In this work we will give a brief introduction to a couple of different approaches on rigorously
discussing what the size of a set is. We compare the sizes of groups in our daily lives all the …

The idea for this vignette comes from mathematician David Hilbert. 1 It is a different
presentation of infinity. Imagine a hotel with 10,000 rooms. Every room is occupied. Should a …

There is a motivation to generalize the notion of length to another concept that could capture
length in a more general setting. In principle, the measure of a set should refer to the size of …

Theorem 1 is reminiscent of the well-known fact that if a1,••., an are arbitrary integers, then
there is a nonempty subset I of {1,..., n} such that L; e 1a;= 0 (modn). However, this result is a …

The purpose of this article is to provide a brief description of the 'Hilbert's Paradox of Grand
Hotel'. It incorporates the concept of transfinite or ordinal numbers (sometimes called …

We try to explain the convictions of mathematicians that the axioms of Russell's theory of
types and of ZFC are consistent. We claim that this is an inductive inference (a …

The pigeonhole principle (also known as Dirichlet's principle) states the “obvious” fact that
n+ 1 pigeons cannot sit in n holes so that every pigeon is alone in its hole. More generally …

In this chapter we shall give some basic notions and related results from the classical
measure theory in a very condensed form. The main purpose is to make it easier to follow …

Many problems in mathematics, specifically in combinatorics, can be simplified by a simple
change of perspective. Often a difficult counting problem can be solved when the sets are …

First, if there are only apples, and we would like to choose 50 boxes so that together they
contain at least half of the apples, that's easy. We just choose the 50 boxes with the greatest …