The aim of the present paper is to provide a robust classification of valid sentences in set theory by means of existence and related notions and, in this way, to capture similarities and dissimilarities among the axioms of set theory. In order to achieve this, precise definitions for the notions of productive and nonproductive assertions, constructive and nonconstructive productive assertions, and conditional and unconditional productive assertions, among others, will be presented. These definitions constitute the result of a semantical analysis of (...) the notions involved. The conceptual clarification developed here results in a classification of valid sentences of set theory that goes against the standard view that extensionality is not an existence assertion. (shrink)