On Oct 7, 2009, at 1:27 PM, John F. Sowa wrote: (01)
> Pat,
>
> I agree with your comments, but I'll add a quibble. (02)
OK, and a quibble back... (03)
>
> It is certainly true that the identity criterion for sets
> (namely, having the same members) is simple.
>
> But the identity criterion for types is also easy to state:
> logically equivalent definitions. But the proof of
> equivalence might be nontrivial.
>
> For mathematical sets, (04)
There is no such thing as a 'mathematical' set. Sets are sets,
whatever they are sets of. Set theory is ontologically neutral. (05)
> the "simple" criterion usually
> involves a proof of equivalence: for anything other than
> small finite sets, it is necessary to prove that the two
> specifications determine exactly the same elements. (06)
That is needed in order to SHOW that two sets are identical, yes. (07)
> So there is not much difference for mathematical sets
> and types. For real world entities, both sets and types
> have to address the same kinds of messy details. (08)
If A is the set of all foodles, and B is the set of grongles, and if
every foodle is a grongle and vice versa, then A is the same set as B.
This argument works, and for the same reasons, whether foodles are a
mathematical abstraction, an kind of elementary particle, a
measurement of astronomical brightness or a way of cooking fish. LIke
I say, set theory is ontologically neutral. (09)
Pat (010)
>
> John
>
>
>
>
>
>
>
>
>
> (011)
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