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Full Idea
The interest of Frege's Theorem is that it offers us an explanation of the fact that the numbers satisfy the Dedekind-Peano axioms.
Clarification
Frege's Theorem says arithmetic rests on one-one correspondence
Gist of Idea
Frege's Theorem explains why the numbers satisfy the Peano axioms
Source
Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 6)
Book Ref
-: 'Notre Dame Journal of Formal Logic' [-], p.204
A Reaction
He says 'explaining' does not make it more fundamental, since all proofs explain why their conclusions hold.
| 17448 | In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck] |
| 17449 | We can understand cardinality without the idea of one-one correspondence [Heck] |
| 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] |
| 17454 | Children can use numbers, without a concept of them as countable objects [Heck] |
| 17456 | Counting is the assignment of successively larger cardinal numbers to collections [Heck] |
| 17455 | Is counting basically mindless, and independent of the cardinality involved? [Heck] |
| 17453 | The meaning of a number isn't just the numerals leading up to it [Heck] |
| 17457 | A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck] |
| 17458 | Equinumerosity is not the same concept as one-one correspondence [Heck] |
| 17459 | Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck] |