From: amnell@klaava.Helsinki.FI (Marko Amnell)
Newsgroups: sci.philosophy.meta,sci.logic
Subject: Re: Is There a Set of All Truths?
Date: 17 Aug 92 06:37:56 GMT
Organization: University of Helsinki

In article <1992Aug17.031215.25189@tamsun.tamu.edu> cmenzel@tamsun.tamu.edu (Christopher P Menzel         ) writes:
>In article <1992Aug16.134337.14791@husc3.harvard.edu zeleny@husc9.harvard.edu (Michael Zeleny) writes:
>>It may be that Forster's theorem allows NF to escape
>>(barely) from the full force of the accusation of _jonglerie_, made
>>originally by D.A.Martin in his 1968 review of Quine's book _Set Theory and
>>its Logic_; however it in no way vitiates his contention that NF is not to
>>be considered as a desirable alternative to ZF, at least until such time as
>>an equally solid interpretation is made on the basis of permutation models.
>
>...enthusiastic applause...
>
>>I find
>>Grim's argument against the possibility of omniscience quite unconvincing,
>>not in the least because it underestimates the possibilities offered by ZF
>>-Power + suitable restrictions on Separation.  Perhaps Chris Menzel would
>>have more to say on Grim's attempt to dismiss his suggestions.
>
>In a note in {\it Analysis} in 1985 I defended a certain set theoretic
>conception of possible worlds--and in particular the existence of a
>set of all truths--against an objection by Selmer Bringsjord that
>depended on using a set theory with full power set.  I suggested it
>would not be untoward for the world theorist to scrap the axiom in
>favor of a weaker set theory.  Grim pointed out in reply in a later
>issue (and, I believe, in his book) that a paradox could still be
>generated even on the weaker theory.  His argument depended on a
>couple of important assumptions--though ones acceptable to the world
>theorists in question: first, that for every set M there is the truth
>S(M) that M is a set; and second, that truths are sufficiently
>"fine-grained," so that, in particular, if M and M' are distinct sets,
>then S(M) \neq S(M').  Given this, one can generate the following
>Cantor-style paradox on the assumption that there is a set of all
>truths.  Suppose there is such a set T.  By separation, let A be the
>set of all truths S(M), for any M.  Now, by fine-grainedness, S has an
>inverse F on the members of A, i.e., F(S(M)) = M.  Thus, F, so to say,
>"extracts" the set that a given member of A is "about."  So by
>separation again, let A* be the set {t \in A | t \not\in F(t)}.  A*,
>of course, is a set, and so the truth S(A*) \in A.  Question: S(A*)
>\in A*?  If so, then by definition of A*, S(A*) \not\in F(S(A*)); but
>F(S(A*)) is A*, hence S(A*) \not\in A*.  If not, on the other hand,
>then S(A*) \not\in F(S(A*)), and hence S(A*) satisfies the condition
>for membership in A*, i.e., S(A*) \in A* after all, contradiction.
>
>The argument is airtight given the assumptions, which most believers
>in a robust theory of intensional entities will accept, and it does, I

See what I mean folks, ZF just won't do the job!

(Sorry about the repeated reply above, I've got a chimp with a penchant
for Shakespeare running loose here...  Actually, readers of sci.logic
will have to imagine my replies until I figure out how to follow-up
there.)

Marko Amnell
Newsgroups: sci.philosophy.meta,sci.logic