Assume that we are working in a set theory T, formalized in the language of ZFC, whose axioms---in addition to those of ZFC---include also the negation of "V=OD".
For any set X, let CARD(X) denote the cardinal number of X, let ORDEF(X) denote the set of ordinal definable elements of X and let P(X) denote the the set of all subsets of X. As usual, N denotes the set of all non-negative integers and R denotes the set of all real numbers.
It is known that T+"CARD(ORDEF(P(N)))=CARD(N)" is consistent if ZFC is.
I would like to know whether this phenomenon---that the consistency of ZFC implies the consistency of T+"CARD(ORDEF(P(X)))=CARD(X)"---is true for many sets X? In particular, is it true for X=R?
My reason for being interested in this question is that, for many uncountable sets X, it becomes difficult to work in P(X) because so many sets---most of which we do not care about---belong to P(X).
If CARD(ORDEF(P(X)))= CARD(X) and if no inconsistency is introduced, we could work in ORDEF(P(X)) instead of in P(X). Most of the sets belonging to P(X) in which we are interested should still belong to ORDEF(P(X)) and we would be working in a set is no greater than that of X.
Just consider how few of the sets belonging to P(R) are of any interest to anybody. Most of them are literally indescribable in any conceivable language.