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Luca Incurvati’s Conceptions of Set, 4

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On now to the very interesting second half of the second chapter, where we are still considering the iterative conception in an initial way. So, quoting Luca, According to the iterative conception, then, sets can be arranged in a cumulative hierarchy divided into levels. This conception sanctions (at least) most of the axioms of standard set theory and provides a convincing explanation of the paradoxes; but is it correct? What reasons, then, can be offered in support of endorsing the iterative conception? Luca first discusses the idea that we should take literally the metaphor of construction that comes to us so readily in describing the iterative conception. So, the idea is, sets really are formed in a stage-by-stage process, where at each stage we can only collect together in various ways what is already available. But how do we make better-than-metaphorical sense of this idea of forming sets in a process when we are supposed to be dealing (aren’t we?) with abstract items which (i) exist independently of our activities (aren’t really formed) and (ii) in a timeless way (so there’s no real process of level-building). Arguably, the constructionist metaphor at best gives colour but no real underpinning to the iterative conception. Suppose, however, we do try to push the metaphor harder. Then, Luca argues, [my numbering] (i) it seems part of the constructivist doctrine that, at any point in the construction process, we can only construct sets specifiable by reference to sets already constructed. (ii) This, however, seems to sanction only a predicative version of Z’s Separation Schema … which cuts down the strength of our set theory. Now, (i) gives us one way of elaborating what the ‘the constructionist doctrine’ might be supposed to be. Though we could, I suppose, pause to ask whether is it compulsory to construe ‘construct sets from sets that are already constructed’ as implying ‘construct sets specifiable by reference to sets already constructed’. Be that as. . .

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News source: Logic Matters

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