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                  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
  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  I-  (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  I=  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.


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.