Peter Smith's Blog, page 58

Lockdown listening: online opera etc.

I was late to notice that the terrific Bachtrack site is keeping a very well organized list of online streaming opera, dance, concerts, etc. So just in case you too had missed it, here’s a link. The Guardian also has an excellent and frequently updated guide to Lockdown Listening.

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Bachtrack’s guide to online opera etc.

I was late to notice that the terrific Bachtrack site is keeping a very well organized list of online streaming opera, dance, concerts, etc. So just in case you too had missed it, here’s a link. 

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From a small corner of Cambridge, 4

Chez Logic Matters, we are fine: the wider world, not so much. It is becoming clear that the early days of the pandemic in the UK were mismanaged, not listening to the scientists, with unconscionable delays in reaching lockdown that will have cost many lives. It could well be that this becomes the worst affected country in Europe. Then who  knows how things will go from here, except that it will surely be a very long haul. Afterwards — and what will “afterwards” mean? — the world is going to be a different place in lots of foreseeable and no doubt even more unforeseeable ways.

Meanwhile, however, another two weeks have rattled by in this small corner of Cambridge. We are no longer completely housebound, as we have just started walking out on Midsummer Common (very quiet). And we are more than comfortable at home, finding plenty to do. FaceTime and Zoom keep us connected with friends and family, who all seem safe and well. Compared with so many, we are extremely lucky, and we are very conscious of that.

And there has been much to divert us — a nice mix of the occasional streamed High Culture and fun (the Così from the Royal Opera House, still available, scored very high on both counts!). Hilary Mantel’s The Mirror and the Light arrived a couple of weeks ago, but we haven’t yet tackled that, because we both decided we wanted to re-read from the beginning of the trilogy. So Mrs Logic Matters has been diving back into Wolf Hall with great enjoyment. And I’ve just finished re-reading another weighty book — Arnold Bennett’s The Old Wives Tale. Is Bennett much regarded these days? But he writes so well — with irony yes, but insight — about ordinary lives, the way our upbringings so constrain us, about the passing of the years, the compromises we make. He creates a very human world that you find yourself swept up in.  This week’s warm recommendation, then.

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Live from lockdown: Boris Giltburg plays Beethoven



The very engaging Boris Giltburg has been playing some live lunchtime concerts from his living room. Here’s one of the sunniest of Beethoven Sonatas (Op. 22 in B-flat major).

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Luca Incurvati’s Conceptions of Set, 5

We have now reached Chapter 3, which can be conveniently divided into three parts. The first part (§§3.1–3.2) discusses two initial, and very general, challenges to the iterative conception, challenges which (it seems to me) can be fairly readily met. These are the topic of this post. The second part of the chapter (§§3.3–3.5) discusses another very general challenge, to my mind a rather more interesting one: I’ll consider that in my next post. The final part (§3.6) discusses a more specific challenge (meaning one that arises from focused technical questions, about the status of replacement, rather than from sweeping conceptual considerations). I’ll need to revise my homework on replacement! — but hope to have something sensible to say in a third post. So to begin …

The first challenge to the iterative conception we’ll discuss is what Luca calls the missing explanation objection. In brief,

[I]f we take all sets to be those in the hierarchy, we cannot explain the appeal of the naïve conception of set, as embodied in the Axioms of Comprehension and Extensionality.

This supposed objection has been pushed by Graham Priest, but it has always struck me as pretty feeble. Assume we have distinguished the concept of set (a unique object over and above its members) from the idea of a class-as-many. Now we have this intended concept of set in play, there is room to further distinguish the following two claims: (i) [Naive property comprehension] for any property, there is a set of all and only the objects with that property, and (ii) for any determinate plurality of objects, there is a set of all and only them. Now, (i) gives us e.g. a set of all sets, while (ii) doesn’t — because no determinate plurality of sets is all the sets (since given that plurality of sets there is, by (ii), another set, namely the set of them). The defender of the iterative conception, who will reject (i) but can accept a version of (ii), can reasonably say that once we’ve distinguished sets from classes-as-many, the remaining appeal of (i), such as it is, comes from confusing it with (ii). And once the distinction is made, the appeal will vanish.

That, at any rate, is the sort of story I would have given. Luca says rather more over six and a half pages. In the first past of his discussion, he presses the distinction between (i) and the claim (i*) for any property of individuals, there is a set of all and only the individuals with that property [where the individuals are the non-sets]; and he suggests that part of the appeal of (i) comes from confusing it with, or recklessly generalizing from, the harmless (i*). Perhaps there is something in that, though I’m not very sure. The second part of Luca’s discussion then gives a more careful treatment related to — though not, I think, quite the same as — the response that I sketched.

The second challenge to the iterative conception, again pressed by Priest but also encountered elsewhere is what Luca calls the circularity objection. This arises from the suggestion that iterative conception of the set-theoretic is “parasitic on a prior notion of an ordinal” and, if we are not going to go round in circles, that’s will need to be derived from a different notion of set (so the iterative conception can’t be fundamental).

But this too has always struck me as feeble (roughly: it depends on forgetting that the von Neumann ordinals are a handy implementation, not the one-and-only possible story about ordinals-as-indexers-for-tranfinite-processes). After all, we can get a long way into the theory of at least countable ordinals without talking about sets at all — we just need numbers (as individuals) and order-relations on them. If you insist on treating relations as sets of pairs which are themselves sets of sets, you still only need a few levels of sets. So: start with the numbers and a few levels. Develop a theory of countable ordinals. Use them to index more levels (lots of levels!) to get a very rich universe. In this universe we can define many more ordinals. OK, so we can now lever ourselves us by indexing more levels with these new ordinals. And so on upwards … There’s no circularity. When adding stages of the hierarchy, we already can define the ordinals we need to index the additional stages. This sort of idea was already being explained by Gödel in 1933.

Ok, that’s a bit arm-waving, but they basic idea is probably familiar. Turning to Luca’s discussion, he first gives a considerably more careful and more developed version of this Gödelian pre-emptive response to the challenge.

But he then adds a very important additional point which is worth highlighting here:

The axiomatization given by Scott (1974), of which [Scott-Potter] SP is a descendant, shows that the worry that the notion of a well-ordering is needed to grasp the iterative conception is really just an idle concern. In particular, what Scott provided is an axiomatization of set theory which, albeit sanctioned by the iterative conception, does not assume a previous conception of the hierarchy as constituted by levels ordered by the ordinals. Rather, starting from certain elementary facts about levels, which … he called ‘partial universes’, he established facts about sets and levels. Notably, what is assumed about the levels does not include that the levels are well-ordered. More specifically, he showed that, assuming the Axioms of Restriction and Accumulation, we can prove, together with the Axioms of Separation and Extensionality, that all axioms of Z except for Infinity hold, that every set is well-founded and, crucially, that the levels are well-ordered by membership. … The upshot is that the fact that the levels of the hierarchy are well-ordered is not required to grasp the iterative conception, but is a consequence of it. I conclude that we do not need a prior and different notion of set to make sense of the notion of the cumulative hierarchy, and the circularity objection fails.

That seems conclusive.

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Luca Incurvati’s Conceptions of Set, 4

On now to the very interesting second half of the second chapter, where we are still considering the iterative conception in an initial way.

So, quoting Luca,

According to the iterative conception, then, sets can be arranged in a cumulative hierarchy divided into levels. This conception sanctions (at least) most of the axioms of standard set theory and provides a convincing explanation of the paradoxes; but is it correct?

What reasons, then, can be offered in support of endorsing the iterative conception?

Luca first discusses the idea that we should take literally the metaphor of construction that comes to us so readily in describing the iterative conception. So, the idea is, sets really are formed in a stage-by-stage process, where at each stage we can only collect together in various ways what is already available.

But how do we make better-than-metaphorical sense of this idea of forming sets in a process when we are supposed to be dealing (aren’t we?) with abstract items which (i) exist independently of our activities (aren’t really formed) and (ii) in a timeless way (so there’s no real process of level-building). Arguably, the constructionist metaphor at best gives colour but no real underpinning to the iterative conception.

Suppose, however, we do try to push the metaphor harder. Then, Luca argues, [my numbering]

(i) it seems part of the constructivist doctrine that, at any point in the construction process, we can only construct sets specifiable by reference to sets already constructed. (ii) This, however, seems to sanction only a predicative version of Z’s Separation Schema …

which cuts down the strength of our set theory. Now, (i) gives us one way of elaborating what the ‘the constructionist doctrine’ might be supposed to be. Though we could, I suppose, pause to ask whether is it compulsory to construe ‘construct sets from sets that are already constructed’ as implying ‘construct sets specifiable by reference to sets already constructed’. Be that as it may, it would have been good if Luca had then paused longer over the implications of (ii), saying more about impredicative set theories. Just how weak are they? If we can live with weak impredicative set theories for ordinary mathematical purposes (as Feferman claimed, for example), then why not treat them as what is, on second thoughts, warranted by a rather strictly interpreted iterative conception? Some readers might have wanted rather more here.

However, with the iterative conception so understood, we’ll have to back off from our original thought that the iterative conception sanctions (most of) standard set theory. And Luca takes this in itself to be a reason to resist the constructivist gloss on the iterative conception.

Moving on — we’ve got to §2.4 of the book — Luca next considers the idea that we can underwrite the iterative conception, not by saying that the sets are literally ‘formed’ stage by stage, but by invoking a [now timeless] relation of metaphysical dependence between a set and its members: the hierarchy reflects this structure of metaphysical dependence.

Not surprisingly — or at least, not surprisingly to someone as sceptical about such metaphysical notions as I am — Luca has little trouble in showing that various attempts to elucidate this supposed relation of metaphysical dependence in terms of other metaphysical notions (like that of essential property) either go round in very tight circles, or pretend to explain the obscure in terms of the even more obscure. Moreover it is quite unclear, as Luca also argues, that even if we could make good a suitable notion of metaphysical dependence, that this would underpin an iterative hierarchy of the right structure (can’t there be, for a start, circles of metaphysical dependencies?). The critical discussion in §2.4 seems pretty conclusive to me.

So where does that leave us? We can’t, it seems, underwrite the iterative conception (or at least an iterative conception that will sanction something like standard set theory) by trying to cash-out a construction metaphor or a metaphor about dependence. But then recall this well-known remark from George Boolos about the iterative conception, aptly quoted by Luca:

[F]or the purpose of explaining the conception, the metaphor is thoroughly unnecessary, for we can say instead: there are the null set and the set containing just the null set, sets of all those, sets of all those, sets of all Those, … There are also sets of all THOSE. Let us now refer to these sets as ‘those’. Then there are sets of those, sets of those, … Notice that the dots ‘…’ of ellipsis, like ‘etc.,’ are a demonstrative; both mean: and so forth, i.e. in this manner forth.

Luca picks up on Boolos’s thought, and argues that we should indeed be content with what he calls a minimalist account of the iterative conception (we are supposed to hear echoes here of talk about a minimalist account of truth — I’m not entirely convinced that’s helpful, given that minimalism about truth is deflationist in spirit while Luca’s iterative conception remains very robust; but let that pass). He finds such a conception already in Gödel, quoting a remark where he talks of a concept of set

according to which a set is anything obtainable from the integers (or some other well-defined objects) by iterated application of the operation (“set of”).

And that, the suggestion goes, is where the iterative conception bottoms out, just in the idea of iterated applications of ‘set of’.

Note, in passing, that if what crucially matters is the set of operation, and (as Boolos’s words suggest) this operation takes zero, one, or more things (plural), and yields a set of them, then arguably the natural logical home for set theory would seem not to be standard first-order logic which has no place for plurals, but a plural logic which can treat operations mapping many to one. We’ll have to see if this thought is taken up later.

Anyway, Luca proposes that we take the iterative conception ‘neat’ (so to speak), without the supposed support of further thoughts about constructions or dependencies. But without those illusory further supports, why should we think it is a good conception? Well, this is what the rest of the book is going to be about … showing on the hand that other conceptions won’t give us what we want, and on the other hand that the iterative conception, minimally construed, can resist various critical attacks. So we get (in Luca’s words) an ‘inference to the best conception’. We’ll have to see how this promised inference pans out!

To be continued: the next chapter is on Challenges to the Iterative Conception.

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Live from lockdown: Pavel Haas Quartet and Ivo Kahánek play Dvorak

A live concert earlier today, from a series of charity concerts in Prague. The Pavel Haas playing the second Dvorak Piano Quintet at the Rudolfinium, with the fine Czech pianist Ivo Kahánek (while doing some social distancing!). This struck me as a particularly heartfelt performance of music which means a lot to them. It starts at 3 minutes in.

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Pavel Haas Quartet and Ivo Kahánek play Dvorak

A live concert earlier today, from a series of charity concerts in Prague. The Pavel Haas playing the second Dvorak Piano Quintet at the Rudolfinium, with the fine Czech pianist Ivo Kahánek (while doing some social distancing!). This struck me as a particularly heartfelt performance of music which means a lot to them. It starts at 3 minutes in.

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The Barber of Seville

Freely available, streaming until April 12, a really fun Barber of Seville from Paris Opera. Makes for a delightfully distracting evening in these troubled times. Warmly recommended. Enjoy!

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Luca Incurvati’s Conceptions of Set, 3

Well, I’m half-way through the task of writing up answers to the Exercises for Chapter 41 of ILF2 (and since I have the space for a few additional exercises, I’ll be trying to think up some more). But there is only so much excitement I can take! So let me return for a bit to reading Conceptions of Set. And by the way, do note that Luca has now commented on my first tranche of comments.

Chapter 2 is called ‘The Iterative Conception’, and really divides into two parts. The first part outlines this conception (and explains its relation to [some of] the axioms of set theory). The second critically considers whether the conception can be grounded (as some have supposed) in the thought that there is a fundamental relation of metaphysical dependence between collections and their members. More on this very interesting second part in my next posting. For now, let’s just think a bit about the iterative conception itself, mention some issues about the height and width of the cumulative hierarchy, and then say something about some set theories which tally with this conception.

Luca’s discussion starts like this:

On the iterative conception, sets are formed in stages. In the beginning we have some previously given objects, the individuals. At any finite stage, we form all possible collections of individuals and sets formed at earlier stages, and collect up the sets formed so far. After the finite stages, there is a stage, stage ω. The sets formed at stage ω are all possible collections of items formed at stages earlier than ω – that is, the items formed at stages 0, 1, 2, 3, etc. After stage ω, there are stages ω + 1, ω + 2, ω + 3, etc., each of which is obtained by forming all possible collections of items formed at the preceding stage and collecting up what came before. …

Of course, that’s exactly the usual story! But perhaps we should discern two thoughts here. There’s the core iterative idea that sets are built up in stages, and that after each stage there is another one where we can form new sets from individuals and/or the sets we have formed before. This captures an idea of indefinite extensibility, while rejecting the idea that at any stage we have formed all the sets (so we develop this thought, it looks as if we are going to avoid entangling ourselves with the familiar paradoxes). Then we have the further idea that we can iterate the set-building transfinitely; there are set-building stages indexed by limit ordinals, where we can collect together everything formed so far.

Luca of course stresses that the iterative conception itself leaves it open how far the cumulative hierarchy goes (what the ‘height’ of the universe is). But I think he is more concerned with how far into the transfinite we should go, while I would have liked him to pause longer here at the start, over the question of why we need to go into the transfinite at all. After all, it might be said, if we are allowing individuals, then a set universe where we have the natural numbers as individuals and then the finite levels of the hierarchy gives us a capacious setting in which arguably most mathematics can be carried out. So someone might ask: why commit ourselves to more, why go transfinite? But we’ll no doubt be coming back to issues of ‘height’

The iterative conception also leaves it open what exactly we are to make of forming ‘all possible collections of items’ from earlier stages. How ‘wide’ or ‘fat’ is each stage? ‘All possible’ certainly seems intended to be more generous than e.g. ‘all describable’; which is why we think the axiom of constructibility V = L gives us a cumulative hierarchy of rather anorexic stages, less than we intended, and why the axiom of choice can seem so natural. We are tempted to say: if all (banging the table, yes ALL!) sets are formed at each stage, then surely the needed choice sets are formed in particular. But as Luca nicely points out, following Boolos, that tempting thought is on second thoughts not so convincing, unless we build in another thought which is not itself part of the core iterative conception. The extra we seem to need is the combinatorial conception’s thought that “the existence of a set does not depend on the existence of a condition satisfied by all the members or of a rule for selecting them, [so] nothing seems to stand in the way of the choice sets being formed”. But again, we’ll need to come back to issues of ‘width’.

And what about the individuals at the ground level of the hierarchy? Do we need to consider set theories with urelements? Luca makes a familiar point:

From the mathematician’s perspective, starting with no individuals makes a lot of sense: mathematicians tend to be interested in structures up to isomorphism, and it is usually assumed that — no matter how complex or big a putative set of individuals might be — there will always be a corresponding set in the hierarchy of the same [size].

(Actually, Luca writes ‘order type’ rather than ‘size’; I’m not sure why.) So for many mathematical purposes we can do without individuals, and Luca proposes to typically focus his attention on pure set theories.

OK, so far so good: now turn to the question of what set theories the iterative conception might give its blessing to.

There are familiar worries about replacement and choice, so Luca shelves those for later consideration. And set aside extensionality as already underwritten by our very concept of set. Then Luca argues — in a familiar way — that the iterative conception sanctions the other axioms of Zermelo set theory Z. But he discusses other theories too: the stage theory ST of Shoenfield and Boolos; the theory Z+ which you get by replacing the Axiom of Foundation with an axiom which asserts that every set is the subset of some level of the hierarchy; and SP (a version of) Scott-Potter set theory. Luca argues, plausibly enough, that the iterative conception underwrites not only ST (which implies the axioms of Z leaving aside extensionality), but also Z+ and SP (those two theories in fact being equivalent).

Those claims are all persuasive. If I have a comment, then, it is about presentation rather than content. Luca’s Chapter One finishes with a couple of Appendices, two pages on cardinals and ordinals, Cantor/Frege/Russell vs the standard ZFC treatment, and one page on Cantor’s Theorem. Fine. But if a reader needs those explanations of some absolute basics, then I suspect they are going to need significantly more explanation here. For many a reader will only have encountered standard Zermelo Fraenkel set theory, and would surely have welcomed a less rushed treatment (or another chapter Appendix) elaborating on those neighbouring alternatives — especially given that some of these embody the iterative conception in a particularly direct and appealing way.

To be continued, with a discussion of Luca on grounding (or not grounding) the iterative conception in some idea of collections ‘depending’ on their members.

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