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!