We consider a Cantor-like set as a geometric projection of a Bernoulli process. P. Billingsley (1960) and C. Dai and S.J. Taylor (1994) introduced dimension-like indices in the probability space of a stochastic process. Under suitable regularity conditions we find closed formulae linking the Hausdorff, box and packing metric dimensions of the subsets of the Cantor–like set, to the corresponding...

A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set K ⊆ R is constructed such that every set definable in (R, <,+, ·,K) is Borel. In addition, we prove quantifierelimination and completeness results for (R, <,+, ·,K), making the set K the first example of a modeltheoretically tame Cantor set. This answers questions raised by Friedma...

A general method is described which gives rise to highly symmetric tessellations of the Cantor sphere, i.e., the 2-sphere with the Cantor set removed and endowed with the hyperbolic geometry with constant negative curvature. These tessellations correspond to almost vertextransitive planar graphs with infinitely many ends. Their isometry groups have infinitely many ends and are free products wit...

An AF C*-algebra has a natural filtration as an increasing sequence of finite dimensional C*-algebras. We show that it is possible to construct a Dirac operator which relates to this filtration in a natural way and which will induce a metric for the weak*-topology on the state space of the algebra. In the particular case of a UHF C*algebra the construction can be made in a way, which relates di...

Let X be the Cantor set and φ be a minimal homeomorphism on X×T. We show that the crossed product C∗-algebra C∗(X×T, φ) is a simple AT-algebra provided that the associated cocycle takes its values in rotations on T. Given two minimal systems (X × T, φ) and (Y × T, ψ) such that φ and ψ arise from cocycles with values in isometric homeomorphisms on T, we show that two systems are approximately K-...

In 1885, Georg Cantor published his review of Gottlob Frege’s Grundlagen der Arithmetik. In this essay we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor’s review, and a new translation of Frege’s brief response to Cantor. In recent years it has entered philosophical folklore that Cantor’s 1885 review o...