From: amnell@klaava.Helsinki.FI (Marko Amnell)
Newsgroups: sci.philosophy.meta
Subject: Re: Is There a Set of All Truths?
Date: 17 Aug 92 05:38:08 GMT
Organization: University of Helsinki


In article <1992Aug16.134337.14791@husc3.harvard.edu>
zeleny@husc9.harvard.edu (Michael Zeleny) writes:

>In article <1992Aug14.115223.16515@klaava.Helsinki.FI>
>amnell@klaava.Helsinki.FI (Marko Amnell) writes:

[...]

MA:
>>  Grim holds  that "any  adoption of  an NF-like theory to save a
>>'set of  all truths' proves to be a very desperate move.  For the
>>costs of  NF turn  out to  be enormous."2  But, as Thomas Forster
>>notes, having  a universal set  "does not exclude the possibility
>>that ... the small sets in such a system form a model of ZF.  ...
>>Thus the extra generality of V e V need not cost anything."3  The
>>choice  between  alternative systems  for  various purposes boils
                                        ^^^^^^^^^^^^^^^^^^^^^
>>down to  the problem  of how  the axioms  are  justified.    NF's
>>stratification condition  for set  abstraction can be defended to
>>be more than just a syntactic trick.4
>
MZ:
>I'll go on record saying that Forster's book referenced below is the best
>treatment of set theories with a universal set known to me, though this may
>be a consequence of my not having read Randall Holmes' thesis.  Having said
>that, I must protest against his interpretation of the theorem stating that
>a sentence is equivalent to a stratified sentence iff it is preserved under
>all permutation models with a setlike permutation.  This is a neat
>classification result, but no more than that, whereas the competing,
>ZF-like conception of sets is founded on a solid conception of the
>iterative hierarchy.

For myself, I've never suggested that NF ought to replace ZF as the foundation
of mathematics (for which purpose the iterative hierarchy is essential).
I'm investigating to what extent NF might be helpful in this context as an
alternative logic for dealing with what I've called alethic totalities
(including possible worlds considered as maximal consistent sets of
propositions -- the permutation models, incidentally, produce a system of
S5 modal logic).

MZ:  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.

Forster also suggests other philosophical uses for set theories with
a universal set, such as 'sets as natural kinds'.  See his book for
details. (yes, that was a plug)

[...]

MA:
>>What about  the true propositions that are left out of USC(T)?  I
>>suggest that we interpret these as _noumenal truths_ -- ineffable
>>but  true propositions,  such as  facts about God.   The question
>>of whether there actually is  an alethic totality  made up of all
>>true _propositions_ is, of course, left unanswered.
>

MZ:
>I have nothing to say on the subject of "noumenal truths", and include this
>part of the article only for the new readers' benefit.  However 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.

I wish he would too, although I'm sure readers would agree that dropping
an axiom by fiat is liable to teach us less about what a set of all
truths would mean than a logic tailored to include such totalities.

MZ:
>Having said that, I find that the real argument against divine omniscience
>consists in its incompatibility with omnipotence and benevolence.  Or the
>other way around.

'Point well taken, although I'm sure you're aware that theologians have
made a living for centuries by finding refutations to this argument.
The best known is probably that the possibility of evil is a necessary
condition for the existence of human freedom, so that it is the individual
who committs an evil act (and _actualizes_ the possibility of evil) who
assumes the responsibility for it (and not God).  I won't appeal to such
defences, however, since I have no desire to defend classical theism.
My interest in God is more Spinozan (although Bento will spin in his
grave to hear what follows).  Like Barrow and Tipler in _The Anthropic
Cosmological Principle_ I want to speculate about the existence of a 
being with (near) total knowledge at some indeterminate point in the
evolution of the physical universe.  Given the way the world is (just
look to Bosnia to temper any cosmic optimism) I see no reason why such
a being would be all-good.  And given the way physics is, it wouldn't be
all-powerful either.

[...]

>>References:
>>
>>[1] Forster, T.E. (1992)   _Set  Theory  with  a  Universal  Set:
>>    Exploring an Untyped Universe_. Oxford University Press.
>>
>>[2] Grim, P. (1984)  "There is no Set of All Truths".
>>    _Analysis_ 44 pp.206-8.
>>
>>[3] Grim, P. (1991) _The Incomplete Universe: Totality, Knowledge
>>    and Truth_. The MIT Press.
>>
>>[4] Jensen,  R.B. (1969)  "On the  Consistency of  a  Slight  (?)
>>    Modification  of  Quine's  New  Foundations".   _Synthese_ 19
>>    pp.250-63.
>
>cordially,
>mikhail zeleny@husc.harvard.edu
>"Un de mes plus grands plaisirs est de jurer Dieu quand je bande."

Oui, mais c'est Dieu qui vas rire finalement,

Marko Amnell