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Concept
Zero sharp
Summary
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 , 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.
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 ci interpreted as the uncountable cardinal .
(Here means in the full universe, not the constructible universe.)
If there is in V an uncountable set of Silver order-indiscernibles in the constructible universe L, then 0# is the set of Gödel numbers of formulas θ of set theory such that
where ω1, ... ωω are the "small" uncountable initial ordinals in V, but have all large cardinal properties consistent with V=L relative to L.
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.
Official source
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