Zero sharp

Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols c₁, c₂, ... 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 ${\displaystyle \aleph _{i}}$. (Here ${\displaystyle \aleph _{i}}$ means ${\displaystyle \aleph _{i}}$ in the full universe, not the constructible universe.)

If there is in V an uncountable set of Silver order-indiscernibles in the constructible universe L, then 0^# is the set of Gödel numbers of formulas θ of set theory such that

${\displaystyle L_{\omega _{\omega }}\models \theta (\omega _{1},\omega _{2},...\omega _{n})}$

where ω₁, ... ω_ω are the "small" uncountable initial ordinals in V, but have all large cardinal properties consistent with V=L relative to L.

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.

The condition about the existence of a Ramsey cardinal implying that 0^# exists can be weakened. The existence of ω₁-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^#.

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^#.

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 Δ¹
₃ set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ¹
₂ and Π¹
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 ${\displaystyle \omega _{1}}$ 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 L. Then no set in L of L-size smaller than ${\displaystyle \omega _{2}^{L}}$ (which is uncountable in V, since ${\displaystyle \omega _{1}}$ is preserved) can cover ${\displaystyle G}$, since ${\displaystyle \omega _{2}}$ is a regular cardinal.

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.

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

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