My comments on Ch. 6 ended inconclusively. But I’ll move on to say just a little about the final chapter of the book, Ch. 7 ‘The Graph Conception’.
Back to the beginning. On the iterative conception, the hierarchy of sets is formed in stages; at each new stage the set of operation is applied to some objects (individuals and/or sets) already available at that stage, and outputs a new object. This conception very naturally leads to the idea that sets can’t be members of themselves (or members of members of themselves, etc.), which in turn naturally gives us the Axiom of Foundation.
But now turn the picture around. Instead of thinking of a set as (so to speak) lassoing already available objects, what if we think top down of a set as like a dataset pointing to some things (zero or more of them)? On this picture, being given a set is like being given a bundle of arrows pointing to objects (via the has as member relation) — and why shouldn’t one of these arrows loop round so that it points to the very object which is its source (so we have a set one of whose members is that set itself)?
Elaborating this idea a bit more, we’ll arrive at what we might call a graph conception of set. Roughly: take a root node with directed edges linking it to zero or more nodes which in turn have further directed edges linking them to nodes, etc. Then this will be a picture showing the membership structure of a pure set with its members and members of members etc. (a terminal node with no arrows out picturing the empty set); and any pure set can be pictured like this. But there is nothing in this conception as yet which rules out edges forming short or long loops. So on this conception, the Axiom of Foundation will fail.
Talking of graphs in this way takes us into the territory of the non-well-founded set theories introduced by Peter Aczel. And these are the focus of Luca’s interesting chapter. I’m not going to go into any real detail here, because much of the chapter is already available as a standalone paper, The Graph Conception of Set, J. Philosophical Logic 2014. But in §7.2, Luca explains — a bit too briskly for some readers, I imagine — Aczel’s four systems. Then in §7.3, he argues that these systems are not mere technical curiosities but arise out of what we’ve just called a graph conception of set: think of sets top down as objects which may have members (as I put it, point to some objects) which may have members etc., and “one can just take sets to be what is depicted by graphs of the appropriate form”. More specifically, in §7.4 Luca argues that the particular anti foundation axiom AFA is very naturally justified on the graph conception. In §7.5, it is argued that some other core set theoretic axioms are equally naturally justified on the graph conception, while Replacement and Choice remain outliers as on the iterative conception.
In the end, then, the claim is that ZFA (ZFC with Foundation replaced by AFA) is as well justified by graph conception as ZFC is justified by the iterative conception. This is among the most original claims in the book, I think, and seems to be to be very well defended. In §§7.6-7.8, Luca fends off some objections to a set theory based on the graph conception, though he argues that the graph conception doesn’t so naturally accommodate impure sets with urelements. But §7.9 then worries that “ultimately, non-well-founded set theory must be justified by appealing to considerations that come from the theory of graphs. Thus, in this sense, non-well-founded set theory is not justificatory autonomous from graph theory. … [I]f set theory is to provide a foundation for mathematics in the sense … endorsed by many a set theorist, the iterative conception fares better than the graph conception.”
Now “faring better” isn’t ”beating hands down”: but Luca will live with that. In the very brief Conclusion which follows Chapter 7, he is open to a “a moderate form of pluralism about conceptions of set, according to which, depending on the goals one has, different conceptions of set might be preferable”. Nonetheless, Luca’s final verdict is conservative: “[W]hen it comes to the concept of set, and if set theory is to be a foundation for mathematics, then the iterative conception fares better than its currently available rivals. This vindicates the centrality of the iterative conception, and the systems it appears to sanction, in set theory and the philosophy of mathematics.”
“We shall not cease from exploration/And the end of all our exploring/Will be to arrive where we started/And know the place …” well, not perhaps for quite the first time, but better than we did! And when I’ve had the chance to think about things a bit more, I’ll edit these blogposts into a single booknote, and may also add some afterthoughts. But I’ve enjoyed reading Conceptions of Set and blogging about it a good deal, and I hope I’ve encouraged some of you to read the book (and all of you to ensure it is in your university library). Congratulations again to Luca!
Re
I don’t see all that big a difference between ‘lassoing’ objects and pointing to them. In both cases, the intuitive picture is of objects already available to be lassoed or pointed to. And if the objects are sets, and a set is a bundle of arrows, an object you can point to is already a bundle containing whatever arrows it does. You can’t give it a new arrow that points to itself, because sets aren’t modifiable. There might be a conception of sets based on that picture, but the sets / graphs would be acyclic.
(On p 202, Incurvati speaks of constructing a new graph by “drawing a new edge” from the top node of an existing graph, ?, which can certainly look like it’s modifying ?. But he immediately follows that by saying “more intuitively”, you make two copies of ?, one of which has an additional edge that points to the other. Even in the “drawing a new edge” version, for this particular case, the edge is drawn to what is effectively a copy of the unmodified ?, and so (using a suitable notion of graph identity) perhaps the unmodified ? still exists, though it looks like it no longer has the same top node. No wonder he finds the “copy” version more intuitive!)
On the other hand, it’s easy to imagine a graph being constructed by first making all the nodes and then adding edges, or by interleaving those steps. An edge could then lead from a node to itself. That’s an intuitive picture for graphs, though, not sets.
We can then say that if we treat graphs of a particular sort as sets (accessible pointed graphs), we get a reasonable set theory, ZFA, and even a hierarchical construction that gives us a picture of the ZFA universe. I think Incurvati does a good job of explaining and motivating this.
However, getting to a successor stage in the graph-theoretic hierarchy involves constructing “all sets depicted by apgs whose nodes are taken from apgs depicting subsets of (the preceding stage)” (p 211). And so (p 212)
So understanding the construction relies on an understanding of graphs. It’s not conceptually autonomous.
Incurvati seems to be trying to set that aside at the end of §2.7 by saying “the hierarchical construction of the AFA universe … is only intended to provide us with a picture of this universe, not with a conception of what sets are.” However, (1) it’s reasonable to think that understanding a picture of the universe is part of understanding the conception, and (2) understanding the conception also relies directly on an understanding of graphs, so it’s not conceptually autonomous anyway.
In §2.9 on autonomy, Incurvati discusses three kinds of autonomy: logical, conceptual, and justificatory. He argues that non-well-founded set theories have all three sorts with respect to well-founded set theories — but lack justificatory autonomy from graph theory.
However, he doesn’t consider whether non-well-founded set theories are logically or conceptually autonomous from graph theory. I think it’s pretty clear they aren’t conceptually autonomous.
Thanks for this, which digs down at a couple of points more than I did in my comments. An initial thought, though: I’m not sure I see why (on the “pointing” picture) the sets/graphs would be acyclic: follow the arrows where they go and we could (it seems) find ourselves going round in circles. But I agree if we get a “conception of set” at all this way, it is not conceptually autonomous — we are in the end relying on thoughts about graphs. Though if asked, I’d have said that that was Luca’s implied view.
I’ll try to explain how I was thinking when I said “the sets / graphs would be acyclic.”
I think there are intuitive “pointing” pictures for graphs that allow cycles. For instance, we can imagine a graph-construction kit as providing nodes and edges as parts, and (with some constraints depending on the type of graph) we can connect them any way we want, which includes connecting a node to itself. There are construction toys quite like that.
So someone could get out their graph construction kit, construct a cyclic graph while obeying the constraints for accessible pointed graphs, and then say “let’s treat this as a set”. (“Here’s one I made earlier.”)
However, is there an intuitive picture that doesn’t involve graph-construction kits (and the like) and that doesn’t treat the graph-sets as pre-made objects that are just given to us without an intuitive picture of how they’re made?
The “bundle of arrows” idea looks like it might do this. It at least doesn’t go all the way to being a graph-construction kit, because it doesn’t involve nodes, because it’s quite natural to see the has-as-member relation as an arrow, and because sets naturally involve some idea of collecting together into one thing, and a bundle would do that.
The intuitive picture that suggests to me is that, to make a set, we have some things, we make arrows pointing to those things, we gather the arrows as a bundle, and that bundle of arrows is a new set.
How could the bundle point to itself? A arrow pointing to the bundle can’t be one of the arrows we bundled, because when we made those arrows, the bundle didn’t yet exist to be pointed-to as one of “those things”; and it can’t be added to the bundle after we’ve made it, because if the bundle is a set, bundles aren’t modifiable (because sets aren’t).
So, in that picture, bundle sets are acyclic.
– – – – –
Sets in a programming language like Java are unlike mathematical sets in some respects, and one of those respects is that they’re modifiable. Here’s an example using JShell that constructs a Set that contains itself as a member:
$ jshell
| Welcome to JShell — Version 11.0.2
| For an introduction type: /help intro
jshell> Set⟨Object⟩ s = new HashSet⟨⟩() // make a set
s ==> []
jshell> s.add(1) // add 1 to the set
$2 ==> true
jshell> s // look at a textual rep of the set
s ==> [1]
jshell> s.add(s) // add s to itself
$4 ==> true
jshell> s // look at a textual rep of the set
s ==> [1, (this Collection)]
jshell> /exit
| Goodbye