Zero sharp

Zero sharp is a sharp for the constructible universe \(L\), which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\):

• \(|I \cap \kappa| = \kappa\)
• For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\)
• For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\).

Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles.

It is known that the following are equivalent:

• \(0^\sharp\) exists.
• There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (\(L\) does not have the covering property).
• For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\).
• Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is determined.
• \(\aleph_\omega^V\) is regular in \(L\).
• There is a nontrivial elementary embedding \(j: L \to L\).
• There is a proper class of nontrivial elementary embeddings \(j: L \to L\), all with different critical points.
• For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\).
• Every uncountable cardinal is inaccessible in \(L\).
• There is a singular cardinal \(\gamma\) so that \((\gamma^+)^L < \gamma\).

While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following:

• \(\aleph_\omega\) is totally stable (stable for first-order formulae?) - in fact, every uncountable cardinal is totally stable.
• If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\).
• There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable.

And "\(0^\sharp\) exists" is strictly implied by the following:

• Chang's conjecture
• There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\).
• There is a weakly compact cardinal \(\kappa\) so that \((\kappa^+)^L < \kappa\).
• There is a Ramsey cardinal.

If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles.

Alternatively, \(0^\sharp\) may be defined as a sound mouse (iterable premouse), or as an Ehrenfeucht-Mostowski blueprint.