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
1 => 2
2 => 4
3 => 6
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.