Full Idea
Frege assumed that since infinite collections cannot be counted, he needed a theory of number that is independent of counting. He therefore took one-to-one correspondence to be basic, not well-orderings. Hence cardinals are basic, not ordinals.
Gist of Idea
Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Shaughan Lavine - Understanding the Infinite III.4
Book Reference
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.55
Related Ideas
Idea 15915 Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
Idea 15912 Counting results in well-ordering, and well-ordering makes counting possible [Lavine]