Editing Zero sharp (section)

== Definition ==

Zero sharp was defined by Silver and [[Robert M. Solovay|Solovay]] as follows. Consider the language of set theory with extra constant symbols <math>c_1</math>, <math>c_2</math>, ... for each nonzero natural number. Then <math>0^\sharp</math> is defined to be the set of [[Gödel number]]s of the true sentences about the constructible universe, with <math>c_i</math> interpreted as the uncountable cardinal <math>\aleph_i</math>.
(Here <math>\aleph_i</math> means <math>\aleph_i</math> in the full universe, not the constructible universe.)

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. More generally, the definition of <math>0^\sharp</math> works provided that there is an uncountable set of indiscernibles for some <math>L_\alpha</math>, and the phrase "<math>0^\sharp</math> exists" is used as a shorthand way of saying this.

A closed set <math>I</math> of [[indiscernibles|order-indiscernibles]] for <math>L_\alpha</math> (where <math>\alpha</math> is a limit ordinal) is a set of ''Silver indiscernibles'' if:
*<math>I</math> is unbounded in <math>\alpha</math>, and
*if <math>I\cap\beta</math> is unbounded in an ordinal <math>\beta</math>, then the [[Skolem hull]] of <math>I\cap\beta</math> in <math>L_\beta</math> is <math>L_\beta</math>. In other words, every <math>x\in L_\beta</math> is definable in <math>L_\beta</math> from parameters in <math>I\cap\beta</math>.

If there is a set of Silver indiscernibles for <math>L_{\omega_1}</math>, then it is unique. Additionally, for any uncountable cardinal <math>\kappa</math> there will be a unique set of Silver indiscernibles for <math>L_\kappa</math>. The union of all these sets will be a proper class <math>I</math> of Silver indiscernibles for the structure <math>L</math> itself. Then, <math>0^\sharp</math> is defined as the set of all Gödel numbers of formulae <math>\theta</math> such that

<math>L_\alpha\models\theta(\alpha_1,\alpha_2\ldots\alpha_n)</math>

where <math>\alpha_1 < \alpha_2 < \ldots < \alpha_n < \alpha</math> is any strictly increasing sequence of members of <math>I</math>. Because they are indiscernibles, the definition does not depend on the choice of sequence.

Any <math>\alpha\in I</math> has the property that <math>L_\alpha\prec L</math>. This allows for a definition of truth for the constructible universe:

<math>L\models\varphi[x_1...x_n]</math> only if <math>L_\alpha\models\varphi[x_1...x_n]</math> for some <math>\alpha\in I</math>.

There are several minor variations of the definition of <math>0^\sharp</math>, which make no significant difference to its properties. There are many different choices of Gödel numbering, and <math>0^\sharp</math> depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode <math>0^\sharp</math> as a subset of formulae of a language, or as a subset of the hereditarily finite sets, or as a real number.