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.
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 the numeral '0').
Zero sharp was defined by Silver and Solovay as follows.
Consider the language of set theory with extra constant symbols
is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with
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.
works provided that there is an uncountable set of indiscernibles for some
there will be a unique set of Silver indiscernibles for
The union of all these sets will be a proper class
is defined as the set of all Gödel numbers of formulae
is any strictly increasing sequence of members of
Because they are indiscernibles, the definition does not depend on the choice of sequence.
, which make no significant difference to its properties.
There are many different choices of Gödel numbering, and
Instead of being considered as a subset of the natural numbers, it is also possible to encode
The condition about the existence of a Ramsey cardinal implying that
Donald A. Martin and Leo Harrington have shown that the existence of
is equivalent to the determinacy of lightface analytic games.
In fact, the strategy for a universal lightface analytic game has the same Turing degree as
It follows from Jensen's covering theorem that the existence of
Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of
implies that every uncountable cardinal in the set-theoretic universe
and satisfies all large cardinal axioms that are realized in
This is in some sense the simplest possibility for a non-constructible set, since all
is the core model—that is, the canonical inner model that approximates the large cardinal structure of the universe considered.
In that case, Jensen's covering lemma holds: This deep result is due to Ronald Jensen.
For example, consider Namba forcing, that preserves
does not exist, it also follows that the singular cardinals hypothesis holds.[1]p.