Abstract: Pursues the theoretical level of the two-tiered epistemology of set theoretic realism, the level at which more abstract axioms can be justified by their consequences at more intuitive levels. I outline the pre-axiomatic development of set theory out of Cantor's researches, describe how axiomatization arose in the course of Zermelo's efforts to prove Cantor's Well-ordering Theorem, and review the controversy over the Axiom of Choice. Cantor's Continuum Hypothesis and various questions of descriptive set theory were eventually shown to be independent of the standard axioms, and new axiom candidates—Gödel's Axiom of Constructibility, on the one hand, determinacy and large cardinal axioms, on the other—offer dramatically different solutions. This situation presents a new epistemic challenge to the set theoretic realist: on what grounds can we adjudicate between new axiom candidates?

Keywords: Axiom of Choice, axiomatic set theory, Cantor, Cantor's Continuum Hypothesis, descriptive set theory, determinacy axioms, Gödel's Axiom of Constructibility, large cardinal axioms, Zermelo,