minor variations of the definition of 0#, which make no significant difference to its properties. There are many different choices of Gödel numbering, and 0# depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode 0# as a subset of formulae of a language, or as a subset of the hereditarily finite sets, or as a real number. Statements implying existence The condition about the existence of a Ramsey cardinal implying that 0# exists can be weakened. The existence of ω1-Erdős cardinals implies the existence of 0#. This is close to being best possible, because the existence of 0# implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot be used to prove the existence of 0#. Chang's conjecture implies the existence of 0#. Statements equivalent to existence Kunen showed that 0# exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe L into itself. Donald A. Martin and Leo Harrington have shown that the existence of 0# is equivalent to the determinacy of lightface analytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as 0#. It follows from Jensen's covering theorem that the existence of 0# is equivalent to ωω being a regular cardinal in the constructible universe L. Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of 0#. Consequences of existence and non-existence Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L and satisfies all large cardinal axioms that are realized in L (such as being totally ineffable). It follows that the existence of 0# contradicts the axiom of constructibility: V = L. If 0# exists, then it is an example of a non-constructible Δ set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ and Π sets of integers are constructible. 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, Jensen's covering lemma holds: For every uncountable set x of ordinals there is a constructible y such that