between properties of the considered objects with properties of known physical objects and fundamental interactions both in micro-, and macro-scales.Comment: LaTeX2e, 11 pages, 2 figure

On (1,ω1)\left( 1,\omega_{1}\right) \emph{-}weakly universal functions

A function $U:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ is called $\left( 1,\omega_{1}\right)$\emph{-weakly universal }if for every function $F:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ there is an injective function $h:\omega_{1}\longrightarrow\omega_{1}$ and a function $e:\omega \longrightarrow\omega$ such that $F\left( \alpha,\beta\right) =e\left( U\left( h\left( \alpha\right) ,h\left( \beta\right) \right) \right)$ for every $\alpha,\beta\in\omega_{1}$. We will prove that it is consistent that there are no $\left( 1,\omega_{1}\right)$\emph{-}weakly universal functions, this answers a question of Shelah and Stepr\={a}ns. In fact, we will prove that there are no $\left( 1,\omega_{1}\right)$\emph{-}weakly universal functions in the Cohen model and after adding $\omega_{2}$ Sacks reals side-by-side. However, we show that there are $\left( 1,\omega _{1}\right)$\emph{-}weakly universal functions in the Sacks model. In particular, the existence of such graphs is consistent with $\clubsuit$ and the negation of the Continuum Hypothesis