Why be a Height Potentialist?

According to height potentialism, the height of the universe of sets is “potential” or “indefinitely extensible,” and this is something that a (formal) theory of sets should capture. Height actualism is the rejection of height potentialism: the height of the universe of sets isn’t potential or indefinitely extensible, and our standard non-modal theories of sets don’t need to be supplemented with or reinterpreted in a modal language. In this paper, I examine and (mostly) criticize arguments for height potentialism. I first argue that arguments for height potentialism that appeal to its explanatory powers are unsuccessful. I then argue that the most promising argument for height potentialism involves the claim that height potentialism follows from our intuitive conception of sets. But, as I explain, on the most plausible way of developing this argument from an intuitive conception of sets, it turns out that whether height potentialism or height actualism is true is a verbal dispute, i.e., a matter of what meanings we choose to assign to our set-theoretic expressions. I explain that only pragmatic considerations can settle such a dispute and that these weigh in favor of actualism over potentialism. My discussion is also intended to serve two broader aims: to develop what I take to be the most promising line of argument for height potentialism, and to elaborate the height actualist position in greater detail than is standardly done.