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Luca Incurvati’s Conceptions of Set, 15

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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. . .

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News source: Logic Matters

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