This paper is the sequel to Friedman (1974), in which it was shown that the set of finitely ranked sets (V~) has a Spinozistic partitioning (defined below).
Moreover, the following conjectures were left as open problems:
C 1 : The universal class (V) has a Spinozistic partitioning.
C2. V has a Spinozistic partitioning such that each partition class is a model of set theory (in particular, an e-model of ZF (Zermelo-Fraenkel set theory)).
(Note that V is the class of all sets but is not itself a set.) In this paper, both conjectures are answered affirmatively by the proofs of Ca and C2, respectively 1. I shall be working in NBG (von Neumann-Bernays-G6del set theory (see G6del, 1940)). Thus again, it will be seen that non-trivial settheoretical theorems can be directly suggested by a metaphysical system. DEF: A is a Spinozistic partitioning ~ (Sp) of Y if and only if A is an infinite sequence satisfying the following conditions:
(i) Y= U~<~A~, and Ap~A~=O, for all ordinals fl, y<A, where fl-#7 (and where A=the cardinality (or length) of the sequence A); in other words, A is a partitioning of Y.
(ii) Aa = A, for every fl <,4.
(iii) Aa~-A for every fl, 7<A; in other words, Aa is e-isomorphic to At, which means that there exists a one-one mapping f from Ap ontoA~ such that (Vx, y~Aa) (xe y= f (x)ef (y)).