ABSTRACT
It is well known that one of the first mathematical results of Cantor (which turned out to be rather surprising to him) was the discovery of the existence of a bijection between the set R of all real numbers and the corresponding product set R2 = R ×R (i.e., the Euclidean plane). For a time, Cantor did not believe that such a bijection exists and even wrote to Dedekind about his doubts in this connection. Of course, Cantor already knew of the existence of a bijection between the setN of all natural numbers and the product set N ×N. A simple way to construct such a bijection is the following one. First, we observe that a function
f : N→ N \ {0} defined by the formula
f(n) = n+ 1 (n ∈ N) is a bijection between N and the set of all strictly positive natural numbers. Then, for each integer n > 0, we have a unique representation of n in the form
n = 2k(2l + 1)
where k and l are some natural numbers. Now, define a function
g : N \ {0} → N×N by the formula
g(n) = (k, l) (n ∈ N \ {0}). One can immediately check that g is a bijection, which also yields the corresponding bijection between N and N×N.
