x ⊂ y and y has the same cardinality as x. This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannot be removed. For example, consider Namba forcing, that preserves and collapses to an ordinal of cofinality . Let be an -sequence cofinal on and generic over L. Then no set in L of L-size smaller than (which is uncountable in V, since is preserved) can cover , since is a regular cardinal. Other sharps If x is any set, then x# is defined analogously to 0# except that one uses L[x] instead of L. See the section on relative constructibility in constructible universe. See also 0†, a set similar to 0# where the constructible universe is replaced by a larger inner model with a measurable cardinal. References Real numbers Determinacy Large cardinals Constructible universe