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 {\displaystyle c_{1}} , c 2 {\displaystyle c_{2}} , ... for each nonzero natural number. Then 0 ♯ {\displaystyle 0^{\sharp }} is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with c i {\displaystyle c_{i}} interpreted as the uncountable cardinal ℵ i {\displaystyle \aleph _{i}} . (Here ℵ i {\displaystyle \aleph _{i}} means ℵ i {\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 0 ♯ {\displaystyle 0^{\sharp }} works provided that there is an uncountable set of indiscernibles for some L α {\displaystyle L_{\alpha }} , and the phrase " 0 ♯ {\displaystyle 0^{\sharp }} exists" is used as a shorthand way of saying this. A closed set I {\displaystyle I} of order-indiscernibles for L α {\displaystyle L_{\alpha }} (where α {\displaystyle \alpha } is a limit ordinal) is a set of Silver indiscernibles if:
I {\displaystyle I} is unbounded in α {\displaystyle \alpha } , and if I ∩ β {\displaystyle I\cap \beta } is unbounded in an ordinal β {\displaystyle \beta } , then the Skolem hull of I ∩ β {\displaystyle I\cap \beta } in L β {\displaystyle L_{\beta }} is L β {\displaystyle L_{\beta }} . In other words, every x ∈ L β {\displaystyle x\in L_{\beta }} is definable in L β {\displaystyle L_{\beta }} from parameters in I ∩ β {\displaystyle I\cap \beta } . If there is a set of Silver indiscernibles for L ω 1 {\displaystyle L_{\omega _{1}}} , then it is unique. Additionally, for any uncountable cardinal κ {\displaystyle \kappa } there will be a unique set of Silver indiscernibles for L κ {\displaystyle L_{\kappa }} . The union of all these sets will be a proper class I {\displaystyle I} of Silver indiscernibles for the structure L {\displaystyle L} itself. Then, 0 ♯ {\displaystyle 0^{\sharp }} is defined as the set of all Gödel numbers of formulae θ {\displaystyle \theta } such that
L α ⊨ θ ( α 1 , α 2 , … , α n ) {\displaystyle L_{\alpha }\models \theta (\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})}
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