Editing Zero sharp (section)

==Statements equivalent to existence==
Kunen showed that <math>0^\sharp</math> exists if and only if there exists a non-trivial elementary embedding for the [[Gödel constructible universe]] <math>L</math> into itself.

[[Donald A. Martin]] and [[Leo Harrington]] have shown that the existence of <math>0^\sharp</math> is equivalent to the determinacy of [[lightface analytic game]]s.  In fact, the strategy for a universal lightface analytic game has the same [[Turing degree]] as <math>0^\sharp</math>.

It follows from [[Jensen's covering theorem]] that the existence of <math>0^\sharp</math> is equivalent to <math>\omega_\omega</math> of ''V'' being a [[regular cardinal]] in the constructible universe <math>L</math>.

Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of <math>0^\sharp</math>.