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.

2 thoughts on “Luca Incurvati’s Conceptions of Set, 3”

  1. 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 … 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?

    I think it might usefully be approached in 3 steps:

    1. Why did we (set theorists, other mathematicians) go transfinite initially?
    2. What did we do / discover while we were there?
    3. Looking at all of that, do we now want to stay, or to leave?

    One reason to stay is that otherwise set theory is boring. It’s set theory for people who aren’t interested in set theory, who — like the readers Halmos expected for his Naive Set Theory — are “anxious to study groups, or integrals, or manifolds” instead. Then

    The student’s task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort. In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it.

    Compare that to Judith Roitman in the Preface to her Introduction to Modern Set Theory:

    When, in early adolescence, I first saw the proof that the real numbers were uncountable, I was hooked. I didn’t quite know on what, but I treasured that proof, would run it over in my mind, and was amazed that the rest of the world didn’t share my enthusiasm. Much later, learning that set theorists could actually prove some basic mathematical questions to be unanswerable, and that large infinite numbers could effect the structure of the reals — the number line familiar to all of us from the early grades — I was even more astonished that the world did not beat a path to the set theorist’s door.

    I know which sounds more interesting and fun to me, and it’s not Halmos.

    And it we confine ourselves to boring set theory, then we can no longer explore the transfinite and continue to find interesting things there.

    Incurvati doesn’t put it that way, but does say something on p 66 that fits with the idea that we should take what we’ve already done in the transfinite into account:

    The second goal set theory has often had is that of providing a tool for understanding the infinite, and this goal too seems to require that our set theory have certain resources such as those needed to develop interesting and fruitful theories of cardinals and ordinals.

    1. Thanks! I’m sympathetic to that. But it does seem to go along nicely with my suggestion that a purely iterative conception may only take us so far, and it is some further thoughts (e.g. as in the quote from Luca) which gets us to take the levels of the hierarchy as reaching transfinitely high.

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