[FOM] Real Numbers
Jasper Stein
jasper at cs.kun.nl
Fri May 16 04:36:43 EDT 2003
Hartley Slater wrote:
[about natural numbers:]
> series. They cannot be *any* series of sets: the theory of number must
> preceed the theory of sets, because one needs the concept of number to
> decide which specific predicates determine sets - the count predicates.
> So the natural numbers are not sets period. You are presuming that the
> Fregean assumption that all predicates are count is correct. But it isn't.
I don't see how the theory of number must preceed the theory of sets (or
theories of sets). The axioms of ZFC (and presumably many other set
theories that I am less familiar with) are written down without
referring to any natural number at all.
Of course we do need a theory of how to interpret "there exist" and "for
all" etc. - but I'd say that's something quite different from a theory
of number, especially of natural numbers.
I may misunderstand your concept of 'count predicates', but anyway I
fail to see the relevance of them to natural numbers. Surely +they+ are
'count'? And surely there +are+ sets (like the von Neumann ordinals)
that are count? So why forbid the usual equating of N and omega?
(By the way, I do agree that numbers are not sets)
Jasper
--
+++ Out of Cheese error +++ MELON MELON MELON +++ Redo from Start +++
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