Subject: Existence of Grothendieck Universes

From: categories <[email-masked]>
Date: Sun, 24 Mar 1996 20:55:07 -0400 (AST)

Date: Sat, 23 Mar 1996 15:50:07 -0800
From: Vaughan Pratt <[email-masked]>

	I once asked Joe Shoenfield that [...]
	Well, Joe said in effect I can tell you everything that goes to
	make up the first Grothendieck universe, except I don't have
	time to finish telling you.

Well, it's no worse than any of my existence "proofs" I suppose.  I
can't shoot it down with the argument that it would equally well prove
the existence of cardinals we already know not to exist, since the
extra time Joe would need to tell us everything that goes into making
up a nonexistent ordinal might disqualify that proof without
disqualifying the other.  :)

Further reflection on this merely seems to lead back to my original
point about there not being enough room in this town for both \in and
U.  If your U is an inaccessible cardinal, and inaccessible cardinals
*do* exist (and some set theorists seem to hope very badly they do,
despite knowing it cannot be proven in ZFC), then rejecting membership
is one reasonable way out.  Becoming an intuitionist is another.  Have
the reasonable alternatives proposed to date been collected in one
place and insightfully classified and compared?

The flip side of this is, if inaccessible cardinals are eventually
proved *not* to exist, then the Cantor-Russell paradox goes away for
those working in a Grothendieck universe.  But in that case the one
reasonable objection I've been able to grasp so far to membership,
namely its incompatibility with existence of one's mathematical home,
goes away too.

Is there any other argument against membership?  Preferably just as
short, but I'll settle for long if that's all that's possible.

Or is the rejection of membership all just touchy-feely stuff based on a
deep-seated feeling for something?

"The von Neumann rigid-epsilon monsters" alluded to by Bill Lawvere
sounds like it might be made into such an argument.  But as I pointed
out, when the only sets are the ordinals, one of each, the "monster" is
tamed to a gentle line +1'ing steadily along, with the occasional
little hiccup at each limit ordinal.  That's no monster.

A more convincing objection is needed.  There must be something else,
ideally something we are all very familiar with.

The smoothness of space, for example.

First off, even if physical space *is* smooth, why should this have any
more bearing on our mathematical spaces than their dimension?
Obviously no one can get away in these enlightened days with
legislating 3, 4, or 26 as an upper bound in mathematics.

Second, it is not even clear that the space we inhabit *is* smooth.
Start with

	http://zebu.uoregon.edu/~imamura/209/may8/may8.html

and look at what Vilenkin and Linde are proposing as an alternative
model of space: foam at fine grain.  Their idea that small distances in
space are chaotic makes much more sense to me than a model based on
extrapolating the smoothness of space-in-the-large down to arbitrarily
fine scales as though space were a mathematical manifold.
That extrapolation applied to the smoothness in the law of large
numbers in statistics for example is well-known to lead to nonsensical
results.

With the statistical analogy in mind, my only question about the
Vilenkin-Linde model is whether the appearance of chaos at fine scales
would be nicely explained by populating unit volumes with only finitely
many points.  It is very easy to construct smooth-in-the-large
manifolds from finite sets, and these will naturally look bumpy in the
small.

We would then have to add another physical constant to the books, the
number of points of space per cubic centimeter.

--
Vaughan Pratt

===================================

PS. I inquired on sci.math.research about my construction of the
nonstandard reals.  Vladimir Pestov at U. Wellington pointed out an
embarrassingly trivial oversight: I'd forgotten that the class of
fields subdivides into smaller elementary classes, and had not thought
to order my field (the standard part leaves no alternative, it can't be
an algebraically closed field).  You can't, at least not without
further restrictions on the sequences.

Luckily the standard part of my field *can* be ordered or I'd have to
come up with some other demonstration to Peter Johnstone that one can
identify the set of reals and its addition morphism without having to
say which ordinals go where.

More generally, any sequence can be named by its elements.  There is no
need to name it by whichever ordinal God picked this time around as the
placeholder in the set of sequences of which that sequence is a
member.

I can't imagine what Peter was thinking.  This sort of construction,
where you have to take down the scaffolding when you're done, goes on
all the time in ordinary mathematics.

---

From: categories <[email-masked]>
Date: Wed, 27 Mar 1996 14:55:46 -0400 (AST)

Date: Tue, 26 Mar 96 13:44:07 -0800
From: [email-masked]

Vaughan Pratt writes:

	If your U is an inaccessible cardinal, and inaccessible cardinals
	*do* exist (and some set theorists seem to hope very badly they do,
	despite knowing it cannot be proven in ZFC), ...

I think the word "despite" here misses the whole point.  It is precisely
*because* the existence of inaccessibles cannot be proven in ZFC, that
the assumption of their existence is interesting.  More exactly, this
is how we know that ZFC+inaccessibles is a proper (if modest) extension
of ZFC itself.

Now of course this is not to say that whenever ZFC fails to prove the
existence of some X that we should immediately assume "X exists"; among
other problems, that would quickly lead to an inconsistent theory.

Inaccessibles, however, have other merits:

	i)	If my model says there are inaccessibles, and
	yours says there aren't, it may just mean that your model
	is an initial segment of mine.  In that case my model
	is clearly better, because it has everything in it that
	yours does, and more besides; moreover, we haven't lost
	any nice features that your model might happen to have, because
	they are still true when prefaced by "in your model..." .

	ii)	Inaccessibles are an example of the intuition
	that says "anything we know how to do too well or too
	precisely, can't possibly tell the whole story; there
	must be more."  In this case what we know too well how to
	do is take exponentials of cardinals and limits of sequences
	of ordinals, where the length of the sequence is a cardinal
	we already have.  Strengthen this intuition to "things
	we can construct in L" and you start to see why 0# should
	exist.

---

From: categories <[email-masked]>
Date: Thu, 28 Mar 1996 15:41:25 -0400 (AST)

Date: Wed, 27 Mar 1996 12:59:32 -0800
From: Vaughan Pratt <[email-masked]>

(I promised Bob I'd give it a rest (my idea, not his), but I can't
resist one more shot.

	i)	If my model says there are inaccessibles,... my model
	is clearly better, because it has everything in it that
	yours does, and more besides; moreover, we haven't lost
	any nice features that your model might happen to have,

Consistency is not a nice feature?  News to me.  My model is more
likely to be consistent than yours.  Inconsistency of yours would be a
beautiful result to most people, like a beautiful building or park.
Inconsistency of mine would be a magnitude-9 earthquake!

Furthermore mathematics has had no trouble imagining my model since
1939 when Goedel showed us how.  It is *very* hard to imagine your
model, so hard that not even the smartest people in the world have been
able to do it in over half a century of trying.  This is *not* a good
sign.

	Strengthen this intuition to "things
	we can construct in L" and you start to see why 0# should
	exist.

This is an argument for assuming the existence of everything whose
nonexistence you can't actually *prove*.  But that forces you to
retreat every time we disprove the existence of yet another ordinal.
Such discoveries happen periodically, and do not astonish working
mathematicians.

A much more stable approach would be to accept the easily imagined, and
reject what is hard to imagine.  Following that strategy, you only have
to retreat in the face of 50-year or even 200-year earthquakes.

Anyway, what are we arguing about here?  What exactly *is* the
advantage of assuming inaccessibles?  (Let's not tempt fate and get too
close to inconsistency by assuming measurables!)  Sure you can shorten
some contrived proofs a lot with inaccessibles, but can you shorten a
proof some mathematician might care about?  Or do anything else useful
with them?

Vaughan Pratt

---

From: categories <[email-masked]>
Date: Fri, 29 Mar 1996 15:59:56 -0400 (AST)

Date: Thu, 28 Mar 96 20:57:48 -0800
From: [email-masked]

>likely to be consistent than yours.

Careful with language.  Models are not consistent or inconsistent;
they simply exist or fail to exist.  Consistency is a property of
*theories*.

If my model exists, then it is better than yours.

>able to do it in over half a century of trying.

I do not know what you might mean by this.  I have no difficulty
whatsoever in imagining inaccessible cardinals.

If you mean I can't imagine everything that might happen *below* an
inaccessible cardinal I plead guilty, and challenge you to imagine
all ordinals below Aleph_1.

>nonexistence you can't actually *prove*.

No, not everything:  For example I don't assume the existence of a proof
of 0=1 from ZFC, even though I can't prove the nonexistence of such
a proof.  In my first article I tried to give an idea of why I assume
one and not the other.

>But that forces you to

-------IF YOU READ NOTHING ELSE IN THE ARTICLE, READ THIS PARAGRAPH---------
Science is a continual process of assuming things that might be proved
wrong, taking the chance that you may later be forced to retreat.  Read
"Conjectures and Refutations" by Karl Popper.

>to retreat in the face of 50-year or even 200-year earthquakes.

Truly, I think you are much overestimating the difference between
ZFC and ZFC+inaccessibles.  To prove a contradiction from ZFC+inaccessibles,
or even ZFC+measurables, would be earthquake enough for me; but even
a contradiction from ZFC would have little effect on most everyday
mathematics as long as it didn't go through in, say, 2nd-order PA.

>advantage of assuming inaccessibles?

Well for example, if inaccessibles exist then we *know* choice is
necessary to prove the existence of a nonmeasurable set of reals.

If measurables exist then every analytic set of reals (i.e. projection
of a Borel set in the plane) is measurable.

---

From: categories <[email-masked]>
Date: Sat, 30 Mar 1996 09:33:02 -0400 (AST)

Date: Fri, 29 Mar 96 12:16:47 -0800
From: [email-masked]

I wrote:

	If measurables exist then every analytic set of reals
	(i.e. projection of a Borel set in the plane) is measurable.

Careless of me.  ZFC is enough on its own to prove that every analytic
set is measurable, either countable or contains a perfect subset, and
has the property of Baire.

Measurables get you the same results one real quantifier higher; i.e.
for projections of co-analytic sets.

For future reference, if I notice an error after making a submission
on lists like this, can I write the moderator and ask to make a correction?