# 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 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 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

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 nonzero natural number. Then $0^{\sharp }$ 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 $\aleph _{i}$. (Here $\aleph _{i}$ means $\aleph _{i}$ 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 $0^{\sharp }$ works provided that there is an uncountable set of indiscernibles for some $L_{\alpha }$, and the phrase "$0^{\sharp }$ exists" is used as a shorthand way of saying this.
A closed set $I$ of order-indiscernibles for $L_{\alpha }$ (where $\alpha$ is a limit ordinal) is a set of Silver indiscernibles if:
* $I$ is unbounded in $\alpha$, and
* if $I\cap \beta$ is unbounded in an ordinal $\beta$, then the Skolem hull of $I\cap \beta$ in $L_{\beta }$ is $L_{\beta }$. In other words, every $x\in L_{\beta }$ is definable in $L_{\beta }$ from parameters in $I\cap \beta$.
If there is a set of Silver indiscernibles for $L_{\omega _{1}}$, then it is unique. Additionally, for any uncountable cardinal $\kappa$ there will be a unique set of Silver indiscernibles for $L_{\kappa }$. The union of all these sets will be a proper class $I$ of Silver indiscernibles for the structure $L$ itself. Then, $0^{\sharp }$ is defined as the set of all Gödel numbers of formulae $\theta$ such that
$L_{\alpha }\models \theta (\alpha _{1},\alpha _{2}\ldots \alpha _{n})$
where $\alpha _{1}<\alpha _{2}<\ldots <\alpha _{n}<\alpha$ is any strictly increasing sequence of members of $I$. Because they are indiscernibles, the definition does not depend on the choice of sequence.
Any $\alpha \in I$ has the property that $L_{\alpha }\prec L$. This allows for a definition of truth for the constructible universe:
$L\models \varphi [x_{1}...x_{n}]$ only if $L_{\alpha }\models \varphi [x_{1}...x_{n}]$ for some $\alpha \in I$.
There are several minor variations of the definition of $0^{\sharp }$, which make no significant difference to its properties. There are many different choices of Gödel numbering, and $0^{\sharp }$ depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode $0^{\sharp }$ 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 $0^{\sharp }$ exists can be weakened. The existence of $\omega _{1}$-Erdős cardinals implies the existence of $0^{\sharp }$. This is close to being best possible, because the existence of $0^{\sharp }$ implies that in the constructible universe there is an $\alpha$-Erdős cardinal for all countable $\alpha$, so such cardinals cannot be used to prove the existence of $0^{\sharp }$.
Chang's conjecture implies the existence of $0^{\sharp }$.

## Statements equivalent to existence

Kunen showed that $0^{\sharp }$ 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 $0^{\sharp }$ 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 $0^{\sharp }$.
It follows from Jensen's covering theorem that the existence of $0^{\sharp }$ is equivalent to $\omega _{\omega }$ 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 $0^{\sharp }$.

## Consequences of existence and non-existence

The existence of $0^{\sharp }$ 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 $0^{\sharp }$ contradicts the axiom of constructibility: $V=L$.
If $0^{\sharp }$ exists, then it is an example of a non-constructible $\Delta _{3}^{1}$ set of natural numbers. This is in some sense the simplest possibility for a non-constructible set, since all $\Sigma _{2}^{1}$ and $\Pi _{2}^{1}$ sets of natural numbers are constructible.
On the other hand, if $0^{\sharp }$ 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 $x\subset y$ 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.
If $0^{\sharp }$ does not exist, it also follows that the singular cardinals hypothesis holds.$^{p.20}$

## Other sharps

If $x$ is any set, then $x^{\sharp }$ is defined analogously to $0^{\sharp }$ except that one uses $L[x]$ instead of $L$, also with a predicate symbol for $x$. See Constructible universe#Relative constructibility.