From: holmes@opal.idbsu.edu (Randall Holmes)
Newsgroups: sci.logic
Subject: Re: The Mathematical Universe (was re: Topos theory for mere mortals (?))
Keywords: inconsistent multiplicities
Date: 29 Sep 92 15:26:25 GMT
Organization: Boise State University

In article <1992Sep28.211315.15922@husc3.harvard.edu> zeleny@husc9.harvard.edu (Michael Zeleny) writes:
>In article <1992Sep28.151315.18034@dcs.qmw.ac.uk>
>arodgers@dcs.qmw.ac.uk (Angus H Rodgers) writes:
>
>>zeleny@husc9.harvard.edu (Michael Zeleny) writes:
>
>MZ:
>>>[...]  However, I wish to go on record as stating that,
>>>though it probably is a sociological fact that most realists who bother
>>>thinking about it, think that there is exactly one universe of all sets,
>>>and that, conversely, some people have gone on record announcing their
>>>conversion to formalism on the grounds of their loss of belief in the
>>>unique existence of such a universe, I do not myself believe that a realist
>>>philosophy of mathematics cannot accomodate a multiplicity of set-theoretic
>>>(or type-theoretic) universes possessed of different size and structural
>>>properties.

But one and the same universe can be described in various different
ways; for instance, a type-theoretical universe can also be described
as a set-theoretical universe for bounded Zermelo set theory.  It is
certainly true, however, that some of the foundational theories might
have the problem that, while theory A and theory B, for example, might
interpret each other, it is not possible for the "universe" of theory
A to be identical with a "universe" for theory B; a B-universe will
contain an A-universe and vice versa, but they cannot be conterminous.
In this case, some decision has to be made on the "platonic" level,
but it remains clear that A and B are equally sensible foundational
theories.  With regard to size, remember that all models _might_ be
externally countable...
 
>
>AHR:
>>Even if we modestly bow to the apparent inevitability of having to
>>face the existence of a plurality of "universes" of "sets", then,
>>unless we give up the use of our rational faculties altogether,
>>we are forced (aren't we?) to develop a "theory of universes", with
>>pretensions to absolute truth.

There is no reason to believe that there is a plurality of universes;
there is one universe, all or part of which can be _interpreted_ as
the universe of any given foundational theory.  The individual
theories play a role vaguely analogous to that of "frames of
reference" in relativity; the status of the absolute truth is not
threatened.


>
>No, we are not forced to do anything of the sort.  Though there may be but
>a single mathematical universe, it does not follow that there ought to
>exist a theory of the same.  As said Spinoza (whose ideas on the infinite
>are known to have inspired Cantor), _definitio est negatio_; and if the
>universe is conceived as encompassing everything, there would seem to be no
>way of meaningfully defining it.

No, it would seem that the definition is trivially easy.  But a
_construction_ is probably quite impossible.  The universe contains
all the things that are (definition); don't ask me to generate a list
of all things in the universe, however.  I agree that there is not a
"theory" of the universe.

>
>AHR:
>>What sort of thing do people mean by the phrase "a universe"? How do
>>we reason about such things? Are they mathematical objects?
>
>Probably not; it is not even clear that they are objects of any sort.
>
>AHR:
>>Perhaps the common noun "universe" here can be replaced by "topos", and
>>perhaps it can't; either way,I'm not technically qualified to say.

A topos is not a universe.  Type theories do not have universes.

>>Be that as it may, my (non-technical) question is whether there is any 
>>way of understanding this word (in the sort of context in which you have
>>just used it) as referring to anything other than a structured set. If
>>there is, my whole argument collapses. (Of course, merely replacing the
>>word "set" here by "class" or "collection" makes no semantic difference.)
>
>There is also a way of interpreting the word in question as _a priori_
>non-referring.
>
>MZ:
>>>There is, after all, no good reason to suppose that everything
>>>in the mathematical universe is a set [...]

Of course, in the _true_ theory [ :-) ] we can _prove_ that not
everything is a set.  (NFU with Choice).

>
>AHR:
>>Naively, I just can't see how the human mind could possibly manage
>>without some such concept of "*the* universe" as you have just
>>appealed to (without irony, I take it).
>
>Note that the same things have been claimed on behalf of the concept of a
>personal God.

I think that most genuinely philosophical arguments for God do very
little to support the notion of a _personal_ God.

  As for the question of my own usage, I use the term in
>question in much the same way as I would talk about a spiffy haircut,
>without thereby committing myself to an ontology of haircuts.  It is, after
>all, an elementary Russellian point that not every singular term of a
>natural language calls for a literal interpretation based on its surface
>grammatical structure.  And in this case I would be very happy to admit the
>meaninglessness of my above remark, by accepting a reasoned argument (some
>are reported in Michael Hallett's _Cantorian Set Theory and Limitation of
>Size_) to the effect that the term `the mathematical universe' is bereft of
>any denotation, on the grounds that the putative totality which it is meant
>to denote is far too large to be a consistent multiplicity in the Cantorian
>sense.

Mikhail, you know perfectly well that "limitation of size" is not the
only way of dealing with the paradoxes, or you wouldn't talk about
type theory the way you do.  Certainly you know by now that NFU is
consistent, so the notion of the universe is not incoherent.  But I
will support you in the assertion that the universe as a totality is
not necessary, either; Zermelo set theory is perfectly coherent [just
wrong :-) ]


>
>>--
>>Angus H. Rodgers  (PhD student), | arodgers@dcs.qmw.ac.uk  
>>Dept. of Computer Science, Queen | [ +44 | 0 ] 71 975 5241   
>>Mary & Westfield College,  Mile  | This space has not been left blank,
>>End Rd., London E1 4NS, England  | intentionally or otherwise.
>
>cordially,
>mikhail zeleny@husc.harvard.edu
>"Un de mes plus grands plaisirs est de jurer Dieu quand je bande."
>
>P.S. The Phallogocentric Cabal will hold its first meeting for the
>1992-3 school year at 7 p.m. on Thursday, October 1 in the Emerson 
>Hall of Harvard University.  Free memberships are still available.


-- 
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
Xref: sparky sci.logic:1670 comp.theory:2147