Full Idea
Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections.
Gist of Idea
Cantor needed Power Set for the reals, but then couldn't count the new collections
Source
report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I
Book Reference
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.4
A Reaction
I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue.
Related Ideas
Idea 15893 Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
Idea 15897 Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Zermelo, by Lavine]