Large cardinals | Determinacy | Constructible universe | Real numbers

Zero sharp

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 Σ, and rediscovered by , 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. (Wikipedia).

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Generic filter | Hereditarily finite set | Set theory | Measurable cardinal | Gödel numbering | Indiscernibles | Uncountable set | Zero dagger | Axiom of constructibility | Regular cardinal | Lightface analytic game | Forcing (mathematics) | Ramsey cardinal | Jensen's covering theorem | Large cardinal | Cardinal number | Constructible universe | List of forcing notions | Tarski's undefinability theorem | Chang's conjecture | Ineffable cardinal | Baire space (set theory) | Cofinal (mathematics) | Cardinality | Turing degree | Erdős cardinal | Cofinality