Zero Sharp - Definition

Definition

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 ℵ_i. (Here ℵ_i means ℵ_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# 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.