# Zero sharp

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 integers (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 Silver (1971), where it was denoted by Σ, and rediscovered by Solovay (1967, p.52), 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[edit]

Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols *c*_{1}, *c*_{2}, ... for each positive integer. Then 0^{#} is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with *c*_{i} interpreted as the uncountable cardinal ℵ_{i}.
(Here ℵ_{i} means ℵ_{i} 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, and showed that with this extra assumption it is possible to define the truth of statements about the constructible universe. More generally, the definition of 0^{#} works provided that there is an uncountable set of indiscernibles for some *L*_{α}, and the phrase "0^{#} exists" is used as a shorthand way of saying this.

There are several 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[edit]

The condition about the existence of a Ramsey cardinal implying that 0^{#} exists can be weakened. The existence of ω_{1}-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[edit]

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 ω_{ω} 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[edit]

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^{1}_{3} set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ^{1}_{2} and Π^{1}_{2} 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*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 and collapses to an ordinal of cofinality . Let be an -sequence cofinal on and generic over *L*. Then no set in *L* of *L*-size smaller than (which is uncountable in *V*, since is preserved) can cover , since is a regular cardinal.

## Other sharps[edit]

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[edit]

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

## References[edit]

- 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 0-444-10535-2. - Harrington, Leo (1978), "Analytic determinacy and 0
^{#}",*The Journal of Symbolic Logic*,**43**(4): 685–693, doi:10.2307/2273508, ISSN 0022-4812, JSTOR 2273508, MR 0518675 - Jech, Thomas (2003).
*Set Theory*. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002. - Kanamori, Akihiro (2003).
*The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings*(2nd ed.). Springer. ISBN 3-540-00384-3. - Martin, Donald A. (1970), "Measurable cardinals and analytic games",
*Polska Akademia Nauk. Fundamenta Mathematicae*,**66**: 287–291, ISSN 0016-2736, MR 0258637 - Silver, Jack H. (1971) [1966], "Some applications of model theory in set theory",
*Annals of Pure and Applied Logic*,**3**(1): 45–110, doi:10.1016/0003-4843(71)90010-6, ISSN 0168-0072, MR 0409188 - Solovay, Robert M. (1967), "A nonconstructible Δ
^{1}_{3}set of integers",*Transactions of the American Mathematical Society*,**127**: 50–75, doi:10.2307/1994631, ISSN 0002-9947, JSTOR 1994631, MR 0211873