Full Idea
Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense.
Gist of Idea
Cardinality strictly concerns one-one correspondence, to test infinite sameness of size
Source
report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1
Book Reference
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.17
A Reaction
It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then?