FOM: natural numbers
Martin Davis
martind at cs.berkeley.edu
Sat Oct 25 12:45:32 EDT 1997
>
>Any adequate theory of number must yield as a theorem each instance
>of the following schema:
>
> Schema N: (E_n x)F(x) if and only if #xF(x)=n*
>________________________________________________________________________
I like to place the issue as follows: Frege and Cantor saw that the notion
of having the same cardinality is definable without specifying what these
cardinalities are. In contemporary terms, they were dealing with an
equivalence relation, and Frege specifically proposed to identify the
numbers with the equivalence classes. This famously collapsed because the
classes are "too large".
Now an alternative, used for exmaple in old-fashioned number theory texts in
dealing with what we'd call residue classes, to using equivalence classes is
to choose some suitable representative from each class. That is exactly what
von Neumann did for cardinals (and of course ordinals as well) and his
treatment has become standard.
Personally, I find this an entirely satisfactory state of affairs, and am
dubious about extracting any more philosophical juice from this issue.
Martin Davis
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