From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Is There a Set of All Truths?
Organization: Boise State University Math Dept.
Date: Tue, 18 Aug 1992 19:57:23 GMT

In article <1992Aug18.135945.7450@klaava.Helsinki.FI> amnell@klaava.Helsinki.FI (Marko Amnell) writes:
>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.

The costs of _NF_ are indeed enormous; not so those of extensions of
NFU.


"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.  ...

I can get the herediatrily strongly Cantorian sets in a model of NFU
to make up a model of ZFC.

>>>>>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

Indeed it can.

>>
>>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).

The iterative hierarchy is definable in NF or NFU as the set of types
of well-founded extensional relations.  Assuming enough axioms of
infinity will give you as much of the iterative hierarchy as desired;
I can describe an extension of NFU in which the strongly Cantorian
types in this set (which do not themselves make up a set) satisfy the
axioms of ZFC.  In NFU, all of this is exactly as justified as the
corresponding extensions of ZFC.  Extensions of NFU could replace ZFC;
it is a matter of taste.


>>>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?

Relative to my remark below, the system of S5 logic is in relation to
permutations which are sets only.

>>
>
>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).

The class of permutations used in this result is broader than the
class of set permutations; there are unstratified propositions (the
axiom of counting, for example) which are invariant under all set
permutations but not under all "set-like" permutations.

>
>>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)

Mikhail, I don't think the book has philosophical intentions.  It is
also avowedly incomplete, and overly preoccupied with NF itself.

>
>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.

God is also aware of all sets which are not members of themselves,
individually, but they do not thereby make up a set (or if they must,
this is a much better disproof of omniscience...)


>
>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.

Again, it is not.  There is a big difference between a predicate and a
set; the "philosophical problems" are probably adequately handled by a
truth predicate (rel what language?).

>
>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.

But in NFU, the "natural" version of Quine's theory, it cannot be
proven.  NF is not a natural set theory (I'll discuss this in more
detail in upcoming papers).  The price of the provability of the axiom
of infinity in NF is terrible.

>
>>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

I think what follows has relatively little to do with Spinoza's views.

 (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.

The God of Spinoza is _eternally_ 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


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
Newsgroups: sci.philosophy.meta