[FOM] Real Numbers

[Jasper Stein]

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
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