Full Idea
Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed.
Gist of Idea
Cantor tried to prove points on a line matched naturals or reals - but nothing in between
Source
report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1
Book Reference
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.43