Zero sharpIn mathematical set theory, 0# (zero sharp, also: 0#) is an important large cardinal number. Formally, 0# is usually defined as follows:
- 0# exists iff there exists a non-trivial elementary embedding j : L → L for the Gödel constructible universe L.
On the other hand, if 0# does not exist, then the constructible universe L, is the core model - that is, the canonical inner model that approximates the large cardinal structure of the universe considered. In that case, the Covering Lemma holds: If x is an uncountable set of ordinals, then there is a constructible y ⊃ x such that y has the same cardinality as x.
Existence of zero sharp is equivalent to determinacy of lightface analytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as 0#.