Georg Cantor defined an infinite

set as a set that could be mapped one-to-one onto a proper part of itself. For example the set of all postive

integers can be mapped onto the set of all

even numbers like this:

1 => 2

2 => 4

3 => 6

etc.

There are many "levels" of infinity. For example, the set of all real numbers cannot be mapped one-to-one onto the set of all integers. There are just too many of them, as can be shown by a diagonal argument.