Abstract:

Gödel gives the name "set of x's" to the operation that, starting with the x's (e.g., the natural numbers), yields the sets whose members are x's (in the example, the sets of natural numbers). Transfinite iteration of this operation yields all sets. In the first version of his paper on the continuum hypothesis, Gödel says, "As opposed to the concept of set in general (if considered as a primitive) we have a clear notion of this concept." I discuss various questions about the concept of set of x's, both the general concept and its instances. Here are some of these questions: Is there an important sense in which the concept is axiomatizable? Is it categorical? Do all first-order statements about instances of the concept have truth-values? Do answers to these questions vary with the instances (with what the x's are)? Does the independence from ZFC of statements like the continuum hypothesis have any bearing on these questions?