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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 Ref
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]
| 15916 | Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine] |
| 17446 | Counting rests on one-one correspondence, of numerals to objects [Frege] |
| 9582 | Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves [Frege] |
| 17444 | Husserl said counting is more basic than Frege's one-one correspondence [Husserl, by Heck] |
| 14118 | We can define one-to-one without mentioning unity [Russell] |
| 9852 | We understand 'there are as many nuts as apples' as easily by pairing them as by counting them [Dummett] |
| 17450 | Understanding 'just as many' needn't involve grasping one-one correspondence [Heck] |
| 17451 | We can know 'just as many' without the concepts of equinumerosity or numbers [Heck] |