Abstract The purpose of this paper is to define number categories as government classes, and to interpret said categories semantically against models which are rich enough to support the desired interpretations. Defining number categories as government classes in terms of their cooccurrence with numerals leads to a system of five pairs of binary categories in complementary opposition: SINGULAR/COSINGULAR, DUAL/CODUAL, PAUCAL/COPAUCAL, MESAL/COMESAL, and UNIVERSAL/COUNIVERSAL. To support the interpretation of these ten categories we focus on Boolean models and compare those that split dualistically into atomistic and atomless portions with those that are uniformly atomless instead. After arguing that models for the interpertation of number should be uniformly atomless, we show how these models can support the interpretation of all count nouns regardless of number category. Particularly their cooccurrence with numerals. Key to such interpretations will be the ability to refer to atomistic structures embedded deeply within atomless models.