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Full Idea
Counting is not mere tagging: it is the successive assignment of cardinal numbers to increasingly large collections of objects.
Gist of Idea
Counting is the assignment of successively larger cardinal numbers to collections
Source
Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 5)
Book Ref
-: 'Notre Dame Journal of Formal Logic' [-], p.202
A Reaction
That the cardinals are 'successive' seems to mean that they are ordinals as well. If you don't know that 'seven' means a cardinality, as well as 'successor of six', you haven't understood it. Days of the week have successors. Does PA capture cardinality?
Related Ideas
Idea 17455 Is counting basically mindless, and independent of the cardinality involved? [Heck]
Idea 17447 Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck]
| 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] |
| 17455 | Is counting basically mindless, and independent of the cardinality involved? [Heck] |
| 17456 | Counting is the assignment of successively larger cardinal numbers to collections [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] |