Zero sharp
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In mathematical set theory, 0# (zero sharp, also: 0#) is defined to be a particular real number satisfying certain conditions. The definition is a bit awkward, because it is consistent (with the system of axioms ZFC of set theory) that in fact there is no real number satisfying the conditions. The proposition "0# exists" is independent of the axioms of ZFC, and is usually formulated as follows:
- 0# exists iff there exists a non-trivial elementary embedding j : L → L for the Gödel constructible universe L.
If 0# exists, then a careful analysis of the embeddings of L into itself reveals that there is a closed unbounded proper class of ordinals that are indiscernible for the structure 
, and 0# is defined to be the real number that codes in the canonical way the Gödel numbers of the true formulas about the indiscernibles in L. 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.
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.
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 ω1 and collapses ω2 to an ordinal of cofinality ω. Let G be an ω-sequence cofinal on 
and generic over L. Then no set in L of L-size smaller than (which is uncountable in V, since ω1 is preserved) can cover G, since ω2 is a regular cardinal.
Donald A. Martin and Leo Harrington have shown that the existence of zero sharp 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#.
[edit] 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.
[edit] References
- Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0444105352.
- Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3540003843.
