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Single Idea 15916

[from 'Grundlagen der Arithmetik (Foundations)' by Gottlob Frege, in 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation ]

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]