Full Idea
Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A).
Gist of Idea
Cantor: there is no size between naturals and reals, or between a set and its power set
Source
report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1
Book Reference
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.19
A Reaction
Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved.