Related papers: Cantor versus Cantor
The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be…
Discussions surrounding the nature of the infinite in mathematics have been underway for two millennia. Mathematicians, philosophers, and theologians have all taken part. The basic question has been whether the infinite exists only in…
We establish various new results on a problem proposed by K. Mahler in 1984 concerning rational approximation to fractal sets by rational numbers inside and outside the set in question, respectively. Some of them provide a natural…
This thesis presents an alternative to Cantor's theory of cardinality, insofar as that is understood as a theory of set size. The alternative is based on a general theory, ClassSize. ClassSize contains all sentences in the first order…
For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness…
Recently it was shown that it is undecidable whether a term rewrite system can be proved terminating by a polynomial interpretation in the natural numbers. In this paper we show that this is also the case when restricting the…
We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the R\'enyi system with one real base. We focus on the so-called…
It is shown how Dedekind cuts can be used to introduce the extended real numbers along with sound arithmetic laws via one simple rule for the addition of sets. The crucial idea is that the use of the lower and the upper part of the cuts,…
We give sufficient conditions for two Cantor sets of the line to be nested for a positive set of translation parameters. This problem occurs in diophantine approximations. It also occurs as a toy model of the parameter selection for…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this…
We give a heuristic for the number of reduced rationals on Cantor's middle thirds set, with a fixed bound on the denominator. We also describe extensive numerical computations supporting this heuristic.
We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen…
For a large class of Cantor sets on the real-line, we find sufficient and necessary conditions implying that a set has positive (resp. null) measure for all doubling measures of the real-line. We also discuss same type of questions for…
This somewhat unusual proof for the fact that the reals are uncountable, which is adapted from one of Bourbaki's proofs in "Fonctions d'une variable reelle", may be of some interest.
The first-order theory of addition over the natural numbers, known as Presburger arithmetic, is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the…
Let C be a Cantor set. For a real number t let C+t be the translate of C by t, We say two real numbers s,t are equivalent if the intersection of C and C+s is a translate of the intersection of C and C+t. We consider a class of Cantor sets…
Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty…
In this paper, we prove the number of countable models of a countable supersimple theory is either 1 or infinite. This result is an extension of Lachlan's theorem on a superstable theory.
The topic of this paper is the subtle interplay between countability and representations. In particular, we establish that the definition of countability of a certain set $X$ crucially hinges on the associated equivalence relation $=_{X}$.…