Existence of Grothendieck Universes
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.
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.
(I promised Bob I'd give it a rest (my idea, not his), but I can't resist one more shot. From: oliver@math.ucla.edu 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: Vaughan Pratt <pratt@cs.Stanford.EDU>
From: oliver@math.ucla.edu
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.
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.
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.
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.
This is an argument for assuming the existence of everything whose 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 retreat every time we disprove the existence of yet another ordinal.
-------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.
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.
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.
Anyway, what are we arguing about here? What exactly *is* the 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.
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?
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oliver@math.ucla.edu -
Vaughan Pratt