Date: Thu, 21 Mar 1996 11:10:22 -0400 (AST) Subject: Choice, inclusions, nonstandard analysis Date: Thu, 21 Mar 1996 03:33:55 -0800 From: Vaughan Pratt Contents: 1. Defense of Choice 2. Toposes with inclusion 3. Nonstandard analysis without tears. (Sorry this got so long.) From: peterj@maths.su.oz.au Vaughan Pratt seems to be challenging the rest of us to find something wrong with his ordinal-based definition of Set, and no-one else has taken up the challenge; so I'll have to break my vow and point out its obvious shortcoming. Peter's points are sufficiently well taken as to deserve both careful reflection and measured response. I apologize for the one-liner on Monday referring to Peter's Sunday message in the middle of a postscript advertising my LL'96 paper, which violated both of these. Cryptic responses only clog everything up. Peter argues that Choice should be avoided in constructions, on the following ground. I suppose only a minority of mathematicians are unhappy about AC to the extent of actually rejecting any construction that can't be done without it. But I think a very large majority would be unhappy about calling upon God to construct things for them (such as the real numbers) for which they know there is a perfectly good man-made construction. Such people are going to be in a near- permanent state of unhappiness if they are condemned to do mathematics inside Vaughan's category. =================== 1. Defense of Choice Before I address the distinction Peter draws here between natural chaos and mathematical artifact, I'd like to put in a word for Choice, which Peter isn't exactly leaping to defend. Using Choice is like wearing eyeglasses. The wearer barely sees them but sees the world more clearly. The *observer* of the wearer on the other hand sees the glasses directly as the wearer's baggage or ornament, and is only indirectly aware of the wearer's improved vision. Using Choice lets you prove more theorems, but they also shorten existing proofs, sometimes significantly. There is no shortage of examples, but just to point to the case under discussion, if you organize a category-with-epsilon-trees along the lines I was suggesting, the construction of epsilon reduces to the equation \epsilon = <. At a fraction of a line, this is significantly shorter than the published constructions, which seem to require a lot more. Long proofs cloud our mathematical vision. A shorter proof shows us the same result but by a path that makes its truth clearer. The proof as the means is a sine qua non, but that does not make it the end, the truth is the end (wearing my Platonist hat for now). The counterpart to the observer of the wearer of eyeglasses is that movie critic of the foundations world, the axiom system critic who worries about excess baggage, ornamentation, and/or legislation. Why interfere with the natural course of things when with just a little accommodation of nature by longer proofs we can leave her unfettered by legislation? Unfortunately all known ways of building a house seem to entail some disturbance of nature, and this commendable environmental concern is in impractical conflict with the requirement that your house have a picture window with a clear view of mathematical truth. Quantum mechanics and Choice are in much the same boat. Neither makes as direct contact with the truth as F=Ma or x+y=y+x, but both are better understood as sharpening vision, providing respectively the physicist and the mathematician with eyeglasses. In time they may come to feel like absolute truth, but this is a slow process. Arguing against Choice, understood internally, applies nonmonotonic logic to restrict what can be deduced. It is fundamentalist in its rejection of sophisticated reasoning supporting clear thinking. (Sorry if that's too cryptic, happy to explain privately if this worries you.) ============= 2. Toposes with inclusion Such people are going to be in a near- permanent state of unhappiness if they are condemned to do mathematics inside Vaughan's category. At first I had interpreted this to mean that mathematicians would be unhappy working with reals that had bits of the membership relation dangling off them. This seemed no objection at all; of course the user doesn't want construction materials on his nice stuff. The builder has to clean up after construction, or the user isn't obliged to pay the bill in full. But even if the builder doesn't do it, the user can always spend a bit extra to do it himself, and the upshot is the same. But a day or so later, after rereading even if [AC] does [hold], you don't have the ability to point to a particular object of it and say "This is the set of real numbers" (still less to point to a particular morphism and say "This is the addition operation on real numbers"); you have to rely on God's (or "the randomizer's", as Vaughan seems to call him) ability to do it for you. it occurred to me that perhaps Peter was interpreting my construction as meaning quite literally that I was proposing to name reals with ordinals. This certainly would make the users permanently unhappy. My interest in this stuff is ultimately as a user, and I would be unhappiest of all because such a user-unfriendly system would have been my fault, and because it would have been a complete waste of effort. I suppose I could just ask Peter which he meant (or was it something else again?). However both are plausible concerns, and I can think of equally good and more or less independent responses to them, which I'll give now. For the first, I'll define what I'll call for the moment an itopos, for topos-with-inclusion, along with a forgetful functor from itoposes to toposes. (This might be better done for categories in general; here I'm only concerned about clarifying one topos.) For the second I'll point to a particular object of my category and say "This is a monoid embedding the monoid of nonnegative reals as its bounded sequences, the rest being nonstandard reals", and point to a particular morphism and say "This is the addition operation on this monoid, whose restriction to the standard nonnegative reals is standard addition"). This will do double duty as a constructive refutation of Peter's objection and as an interesting (to me), short, and I believe new construction of the reals that bypasses *all* the many intermediate constructions in the standard constructions, *and* produces the nonstandard reals more simply than previous constructions (that I'm aware of) as a potentially useful spinoff. A topos with inclusions is no big deal (though it might be if worked out more carefully than here). Its point is to make explicit the identity of the elements of sets, in part because it is useful in its own right in some situations, in part because it clarifies what is being erased when we claim to erase construction details. The identities of a topos are those of a category, defined by a map i:O->M. An *itopos*, or topos with inclusions, extends the notion of identity to a preorder on the objects as a subcategory of the topos. An *extensional* itopos makes this a partial order. This entails an extension not so much of the signature of a topos itself as just an increase in the arity of identity, as a function from objects to morphisms, from one to ordered two, i.e. not just the diagonal but the "upper triangle" of the homfunctor when a partial order, plus squares on the diagonal when a preorder. The inclusions need to be coordinated with the topos structure, not done here (a good thing too if this is all old hat, which seems rather likely given that it is a rather obvious notion). The strict inclusions for the itopos OSet (for Ordinal Set) that I constructed the other day were already described then as the nonidentity maps from i to j that are the left inclusion of i into i+(j-i). In other words just ignore the excess of j over i and map bijectively and monotonely to what remains of j. Application. In an itopos one can define the usual Boolean operations on sets in the (large) distributive lattice of inclusions (omit the downward remainders), knowing that the intersection of two sets contains the very elements that show up in common in those two sets, not mere stand-ins for them. Cleaning up. There is an evident forgetful functor from OSet to Set, obtained by decrementing the arity of identity to leave only the diagonal identities, those in Set(X,X). The practical impact of forgetting the nondiagonal identities is that functors between toposes need preserve only the retained identities. If you find a need to preserve epsilon structure, it means you should be working in the itopos. This addresses Mike Barr's point, which he mentioned as confirming with Makkai, about most applications ignoring the epsilon structure. Erasing the off-diagonal identities makes it official. =============== 3. Nonstandard analysis without tears Definition. The nonstandard nonnegative reals form the set \omega^\omega of sequences of natural numbers, modulo the following two equational universal Horn clauses whose variables range over sequences, with addition understood as pointwise. 1. 2(a/2) = a 2. a + c = b + c -> a = b The function 2a doubles every element of sequence a: (3,2,..) |-> (6,4,...) The function a/2 shifts the sequence right, setting the leading digit, indexed by 0, to 0. (3,2,...) |-> (0,3,2,...). Think of halving a binary numeral by shifting. End of construction. It can easily be seen that the clauses will not identify an unbounded sequence with a bounded one. The standard nonnegative reals are extracted from this monoid as the (equivalence classes of) bounded sequences. The real denoted by the sequence a_i is sum(a_i / 2^i),i<\omega, in other words ordinary binary notation. Example. The ultimately constant sequences (1,0,0,0,...) and (0,1,1,1,...) denote the respective reals 1.00... and 0.111... in binary notation. These are identified by the following computation: 1.111... = 0.222... by 1 1.000... = 0.111... by 2, with c = 0.111... A little work shows that every bounded real is identified by clauses 1 and 2 with a sequence all of whose elements but the leading "digit" are 0 or 1, and which if ultimately constant ends in 0's. This of course is the standard binary representation for real fractions, but with a single digit in "radix" \omega for the integer part. It should be clear that pointwise addition is ordinary real addition for such reals, with infinite carries propagating in finite time as in the example. This construction makes these standard and nonstandard nonnegative reals a monoid. The same technique that extends the natural numbers to the integers extends these nonnegative reals to the standard and nonstandard reals, of which the standard reals then form a ring (define multiplication as all endomorphisms of the monoid, made a binary operation by associating each endomorphism with where it sends 1) and a field as usual, the standard field of reals. Every real in this monoid being bounded away from zero, the whole monoid without the nonstandard reals cannot form a field, though the above definition of the ring satisfies the ring axioms by construction. But we can easily extend it to a field by the same means by which the integers are made the field of rationals: define an ireal (possibly infinite or infinitesimal real) to be a pair a/b of nonstandard reals, modulo the implication ad = bc -> a/b = c/d. This produces with very little fuss a field suitable for nonstandard analysis without tears. These constructions can all be carried out as operations on objects of the category Set, as a topos obtained by my construction from OSet. All terms used in the construction are made explicitly available in the construction I gave, as part of its signature as a bicomplete topos. No trace of the randomness in the choice of well-ordering of \omega^\omega participates in the construction, which should be understood as merely proving existence and properties of the topos thereby constructed, with the proof erased when done. Here is an indication of how this construction could go, at least for the nonstandard reals. Start with two copies of \omega^\omega, for respectively a and b in clause 2 of the definition. Make \omega^\omega copies of \omega^\omega, one for each c. From the copy for c, delete all sequences pointwise less than c. Send maps a+c, b+c from a,b to c respectively. Now make one more copy of \omega^\omega and send maps x-c from c to it. (Note that monus never "underflows", because we deleted those elements.) The colimit of this diagram implements 2. 1 is an easy coequalizer. That's it. For the promised addition morphism, very roughly speaking, start from +:\omega^2->\omega, form +^\omega, and apply to this morphism the same operations that were applied above to the objects, all of which are functorial. Getting from here to the ireals is a matter of constructing the "integers" then the "rationals" with any standard approach, starting with \omega^\omega instead of \omega. This construction was described as an infinite diagram for conceptual simplicity. It can be made finitary with first order logic or adjunctions, however you prefer to look at it. Hopefully all operations needed for this conversion are already in the signature provided for Set; if not the signature needs more oomph. Vaughan Pratt Date: Thu, 21 Mar 1996 13:35:57 -0400 (AST) Subject: Re: Choice, inclusions, nonstandard analysis Date: Thu, 21 Mar 1996 16:53:46 +0000 From: Steve Vickers >Arguing against Choice, understood internally, applies nonmonotonic >logic to restrict what can be deduced. It is fundamentalist in its >rejection of sophisticated reasoning supporting clear thinking. (Sorry >if that's too cryptic, happy to explain privately if this worries you.) There's a sense in which by reasoning non-classically (specifically, with geometric logic) you can eliminate the need for explicit topology: if you define points by a geometric theory, then the topology is implicit if you define a map by geometric constructions, then continuity is automatic From this point of view, the purpose of explicit topology is to correct the errors introduced by classical reasoning. We thus see sophisticated reasoning (classical principles) necessitating complicated thinking (topology), the reverse of what was intended. You don't need dogma to see this could be a bad idea, though you do need hard work to see whether the classical principles really can be dispensed with. Steve Vickers. Date: Fri, 22 Mar 1996 11:28:34 -0400 (AST) Subject: Re: Choice, inclusions, nonstandard analysis Date: Fri, 22 Mar 1996 00:44:30 -0400 From: RJ Wood I would like to comment on Vaughan Pratt's > 1. Defense of Choice ... > Using Choice lets you prove more theorems, but they also shorten > existing proofs, sometimes significantly. There is no shortage of > examples,... Write CD(L) for complete distributivity of a complete lattice L. Write CCD(L) for constructive complete distributivity of L, by which is meant that \/:DL--->L has a left adjoint, where DL is the lattice of down-closed subsets of L, ordered by inclusion. It is easy to show that CD==>CCD but many times in joint work with Fawcett, Rosebrugh and Marmolejo I have been led back to the more interesting AC<==>(CD<==>CCD) * Without choice there is not much that one can prove about CD. In fact, without choice there is a severe shortage of infinite L satisfying CD(L). (Recall that AC is equivalent to `for every set X, CD(PX), where P denotes power set'.) My experience suggests that all theorems about CD follow from constructively proveable theorems about CCD after invoking *. Those that my colleagues and I have discovered have reasonably ``short'' proofs, if one starts with some basic knowledge of adjunctions. Allow me to discharge the facile comment that by *, any theorem that has been proved constructively about CCD (examples exist) proves that ``Using Choice lets you prove more theorems'' because that is not the point of this note. Rather, I think that * and similar results show that some of the definitions and concepts that seem to arise rather ``naturally'' from traditional set-based Mathematics are not particularly natural. Stepping further away from Mathematics, I think that twentieth century Mathematics has frequently sacrificed useful generalization for excessive abstraction. RJ Wood Date: Fri, 22 Mar 1996 14:33:22 -0400 (AST) Subject: Re: Choice, inclusions, nonstandard analysis Date: Fri, 22 Mar 1996 09:18:21 -0800 From: Vaughan Pratt I would like to comment on Vaughan Pratt's Phew, thanks Richard, no fun defending a position alone. Here's another argument for Choice, no more or less a proof than the clarity-of-mathematical-thought argument, it seems to me. "Proof" of AC. It took mathematics about thirty years longer to imagine AC false than it did to imagine it true. This "argument" can be applied in other situations. Applying it to Grothendieck universes (aka inaccessible cardinals) would suggest that they don't exist. We can imagine their nonexistence (their existence is independent of ZFC) but so far we haven't been able to imagine their existence (a proof in ZFC that they don't exist is still on the cards). What arguments exist in *support* of the existence of Grothendieck universes? I see the "cogito ergo sum" argument, what else? Vaughan Pratt Date: Fri, 22 Mar 1996 17:10:02 -0400 (AST) Subject: Re: Choice, inclusions, nonstandard analysis Date: Fri, 22 Mar 1996 16:01:04 -0500 (EST) From: MTHISBEL@ubvms.cc.buffalo.edu Vaughan Pratt says <> I once asked Joe Shoenfield that -- not quite in those terms -- and his answer certainly didn't seem to me "cogito ergo sum". That seems to me, assuming I understand what argument you mean, not radically better than Anselm's proof of the existence of God: we can imagine the best thing in the world, and if it is the best it must exist (otherwise one that existed would be even better), so it exists, QED. Now I don't suppose you mean a White Knight's sort of argument -- the universe imagined me, therefore it exists. You mean I imagine Grothendieck's universe, therefore IT exists. 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. It's the null set, and the singleton of the null set, and [and on. Not just countably, of course; we can describe \omega very satisfactorily, and the union of an omega-sequence of ordinals, and on. This differs from Anselm's word game in being a string of constructions. The first Grothendieck universe is rather large, so it is a fairly formidable kit of constructions. Pass to a second Grothendieck universe, and you used at least one miracle, to produce an individual from the first-universe construction. John Isbell Date: Sun, 24 Mar 1996 20:55:41 -0400 (AST) Subject: Existence of Grothendieck Universes Date: Sat, 23 Mar 1996 15:50:07 -0800 From: Vaughan Pratt 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. Date: Wed, 27 Mar 1996 14:55:46 -0400 (AST) Subject: Re: Existence of Grothendieck Universes Date: Tue, 26 Mar 96 13:44:07 -0800 From: oliver@math.ucla.edu 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. Date: Thu, 28 Mar 1996 15:41:25 -0400 (AST) Subject: Re: Existence of Grothendieck Universes Date: Wed, 27 Mar 1996 12:59:32 -0800 From: Vaughan Pratt (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 Date: Fri, 29 Mar 1996 15:59:56 -0400 (AST) Subject: Re: Existence of Grothendieck Universes Date: Thu, 28 Mar 96 20:57:48 -0800 From: oliver@math.ucla.edu >From: Vaughan Pratt >>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. Date: Sat, 30 Mar 1996 09:33:02 -0400 (AST) Subject: Re: Existence of Grothendieck Universes Date: Fri, 29 Mar 96 12:16:47 -0800 From: oliver@math.ucla.edu 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?