Internal Set Theory Uniqueness Principle

From: David McAnally (D.McAnally_at_i'm_a_gnu.uq.net.au)
Date: 08/01/04 

Date: 1 Aug 2004 16:42:59 GMT

In Edward Nelson's original paper on Internal Set Theory, he gave a proof
that if

        (1) P(x) is a formula in Internal Set Theory with one
            free variable x,

        (2) P(x) is relativized to a standard set V (i.e. all
            quantifiers are of the form "for all x in V", "for
            some x in V", "for all standard x in V", "for some
            standard x in V",

        (3) there is a unique value of x in V for which P(x) holds,

then the unique value of x in V, such that P(x) holds, is standard.

My question is that if

        (1) P(u,v_1,...,v_k) is a formula in Internal Set Theory,

        (2) y_1, ..., y_k are standard,

        (3) there exists exactly one x such that P(x,y_1,..,y_k) holds,

then is it necessarily true that the unique value of x specified in (3)
is standard?

Nelson answered the question in the case where there is only one free
variable and the formula is relativized to a standard set V, with the
answer being in the affirmative. Nelson's proof is easily extendible
to the case where there is more than one free variable. I am curious
about the more general case (i.e. when the formula is not relativized
to a standard set).

If there is a proof in the general case that x must be standard, I would
be grateful to see it. If it is not true in general that x must be
standard, please give an example. Thanks.

David

        "But I'm always true to you, darlin', in my fashion,
         Yes, I'm always true to you, darlin', in my way."
                -- Lois Lane

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