 ∗22 on the “Calculus of Classes” presents the elementary set theory of intersections, unions and the empty set which is often all the set theory used in elementary mathematics of other sorts. The student looking for the set theory of Principia to compare it with, say the Zermelo-Fraenkel system, will have to look at various numbers later in the text. The Axiom of Choice is defined at ∗88 as the “Multiplicative Axiom” and a version of the Axiom of Infinity appears at ∗120 in Volume II as “Infin ax”. The set theory of Principia comes closest to Zermelo’s axioms of 1908 among the various familiar axiom systems, which means that it lacks the Axiom of Foundation and Axiom of Replacement of the now standard Zermelo-Fraenkel axioms of set theory. The system of Principia differs importantly from Zermelo’s in that it is formulated in the simple theory of types. As a result, for example, there are no quantifiers ranging over all sets, and there is a set of all things (for each type). 