Sharp (set theory)

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 integers (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 , where it was denoted by S, and rediscovered by , who considered it as a subset of the natural numbers and introduced the notation O^# (with a capital letter O; this later changed to a number 0).