Editing Zero sharp (section)

{{short description|Concept in set theory}}
In the mathematical discipline of [[set theory]], '''0<sup>#</sup>''' ('''zero sharp''', also '''0#''') is the set of true [[formula (mathematical logic)|formulae]] about [[indiscernibles]] and order-indiscernibles in the [[Gödel constructible universe]]. It is often encoded as a subset of the [[natural number]]s (using [[Gödel numbering]]), or as a subset of the [[hereditarily finite set]]s, or as a [[Baire space (set theory)|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 [[Jack Silver|Silver's]] 1966 thesis, later published as {{harvtxt|Silver|1971}}, where it was denoted by Σ, and rediscovered by  {{harvtxt|Solovay|1967|loc=p.52}}, who considered it as a subset of the natural numbers and introduced the notation O<sup>#</sup> (with a capital letter O; this later changed to the numeral '0').

Roughly speaking, if 0<sup>#</sup> 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.