Definition 1.1. A set M is called finite, if M = ∅, or if there is natural number n such that the elements of M can be numbered 1,. . ., n in such a way that every element of M appears exactly once in the list. Otherwise, M is called infinite.

Example 1. 1. The set {a,b, c, d} is finite. A possible numbering might look like this: Assign 1 to a, assign 2 to b, assign 3 to c, assign 4 to d. Of course there are other possibilities of listing the set M (Which?).

2. The set of all solutions of the equation x² + 23x − 17 = 0 is finite, since the number of solutions of a polynomial is at most equal to the degree.

3. The sets N, Z, Q, R are infinite.

4. The set of all multiples of 5 is infinite.

Sometimes, it is not immediately clear, if a set is finite or infinite. One of the famous problems in Mathematics was the question for which natural numbers n there exist positive natural numbers a,b,c such that aⁿ + bⁿ = cⁿ . This question

is called Fermat’s last problem. If n = 2, then we obtain the Pythagoraic equation,  and e.g. a = 3,b = 4,c = 5 is a solution. It was only shown recently by A. Wiley that 2 is in fact the only integer for which such numbers exist.

Another problem concerns twins of primes. As you will recall, a prime number p is a natural number greater than 1 which is divisible only by 1 and by itself. A pair of prime numbers hp,qi is called a twin pair, if p+2 = q, i.e. they are consecutive odd numbers. Examples of twin pairs are h5, 7i,h11, 13i,h59, 61i. It is not known,  whether there are infinitely many prime pairs.

The following theorem has been known since the time of Euclid (ca. 300 BC):

Theorem 1.1. There are infinitely many prime numbers.

Proof. Following Euclid’s proof, we shall show that to every prime p there is a greater one. Assume that p is the greatest prime number, and let q = 1 · 2 · 3 ·...· p.