Zero sharp

In the mathematical discipline of set theory, 0^# (zero sharp, also 0#) is defined to be a particular real number satisfying certain conditions, namely, to be the real number that codes in the canonical way the Gödel numbers of the true formulas about the indiscernibles in the Gödel constructible universe. It is not known whether a real number satisfying these conditions exists. Its existence is believed to be independent of the axioms of ZFC, the standard form of axiomatic set theory.

Definition

If there exists a non-trivial elementary embedding for the Gödel constructible universe "L" into itself, then there is a closed unbounded proper class of ordinals that are indiscernible for the structure $(L,in)$. 0^# is then defined to be the real number that codes in the canonical way the Gödel numbers of the true formulas about the indiscernibles in "L".

Relation to ZFC

If ZFC itself is consistent, then ZFC extended with the statement "0^# does not exist" is consistent. ZFC extended with the statement "0^# exists" is not known to be inconsistent, and most set theorists believe that it is consistent. See large cardinal property for a discussion.

Consequences of existence and non-existence

0^# exists iff there exists a non-trivial elementary embedding for the Gödel constructible universe "L" into itself. 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 following covering lemma holds:

:For every uncountable set "x" of ordinals there is a constructible "y" such that "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 $omega\_1$ and collapses $omega\_2$ to an ordinal of cofinality $omega$. Let $G$ be an $omega$-sequence cofinal on $omega\_2^L$ and generic over "L". Then no set in "L" of "L"-size smaller than $omega\_2^L$ (which is uncountable in "V", since $omega\_1$ is preserved) can cover $G$, since $omega\_2$ is a regular cardinal.

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^#.

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^†

References

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