INTR00ucT10~ 1.1. The real O# introduced by Solovay [9] has the following wellknown properties:
(1) Let a be an ordinal such that L,[O"] is admissible; then a is a cardinal in the sense of L.
(2) Conversely, let r be a real. Assume that every ordinal a, such that L,[r] is admissible, is a cardinal in the sense of L. Then Ox E L[r],
These results are due to Silver. (For a proof see [S].) In this paper I will prove THEOREM.
Let d(x) be a Z, formula of the ZF set theory. Let M be a transitive model of ZF+ V= L + Va (a cardinal + $(a)). There is an M definable class P of forcing conditions such that the following hold for every P-generic extension N of M:
(1) N and M have the same cardinals and the same cofinality function.
(2) There is a real r in N such that N satisfies
Notes.
(1) The hypothesis Va (a cardinal + L /= &a)) can be weakened in the following way: It is enough that we can find a class generic real x such that L[x] l= a cardinal + L j= d(a).
(2) In view of the previous property of Ox, this theorem is the best possible since "to be a cardinal in L" is defined by a n, formula.
(3) Using the technique developed in [3], it is not difficult to see that