# 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 ${\displaystyle c_{1}}$, ${\displaystyle c_{2}}$, ... for each nonzero natural number. Then ${\displaystyle 0^{\sharp }}$ is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with ${\displaystyle c_{i}}$ interpreted as the uncountable cardinal ${\displaystyle \aleph _{i}}$. (Here ${\displaystyle \aleph _{i}}$ means ${\displaystyle \aleph _{i}}$ 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 ${\displaystyle 0^{\sharp }}$ works provided that there is an uncountable set of indiscernibles for some ${\displaystyle L_{\alpha }}$, and the phrase "${\displaystyle 0^{\sharp }}$ exists" is used as a shorthand way of saying this.

A closed set ${\displaystyle I}$ of order-indiscernibles for ${\displaystyle L_{\alpha }}$ (where ${\displaystyle \alpha }$ is a limit ordinal) is a set of Silver indiscernibles if:

• ${\displaystyle I}$ is unbounded in ${\displaystyle \alpha }$, and
• if ${\displaystyle I\cap \beta }$ is unbounded in an ordinal ${\displaystyle \beta }$, then the Skolem hull of ${\displaystyle I\cap \beta }$ in ${\displaystyle L_{\beta }}$ is ${\displaystyle L_{\beta }}$. In other words, every ${\displaystyle x\in L_{\beta }}$ is definable in ${\displaystyle L_{\beta }}$ from parameters in ${\displaystyle I\cap \beta }$.

If there is a set of Silver indiscernibles for ${\displaystyle L_{\omega _{1}}}$, then it is unique. Additionally, for any uncountable cardinal ${\displaystyle \kappa }$ there will be a unique set of Silver indiscernibles for ${\displaystyle L_{\kappa }}$. The union of all these sets will be a proper class ${\displaystyle I}$ of Silver indiscernibles for the structure ${\displaystyle L}$ itself. Then, ${\displaystyle 0^{\sharp }}$ is defined as the set of all Gödel numbers of formulae ${\displaystyle \theta }$ such that

${\displaystyle L_{\alpha }\models \theta (\alpha _{1},\alpha _{2}\ldots \alpha _{n})}$

where ${\displaystyle \alpha _{1}<\alpha _{2}<\ldots <\alpha _{n}<\alpha }$ is any strictly increasing sequence of members of ${\displaystyle I}$. Because they are indiscernibles, the definition does not depend on the choice of sequence.

Any ${\displaystyle \alpha \in I}$ has the property that ${\displaystyle L_{\alpha }\prec L}$. This allows for a definition of truth for the constructible universe:

${\displaystyle L\models \varphi [x_{1}...x_{n}]}$ only if ${\displaystyle L_{\alpha }\models \varphi [x_{1}...x_{n}]}$ for some ${\displaystyle \alpha \in I}$.

There are several minor variations of the definition of ${\displaystyle 0^{\sharp }}$, which make no significant difference to its properties. There are many different choices of Gödel numbering, and ${\displaystyle 0^{\sharp }}$ depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode ${\displaystyle 0^{\sharp }}$ 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 ${\displaystyle 0^{\sharp }}$ exists can be weakened. The existence of ${\displaystyle \omega _{1}}$-Erdős cardinals implies the existence of ${\displaystyle 0^{\sharp }}$. This is close to being best possible, because the existence of ${\displaystyle 0^{\sharp }}$ implies that in the constructible universe there is an ${\displaystyle \alpha }$-Erdős cardinal for all countable ${\displaystyle \alpha }$, so such cardinals cannot be used to prove the existence of ${\displaystyle 0^{\sharp }}$.

Chang's conjecture implies the existence of ${\displaystyle 0^{\sharp }}$.

## Statements equivalent to existence

Kunen showed that ${\displaystyle 0^{\sharp }}$ exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe ${\displaystyle L}$ into itself.

Donald A. Martin and Leo Harrington have shown that the existence of ${\displaystyle 0^{\sharp }}$ 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 ${\displaystyle 0^{\sharp }}$.

It follows from Jensen's covering theorem that the existence of ${\displaystyle 0^{\sharp }}$ is equivalent to ${\displaystyle \omega _{\omega }}$ being a regular cardinal in the constructible universe ${\displaystyle L}$.

Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of ${\displaystyle 0^{\sharp }}$.

## Consequences of existence and non-existence

The existence of ${\displaystyle 0^{\sharp }}$ implies that every uncountable cardinal in the set-theoretic universe ${\displaystyle V}$ is an indiscernible in ${\displaystyle L}$ and satisfies all large cardinal axioms that are realized in ${\displaystyle L}$ (such as being totally ineffable). It follows that the existence of ${\displaystyle 0^{\sharp }}$ contradicts the axiom of constructibility: ${\displaystyle V=L}$.

If ${\displaystyle 0^{\sharp }}$ exists, then it is an example of a non-constructible ${\displaystyle \Delta _{3}^{1}}$ set of natural numbers. This is in some sense the simplest possibility for a non-constructible set, since all ${\displaystyle \Sigma _{2}^{1}}$ and ${\displaystyle \Pi _{2}^{1}}$ sets of natural numbers are constructible.

On the other hand, if ${\displaystyle 0^{\sharp }}$ does not exist, then the constructible universe ${\displaystyle L}$ 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 ${\displaystyle x}$ of ordinals there is a constructible ${\displaystyle y}$ such that ${\displaystyle x\subset y}$ and ${\displaystyle y}$ has the same cardinality as ${\displaystyle x}$.

This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that ${\displaystyle x}$ is uncountable cannot be removed. For example, consider Namba forcing, that preserves ${\displaystyle \omega _{1}}$ and collapses ${\displaystyle \omega _{2}}$ to an ordinal of cofinality ${\displaystyle \omega }$. Let ${\displaystyle G}$ be an ${\displaystyle \omega }$-sequence cofinal on ${\displaystyle \omega _{2}^{L}}$ and generic over ${\displaystyle L}$. Then no set in ${\displaystyle L}$ of ${\displaystyle L}$-size smaller than ${\displaystyle \omega _{2}^{L}}$ (which is uncountable in ${\displaystyle V}$, since ${\displaystyle \omega _{1}}$ is preserved) can cover ${\displaystyle G}$, since ${\displaystyle \omega _{2}}$ is a regular cardinal.

If ${\displaystyle 0^{\sharp }}$ does not exist, it also follows that the singular cardinals hypothesis holds.[1]p. 20

## Other sharps

If ${\displaystyle x}$ is any set, then ${\displaystyle x^{\sharp }}$ is defined analogously to ${\displaystyle 0^{\sharp }}$ except that one uses ${\displaystyle L[x]}$ instead of ${\displaystyle L}$, also with a predicate symbol for ${\displaystyle x}$. See the section on relative constructibility in constructible universe.