 Most of us agree on what the theories of natural numbers, real numbers, and Euclidean space ought to look like (though constructivist mathematicians will have differences with classical mathematics even here). There was at least initially less agreement as to what a theory of sets ought to look like (or even whether there ought to be a theory of sets). The confidence of at least some mathematicians in their understanding of this subject (or in its coherence as a subject at all) was shaken by the discovery of paradoxes in “naive” set theory around the beginning of the twentieth century. A number of alternative approaches were considered then and later, but a single theory, the Zermelo-Fraenkel theory with the Axiom of Choice (ZFC) dominates the field in practice. One of the strengths of the Zermelo-Fraenkel set theory is that it comes with an image of what the world of set theory is (just as most of us have a common notion of what the natural numbers, the real numbers, and Euclidean space are like): this image is what is called the “cumulative hierarchy” of sets.