From: amnell@klaava.Helsinki.FI (Marko Amnell)
Newsgroups: sci.philosophy.meta
Subject: Re: Is There a Set of All Truths?
Date: 18 Aug 92 13:59:45 GMT
Organization: University of Helsinki

In article <1992Aug17.203857.14834@husc3.harvard.edu> 
zeleny@husc9.harvard.edu (Michael Zeleny) writes:

>In article <1992Aug17.053808.940@klaava.Helsinki.FI> 
>amnell@klaava.Helsinki.FI (Marko Amnell) writes:
>
>>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:
>
>>[...]
>
>A:
>>>>  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.
>
>MA:
>>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:
>In what sense does this classification result hold?
>

I've read through this stuff (Forster,pp.102ff) a couple of times and
don't feel qualified at present to treat it here.  I'll try to remember
to do so later in this newsgroup.  I guess the basic idea is that proofs
of theorems in NF are preserved by permutations of V that are sets, so
that they are valid under all permutations (ie. necessary).

>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.
>
>MA:
>>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)
>
MZ:
>I find the suggestions unconvincing, and the book itself, technically
>accomplished but philosophically naive.  (yes, I own a copy)

What exactly do you find naive about Forster's views?

>>[...]
>
>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:
>Chris Menzel has adequately addressed this claim in the concluding remarks
>of his article on this thread.

He has presented one more argument in ZF -- this hardly settles the
issue.  As to his comments about omniscience I find them absurd.  If
God knows all truths then surely (if the word 'all' is to have any 
proper meaning) there must also _be_ a set of all truths.

MZ:
  Why worry about this any more than you
>worry about the existence of an alethic totality of all sets (in ZF, that
>is)? 

I do worry -- about both.  The existence of a set of all true
propositions is linked to many important philosophical problems.
The articulation of a successful Platonist philosophy of mathematics
is probably not possible with just a set of all true statements since
if Fermat's Last Theorem, eg., is true the proof of this could be
inexpressible.

MZ:
 Now for a counterplug -- look at Hallett's book on limitation of size
>in the same Oxford series.

That book doesn't plug any holes in a ZF account of alethic totalities.

>
>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.
>
>MA:
>>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:
>There are, of course, numerous reasons for dropping full strength Power Set
>axiom, some of which are implicit in the problems associated with the
>notion of an arbitrary subset of an infinite set.  Personally, I find such
>considerations somewhat beside the point, but it's a far cry from "dropping
>an axiom by fiat".

OK.  But the point still is that Zermelo and Fraenkel had good reasons
for including the Power Set Axiom in their systems, while in NF it is
not needed at all.  It's a bit like the observation that while Russell
had to arbitrarily assert the Infinity Axiom in _Principia_, Specker
showed that in NF it can be proven from Comprehension and
Extensionality.

>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.
>
>MA:
>>'Point well taken, although I'm sure you're aware that theologians have
>>made a living for centuries by finding refutations to this argument.
>
MZ:
>Tant pis pour eux.  (yes, I also own a copy of the Adams Family anthology)

You got the reference for that?

>MA:
>>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).
>
MZ:
>What about God's responsibility to the injured party, as a consequence of
>his benevolence?  Also, this fails to account for events like the Lisbon
>earthquake, which originally triggered many refutations of Leibnizian
>optimism.

I guess the theist would say that this is the price we pay for real
freedom with real challenges and adversity (instead of a reified,
deterministic universe).  If pressed he might just say that the problem
of evil is a mystery.  Personally, I come down on the side of Voltaire.

>MA:
>>						  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.
>
MZ:
>Poor Benny... why drag him into a discussion of this sort?

He needs the exercise.  By 'Spinozan' I meant that the omniscient being
in the future might be identical to the universe.  One might even
speculate that its last act would be to crunch the universe and produce
the next Big Bang, thus fulfilling its cosmic _causa sui_ project.

>>[...]
>
>MA
>>>>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."
>
>MA:
>>Oui, mais c'est Dieu qui va rire finalement,
>
MZ:
>Cependant Dieu me pardonnera, car c'est son m\'etier.  Sinon -- "qu'importe
>l'\'eternit\'e de la damnation \`a qui a trouv\'e dans une seconde l'infini
>de la jouissance?"

I doubt you mean our conversation...  Actually, this reminds me of a
passage in the _Dialogo_ of Tullio Regge and Primo Levi where one of them
(I guess Regge) says that entire civilizations might have existed in a
fraction of a second immediately after the Big Bang.

>
>cordially,
>mikhail zeleny@husc.harvard.edu
>"Nous donnons la mort, nous saurons la subir."

Pourquoi?  Vas tu visiter Sarajevo \`apr\`es Paris?

Marko Amnell