From another thread, severad discussions were made - and questions asked - about "infinity" or "infinite".

A complete study of infinite numbers was made in late XIX century by George Cantor - who died crazy.

About finite numbers, nothing special to say: we all know what they are. So, we have so many feet as shoes because it is possible to make a correspondence ( a so called one-to-one function) between feet and shoes: to each foot we assign one shoes, and viceversa.

And, if a set has , say, 3 elements, it cannot have such a one-to-one correspondence with a set having -say -4 elements.


What happens about infinite sets? The first amazing thing is that it is possible to have two sets, one contained in another, and still such it is possible to find such a correspondence:

for example, {even numbers} is a subset of {positive integer numbers}, but there is a one-to-one correspondence:
define, for every even number n, its half n/2.

We say that even numbers are so many as positive integers numbers, or that they have the same "cardinality" = kind of infinity.

The interesting thing is the rational numbers (fractions m/n) are so many as integers numbers (such a correspondence can be built), and real numbers are not (it can be proven that such a correspondence cannot exist)