In this paper, we attempt to construct something analogous to the cumulative hierawhy of sets, for the notion of predicate. Analogously t o set theory, we will view every o b j ~~c ~t in our hicrarchy t o be a predicate, and evrry predicate in our hierarchy to have a ~~~1 1 defined truth value when applied t o any object in our hierarchy.
l ' h c cumulative hierarchies of predicates we consider here arc constructed by means of four principles.
I. The negation of any predicate is a predicate.
IT.
For any predicate x there is a predicate which holds of precisely those predicates 111. Any set y of predicates determines a predicate; i.e., there is a predicate which JV. The union of any set y of predicates is a predicate; i.e., there is a predicate which IVch will also consider the following extensionality principle.
1.. ,Any two predicates which hold of the same predicates are identical.
'I'Ii(> principal drawback in what, we do here is that, on the one hand we see no sigiiificant reason t o limit the construction principles t o the ones considered; on the oth(.i* hand we see no non ad hoe way to extend them. At least, these seem t o form the minimum natural collection that affords a nontrivial theory, and we expect furt JIW progress along these lines. ])(.finition 1 . .4 cumulative hierarchyisa system ({A,}, R), where A, = 0, A , C A , + ~, AA = U A,, and R is a binary relation on A = U A,. We say that ({A,}, R) is a cunr~rlatiw hierarchy of predicates just in case ({A,}, R) is a cumulative hierarchy, (A, R) satisfitis the axiom of extensionality, and each A,,, is the least subset of A obeying w11ic.h hold of x .
holds of precisely the elements of y. holds of precisely those predicates which are held by some member of y.
That is, A,+] is thcb closure of A, under principles I -IV. Two cumulative hierarchies ( { A n ; , R ) and ({Bd), S) are isomorphic just is case there is an isomorphism from (A, R) l ) This rcetbarc.h was partially supported by NSF P038823. JOHN MYHILL has suggested that work done hy MARC KRASXER, ThCorie de la ddfinition, Journal de mathdmatiques puma et appliquees, aer. U. 36 (1957). 325-357. and 37 (1958). 53-101, may be closely related. That paper is reviewed by G.