Counting the Positive Rationals: A Brief Survey

A common question from students on the usual diagonalization proof for the uncountability of the set of real numbers is: when a representation of real numbers, such as the decimal expansions of real numbers, allows us to use the diagonalization argument to prove that the set of real numbers is uncountable, why can't we similarly apply the diagonalization argument to rational numbers in the same representation? why doesn't the argument similarly prove that the set of rational ...

This paper is a supplement to a talk for mathematics teachers given at the 2016 LSU Mathematics Contest for High School Students. The paper covers more details and aspects than could be covered in the talk. We start with an interesting problem from the 2009 Iberoamerican Math Olympiad concerning a particular sequence. We include a solution to the problem, but also relate it to several areas of mathematics. This problem demonstrates the countability of the rational numbers w...

Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.

We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts to recent results, and allowing for a simple comparison-at-a-glance between different constructions.

The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their subsets, unions, and Cartesian products are algorithmically enumerable up to one element as sequences of natural numbers. The method is similar to that of Theory of Numerosities of Benci and Di Nasso 2019) but in comparison, it is motivated by ...

The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The operation that implements addition when applied to sets that represent natural numbers yields both addition and subtraction when used with the sets that encode integers. The encoding of the integers naturally extends beyond them and identifies set...

A proof that the set of real numbers is denumerable is given.

This paper examines the possibilities of extending Cantor's two arguments on the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the set of rational numbers. The paper proves that, unless certain restrictive conditions are satisfied, both extensions are possible. It is therefore indispensable to prove that those conditions are in fact satisfied in Cantor's theory of transfinite sets. Otherwise that theory would be inconsistent.

In prefix coding over an infinite alphabet, methods that consider specific distributions generally consider those that decline more quickly than a power law (e.g., Golomb coding). Particular power-law distributions, however, model many random variables encountered in practice. For such random variables, compression performance is judged via estimates of expected bits per input symbol. This correspondence introduces a family of prefix codes with an eye towards near-optimal cod...

Since the theory developed by Georg Cantor, mathematicians have taken a sharp interest in the sizes of infinite sets. We know that the set of integers is infinitely countable and that its cardinality is Aleph0. Cantor proved in 1891 with the diagonal argument that the set of real numbers is uncountable and that there cannot be any bijection between integers and real numbers. Cantor states in particular the Continuum Hypothesis. In this paper, I show that the cardinality of th...