In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about (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 (somewhat misleadingly) 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 a number 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.
Read more about Zero Sharp: Definition, Statements That Imply The Existence of 0#, Statements Equivalent To Existence of 0#, Consequences of Existence and Non-existence, Other Sharps
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