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Luca Incurvati’s Conceptions of Set, 2

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We are still on Chapter 1 of Luca’s book. Sorry about taking longer than I had intended to get back to this. But I’d promised myself to get the answers to the Exercises for Chs 32 and 33 of IFL2 (on natural deduction for quantifier arguments) done and dusted. Thirty eight pages(!) of work later, they are online! Let’s take it that the concept of set is (at least in part) characterized by Luca’s three conditions — Unity (a set is in some sense a unity, distinct from its members), Unique Decomposition (a set decomposes into its members in just one way), Extensionality. Which leaves more to be said, no doubt. But then there are various possible views of the role of the further story we need. Suppose, for example, that you hold that the concept of set, as pre-theoretically grasped, is governed by the following assumption: that for any coherent predicate there is a set of objects which satisfy it. Then, rapidly, we get to a classically inconsistent naive set theory. Put on hold for now the option of revising your logic as a palliative. Then you’ll want to work with a classically consistent replacement concept of set*. And the further story we need is an elaboration of this replacement concept. Suppose alternatively that, as far as it goes, the concept of set is consistent enough. Then that leaves open a spectrum of possible views (at least I take it there is a spectrum here, though Luca highlights the endpoints). At one end, the idea will be that there is not much more to be said about the basic concept of set. We can go on, though, to sharpen the notion in a number of distinct ways, coming up with different, more refined, concepts — though it may turn out that one sharpening is particularly fruitful, mathematically speaking. [Possible model: we have a rough-and-ready concept of a computable function. This can be refined in various ways, though one direction — giving us the notion of an effectively computable function, where we abstract. . .

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News source: Logic Matters

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