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Re: "Elementary" elements in ZFC.
Posted:
Nov 30, 2001 3:29 AM
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john_correy@excite.ca (John) writes:
> However, I am having trouble with:
>
> One standard way of introducing "atoms" into a set theory is
> by allowing the existence of a bunch of things that have no
> members, and are distinct from each other (only one of which
> is a set: {}). (<20011117095232.15000.00000541@nso-mr.aol.com)
>
> How can we use standard logic to say a bunch of x's are "distinct from
> each another"? We can write Ax~Ey(x = y) or AxAy~(x = y), but neither
> of these makes sense.
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 set & (A y)(y in x -> y is an urelement)).
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)).
Etc. You may wish to have a look at /Vicious Circles/ (Barwise and
Moss) where they use a huge number of urelements. I don't recall
their axiom off the top of my head and my copy of the book is not
handy.
--
Jesse Hughes
"Well, you know as soon as you have a new number I will be happy to
add it to the list. Don't try those childish tit-for-tat games with
me." -- Ross Finlayson on Cantor's theorem.
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