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Re: "Elementary" elements in ZFC.
Posted:
Dec 1, 2001 2:45 PM
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john_correy@excite.ca (John) writes:
> jesse@andrew.cmu.edu (Jesse F. Hughes) wrote in message
> news:<87lmgo1toi.fsf@phiwumbda.localnet>...
>
> >
> > The simplest solution is to introduce a constant for each urelement.
> > Then, for each pair of distinct constants a,b, one says ~(a=b).
> >
> > You can avoid constant proliferation, however, in a couple of ways.
> > If you want a countable set of urelements (without names), you can say
> > (E x)(x is a *countable* set & (A y)(y in x -> y is an urelement)).
> >
>
> It's still not clear (to me) how one can assert--concerning a bunch of
> things whose only salient property is that they are values of
> individual variables--that these are distinct, without also asserting
> that each such thing is distinct from itself. Shouldn't one be able
> to do this in a logic without constant singular terms?
In plain English, the axiom above says that there is a set such that
the set is countable (equinumerable with N) and every element in the
set is an urelement. I did this without adding any constants
(although, note that the axiom as written asserts that there is at
least, not exactly, aleph-null urelements).
Let S be the set whose existence is asserted in the axiom above.
There are aleph-null members of S and every member of S is an
urelement. Thus, there are (at least) aleph-null distinct urelements.
> BTW, I didn't understand your:
>
> > If you want class many, you can write
> > (A x)(x is a set & (A y)(y in x -> y is an urelement)) ->
> > (E z)(z is an urelement and ~(z in x)).
For every set of urelements, there is an urelement not in the set.
Hence, there must be a proper class of urelements.
Note that I could simplify this to the logically equivalent
(A x)(E y)(y is an urelement and ~(y in x)).
I added the junk that required x to be a "set of urelements" just for
intuition's sake. I hope it didn't serve to confuse.
--
Jesse Hughes
"You see 300 of something, anything, and you go `[Man], that's a lot of
stuff.'" -- Jim Bigler, quoted in the Pittsburgh Post-Gazette.
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