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Luca Incurvati’s Conceptions of Set, 3

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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. . .

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News source: Logic Matters

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