between them. A class is infinite iff it is equinumerous with one of its proper subclasses. The axioms of PST are (A1) (extensionality) — Classes that have the same elements are the same. (A2) (class comprehension) — If is a formula, then there exists a class the elements of which are exactly those sets x that satisfy . (A3) (axiom of infinity) — There is an infinite set, and all infinite sets are equinumerous. (inf(x) stands for “x is infinite”; abbreviates that x is equinumerous with y.) (A4) (limitation of size) – A class is a proper class if and only if it is equinumerous with all proper classes. (pr(X) stands for “X is a proper class”.) Remarks on the axioms Although different kinds of variables are used for classes and sets, the language is not many-sorted; sets are identified with classes having the same extension. Small case variables are used as mere abbreviations for various contexts; e.g., Since the quantification in A2 ranges over classes, i.e., is not set-bound, A2 is the comprehension scheme of Morse–Kelley set theory, not that of Von Neumann–Bernays–Gödel set theory. This extra strength of A2 is employed in the definition of the ordinals (not presented here). Since there is no axiom of pairing, it must be proved that for any two sets x and y, the Kuratowski pair {{x},{x,y}} exists and is a set. Hence proving that there exists a one-to-one correspondence between two classes does not prove that they are equinumerous. Pocket set theory is analogous to third order arithmetic, with the sets and classes corresponding to subsets of the natural numbers and subsets of the powerset of the natural numbers. A model for pocket set theory is given by taking the sets of pocket set theory to be the constructible elements of HC (the set of hereditarily countable sets), and the classes to be the constructible subsets of HC. Some PST theorems 1. The Russell class is a proper class. () Proof. cannot be a set by Russell's paradox. ∎ 2. The empty class is a set. () Proof. Suppose (towards a contradiction) that is a proper class. By (A4), must be equinumerous with , in which case is empty. Let i be an infinite set, and consider the class . It is not equinumerous with , thus it is a set. It is finite, but its single element is infinite, thus