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.

See also

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

References

• 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 #". Journal of Symbolic Logic. 43 (4): 685–693. doi:10.2307/2273508. ISSN 0022-4812. 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". Fundamenta Mathematicae. 66 (3): 287–291. doi:10.4064/fm-66-3-287-291. ISSN 0016-2736. MR 0258637.
• Silver, Jack H. (1971). "Some applications of model theory in set theory". Annals of Mathematical Logic. 3 (1): 45–110. doi:10.1016/0003-4843(71)90010-6. MR 0409188.

• Solovay, Robert M. (1967). "A nonconstructible Δ¹
  ₃ set of integers". Transactions of the American Mathematical Society. 127 (1): 50–75. doi:10.2307/1994631. ISSN 0002-9947. MR 0211873.