Journal of Philosophical Logic manuscript No. (will be inserted by the editor) Can All Things Be Counted? Chris Scambler Received: date / Accepted: date Abstract In this paper, I present and motivate a modal set theory consistent with the idea that there is only one size of infinity. Keywords Modal Logic · Set Theory · Forcing · Potentialism 1 Introduction Recent work in mathematical logic and philosophy has seen an increased in- terest in modal set theory. In modal set theory, beyond the usual first order expressions about what sets there are and aren’t, one allows in addition non- trivial assertions about the sets that are possible. Mathematicians have become interested in modal descriptions of their subject matter for the simple reason that certain aspects of modern set theory admit a natural modal description (e.g. [8], [6], [5]); philosophers on the other hand have argued that admission of modal vocabulary may yield a more satisfying solution to the paradoxes of set theory (e.g. [13], [3], [22]). This broadening of expressive resources opens up new conceptual possi- bilities. For example, consider the standard argument that there are multiple infinite cardinalities. The argument has two main components: a version of Hume’s principle gives us a criterion for cardinality, and a version of Cantor’s theorem gives us (at least) two infinite cardinalities on the assumption there is at least one. In slightly more detail: a version of Hume’s principle says that there are at least as many As as Bs exactly when there’s a function on A with all Bs in its range; 1 Cantor’s theorem says there is no function defined on any set A with all subsets of A in its range. It follows that if there is a countably C Scambler 5 Washington Place, New York, NY, USA, 10003 E-mail: cscambler@gmail.com 1 Assuming of course that B is not empty.