Zoom Talk: Chris Scambler (New York University)
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Potentialism in height and width
Roughly speaking potentialism in set theory is the idea that any totality of sets can be extended: that it is (necessarily) possible for there to be more sets than there in fact are. Two kinds of potentialism have been studied in the recent literature: potentialism in height, which concerns the extent of the ordinal spine along which the cumulative hierarchy is constructed, and potentialism in width, which concerns the ‘fatness’ of the various ranks of the cumulative hierarchy.
The nascent literature in this area has seen various suggestions to the effect that height and width potentialism (suitably formalized in modal logic) are inconsistent with one another. However, my recent JPL paper “Can All Things Be Counted?” presents a modal system that combines height and width potentialism. That system has models and therefore is consistent; but the recent arguments suggest its consistency turns on expressive incompleteness, and/or on failing to properly represent width potentialism. In this talk I will sketch the system in question along with these recent inconsistency arguments, and explain where said arguments break down.