Groups and sets

Standard algebra texts define a group to be a set of objects together with a binary operation on those objects (obeying familiar conditions) — or indeed, they define a group to be an ordered pair of a set and a suitable binary function (where ordered pairs and functions are to be treated as sets too, in familiar ways).

But what work is the conventional set talk here actually doing? — is it really doing much more than the set talk back in the days of ‘new math’? Can these sets be treated as Quinean virtual classes? Can we in fact perfectly well think of a group as some objects (plural — or at least one) equipped with a group operation, and drop the set talk in favour of common-or-garden plural locutions?

There are a number of general philosophical issues here about the commitments of various grades of set talk, and about the commitments of plural talk too. But there’s an interesting technical question: how far can we get in group theory before we have to call on substantial ideas from set theory?

Let’s sharpen that question: what issues are there that can readily be understood by a relative beginner in group theory that depend for their answers on set-theoretic matters (e.g. whose answers can depending on the ambient set theoretical principles we countenance?). It is well known, for example, that Shelah showed in 1974 that the Whitehead problem is independent of ZFC. But the beginning student is surely not going to understand the significance of that (indeed picking up a couple of heavyweight standard texts, Dummit and Foote’s Abstract Algebra and Aluffi’s Algebra: Chapter 0, neither even mention the Whitehead problem). So put aside problems at that level. To repeat, then, what student-accessible problems have answers dependent on set theoretic ideas?

One suggestion might be

“Given objects X, is there always a group (X, e, *) with those objects, with one of them serving as the identity e, and some group operation *?”

The answer is positive if and only if the Axiom of Choice is true. Though I suppose here we might now have a serious debate about whether the Axiom of Choice is at bottom an essentially set-theoretic principle at all (after all, some plural logics have versions of choice, and then there are type-theoretic versions).

Here’s a perhaps better example. The following

“Take a group G, its automorphism group Aut(G), the automorphism group of that, i.e. Aut(Aut(G)), the automorphism group of that, i.e. Aut(Aut(Aut(G))), etc. Does this automorphism tower terminate (count it as terminating when successive groups are isomorphic)?”

Joel Hamkins has shown that the answer is yes, but the very same group can lead to towers with wildly different heights in different set theoretic universes. (Why should this be, on reflection, no great surprise? — which is not to diminish Hamkins theorem one jot! Because, as he points out, there’s a sense in which going from a group to its automorphism group is ‘going up a level’; and we can play forcing tricks as we go up the levels.)

OK that’s a lovely example. But what others are there? I asked this on math.stackexchange here and the question got a surprising number of up-votes (suggesting I’m not alone in  finding the issue interesting).

And I didn’t get any more problems which are as immediately student-accessible as the automorphism towers problem, rather re-inforcing my guess that you can get a pretty substantial way into group theory without really tangling with set theory (especially if choice principles are assigned to the logic side of the ledger rather than the intrinsically set-theoretic side). But I did get two nice suggestions a notch or two up in sophistication, which you might be interested to see (if you know some algebra).

5 thoughts on “Groups and sets”

  1. I am answering here not from a philosophical or technical point of view, but rather a pedagogical/cognitive one:

    To me, it meant a great deal to know that all math can be in principle (even if not necessarily in practice) formalized in terms of sets (not only in the sense that a group is a pair of a set and an operation, but also in the sense that an operation is a triple of sets and pairs and triples are also sets) – it meant that math is ontologically simple even if it is not cognitively simple. I knew almost nothing about higher math before I started university, but I think I imagined a much more baroque ontology, with a lot of kinds of objects that could not be put on a single foundation. I would not take that moment of realization away from potential future mathematicians.

    1. But is the fact that we model various areas of mathematics inside set theory ontologically significant?

      Someone might say: Take the simplest of examples: we can model ordered pairs using the familar Kuratowski implementation of the pair (a, b) by the set {{a}, {a, b}}. But note the implementation is not ordered, not always a pair — for (a, a) comes out as {{a}} — and in any case arbitrary, e.g. consider the alternatives {{b}, {a, b}}, or Hausdorff’s {{a, 1},{b, 2}} and so on. We can’t simply say that the Kuratowski implementation tells us, ontologically, what ordered pairs really are! Similarly in other cases.

      It is a good question then, exactly what the role of set theory is as a “foundation”, and in paticular what its ontological significance is. There’s a nice recent discussion by Penelope Maddy which I have a lot of sympathy with: http://www.socsci.uci.edu/%7Epjmaddy/bio/What%20do%20we%20want%20-%20final .

      1. How can the ability to model mathematics inside set theory not be ontologically significant? Suppose instead that it wasn’t possible — that mathematics couldn’t all be modelled in set theory, or even that maths was divided into parts that couldn’t be modelled in any common framework. Would that too not have any ontological significance?

        Anyway, I think Andrei‘s pedagogical/cognitive point about in-principle use of set theory and the moment of realisation is a very good one and is akin to

        (1) Realising that even wildly different-seeming programming languages can be implemented on ordinary, general-purpose digital computers rather than needing special types of machines.

        (For examples of the more different-seeming sort, consider Piet (programs look like abstract paintings), Shakespeare (programs look like plays), Conway’s FRACTRAN (programs are ordered lists of positive fractions), plus less esoteric but still unusual languages such as Lisp, Prolog and Snobol as compared to something like Java.)

        (2) Realising that very different-looking models of computation have turned out to be equivalent to each other in computational power and indeed to something as simple as Turing machines.

        Different programming languages, language implementations, and models of computation can treat lists, for example, in different ways. However, I don’t think I’ve seen anyone dismisses (1) or (2) on the grounds we’re not told what lists really are. What we learn instead is that there are a number of different ways to model / implement lists, rather than lists having to be a new sort of thing introduced as primitive.

        Ordered pairs strike me as similar. The ontological result from set theory is that they can be modelled / implemented in various ways and don’t have to be taken as primitive.

        ***

        There’s a surprising statement in Maddy’s paper. After quoting Voevodsky as saying that using a proof assistant means (among other things) that he no longer has to worry about about how to convince others that his arguments are correct, she writes “I think we can all agree that this is a very attractive picture”.

        I certainly don’t find it attractive. Instead of mathematicians trying to present their findings in understandable, convincing ways, they would in effect just declare “Coq says I’m right: it doesn’t matter what you think.”

        Other Voevodsky quotes show many of the typical attitudes of ideological category- and type-theorists:

        * Antipathy to set theory.

        * A hegemonising ambition to replace ZFC + FOL as a foundation for mathematics.

        * A desire to make things more complicated and harder to understand so they they become the preserve of a more restricted elite (set theory becomes “the theory of types of h-level 2 in Martin-Löf type theory”; they are “developing new type theories more complicated than the standard Martin-Löf type theory”; “such type theories may easily have over a hundred derivation rules”)

        * The assertion that they aren’t just doing something that is good or useful. Instead it’s the right way, and everyone before them was wrong (“the first adequate formalization of the set theory that is used in pure mathematics”).

        1. I wasn’t saying that the fact that we can model (most) mathematics in (standard) set theory has no ontological significance — only that it isn’t obvious what that significance is, and how best to articulate it. You say, for example, that “the ontological result from set theory [re ordered pairs] is that they can be modelled / implemented in various ways and don’t have to be taken as primitive”. But even that isn’t really clear as it stands. Does *X*s can be modelled by *Y*s imply that *X*s aren’t “primitive”? The mathematical modelling we do in physics doesn’t obviously show that elementary particles (or whatever) aren’t “primitive”. So which kinds of modelling reduce the catalogue of the primitive, which don’t? Not an easy question!

          1. I wasn’t making, or relying on, a general claim that Xs that can be modelled by Ys aren’t primitive. I also wasn’t talking about arbitrary forms of ‘modelling’. I was talking about the sort of modelling that’s implementation, which is also what you seemed to be talking about re Kuratowski pairs etc; and I think it is pretty clear that such modelling / implementation of Xs shows that Xs can be modelled / implemented in various ways and don’t have to be taken as primitive, at least when the Xs are things like lists or ordered pairs.

            (That doesn’t mean that it’s not possible to set up a different system in which lists or ordered pairs are treated as primitive.)

            It’s not necessary to answer the general question of “which kinds of modelling reduce the catalogue of the primitive” in order to know that certain forms of modelling show that certain things don’t have to be taken as primitive.

            ***

            When saying “It is a good question then, exactly what the role of set theory is as a “foundation”, and in particular what its ontological significance is”, you pointed to Maddy’s paper “What do we want a foundation to do?”

            I’m not sure why you point to that paper, though, because Maddy doesn’t seem very interested in what the ontological significance might be. Pretty much all she says (so far as I’ve managed to spot) is that Metaphysical insight is a spurious foundational virtue (p 11 n 7) and (p 11):

            For mathematical purposes, the metaphysical claim is beside the point: it doesn’t matter whether we say the von Neumann ordinals are the numbers or the von Neumann ordinals can serve as fully effective mathematical surrogates for the numbers.

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