From: cmenzel@tamsun.tamu.edu (Christopher P Menzel         )
Subject: Re: Is There a Set of All Truths?
Organization: Texas A&M University, College Station
Date: Mon, 17 Aug 1992 03:12:15 GMT

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
think, raise serious difficulties for accounts of modality like Robert
Adams' that seem to accept the assumptions in question and also define
possible worlds to be sets of propositions.  As to whether the
argument creates a logical problem for the idea of omniscience, it's
hard to see what there is in the idea that requires that there be a
*set* of all truths for an omniscient being to know.  Consider a
parallel with quantification.  We seem to have no trouble quantifying
over *all* the sets, even though there is no set of them (according to
our best set theories).  Similarly, it seems quite enough just to say
that an omniscient being knows *all* the truths, even though there's
no set of them either.

--Chris Menzel
  Philosophy Department
  Texas A&M University
Newsgroups: sci.philosophy.meta