In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the natural numbers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as , where it was denoted by Σ, and rediscovered by , who considered it as a subset of the natural numbers and introduced the notation O# (with a capital letter O; this later changed to the numeral '0').

Roughly speaking, if 0# exists then the universe V of sets is much larger than the universe L of constructible sets, while if it does not exist then the universe of all sets is closely approximated by the constructible sets.

Definition

Zero sharp was defined by Silver and Solovay is defined to be the set of Gödel numbers in the full universe, not the constructible universe.)

There is a subtlety about this definition: by Tarski's undefinability theorem it is not, in general, possible to define the truth of a formula of set theory in the language of set theory. To solve this, Silver and Solovay assumed the existence of a suitable large cardinal, such as a Ramsey cardinal exists" is used as a shorthand way of saying this.

of order-indiscernibles is a limit ordinal) is a set of Silver indiscernibles if:

• , and

• , then the Skolem hull.

. The union of all these sets will be a proper class such that

. Because they are indiscernibles, the definition does not depend on the choice of sequence.

. This allows for a definition of truth for the constructible universe:

.

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

-Erdős cardinals.

Chang's conjecture.

Statements equivalent to existence

Kenneth Kunen exists if and only if there exists a non-trivial elementary embedding of the Gödel constructible universe into itself.

Donald A. Martin and Leo Harrington is equivalent to the determinacy of lightface analytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree.

It follows from Jensen's covering theorem of V being a regular cardinal.

.

Consequences of existence and non-existence

implies that every uncountable cardinal and satisfies all large cardinal (such as being totally ineffable contradicts the axiom of constructibility.

sets of natural numbers are constructible.

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:

1. has the same cardinality.

This deep result is due to Ronald Jensen. Using forcing is uncountable cannot be removed. For example, consider Namba forcing to an ordinal of cofinality-sequence cofinal and generic is a regular cardinal.

does not exist, it also follows that the singular cardinals hypothesis holds.p. 20

Other sharps

. See Constructible universe#Relative constructibility.

See also

• 0†, a set similar to 0# where the constructible universe is replaced by a larger inner model with a measurable cardinal.

References

Citations