This paper examines a didactic phenomenon, a perception that the elements of sets have a common property, e.g. that {1, 2, 3} and {a, b, c} are sets but {1, 2, 3, a, b, c} is not a set. The paper reports manifestations of this phenomenon in one country: across a range of school students in textbooks and in curricula and across pre-service and in-service teachers. The paper considers interpretations of the results under various theoretical frameworks and argues that these interpretations can be synthesized around the anthropological theory of the didactic. The paper ends with implications for what is taught under the name 'mathematics'.