Full Idea
Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity.
Gist of Idea
The naturals won't map onto the reals, so there are different sizes of infinity
Source
report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4
Book Reference
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.99