From: zeleny@husc9.harvard.edu (Michael Zeleny)
Newsgroups: sci.philosophy.meta,sci.logic
Subject: Re: Is There a Set of All Truths?
Date: 16 Aug 92 17:43:36 GMT
Article-I.D.: husc3.1992Aug16.134337.14791
Followup-To: sci.philosophy.meta
Organization: The Phallogocentric Cabal

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

MA:
>There have been inquiries to me due to the missing reference
>number of my original note, so I'm re-posting it here unchanged
>as a follow-up:
>
>--The ID number of the original message was:
>
>  <1992Aug7.073316.10742@klaava.Helsinki.FI>
>
>
>                  Is There a Set of All Truths?
>                         By MARKO AMNELL
>
>Patrick Grim's  diagonalization argument  for the  incoherence of
>any notion of a 'set of all truths'1 relies crucially on the kind
>of set  abstraction  used  in  ZF.    Things  look  interestingly
>different in  NF.  Recent work also suggests that the costs of NF
>have been exaggerated.
>  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

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.  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.

MA:
>  In NF  a set x is called _Cantorian_  if it is the same size as
>the set  of all  its singleton subsets USC(x).  It turns out that
>Grim's diagonalization argument works in NF as long as the set of
>all truths is Cantorian.  This is because NF proves the following
>version of Cantor's theorem:
>
>(1)       NF  |-  (x)  P(x) > USC(x).
>
>The proof  is  by  _reductio_.    Suppose  that  there  exists  a
>surjective function f: USC(x) --> P(x) and consider  the sentence
>'y e x & y ~e f({y})'.  Translating  this into  primitive symbols
>shows it  to be  stratified so that f receives type n+4 when y is
>type  n.    A  diagonal  argument  demonstrates   that   the  set
>{y: y e x & y ~e f({y})} cannot be in the image of f.
>  The universal set V  contains all sets as members so it must be
>identical to P(V).  From this it follows by  (1)  that  V is non-
>Cantorian and bigger than USC(V).
>  One  can   modify   the  extensionality  axiom   to  allow  for
>urelements.   Ronald Jensen  showed  that  the consistency of the
>resulting system,  NFU, can be proven in elementary number theory
>and it remains consistent if we add Inf and AC.  There are models
>of NFU+Inf+AC with standard ordinals  in which almost all members
>of V are urelements.5
>  We  can  formulate  a set of all possible true statements T  in
>NFU+Inf+AC by introducing true statements as urelements  and true
>propositions as  singleton sets of urelements.   We interpret the
>primitive symbols _for urelements_ so that  'e' means 'expresses'
>and  '='  means  'expresses  the same proposition as'.   Clearly,
>there are now more urelements than equivalence classes of them:
>
>(2)	  NFU+Inf+AC  |=  T > USC(T).
>
>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.

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.

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.

MA:
>Department of Philosophy, University of Helsinki
>Unioninkatu 40 B, 00170 Helsinki, Finland
>
>
>Acknowledgement:
>
>I wish to thank  Thomas Forster, Patrick Grim,  Ilkka Niiniluoto,
>and John Post  for help and encouragement  in the writing of this
>note.
>
>
>Notes:
>
>1  [2], [3] pp.91-4.
>2  [3] p.100.
>3  [1] p.1.
>4  [1] ch.3.
>5  [4], [1] pp.66-9.
>
>
>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."