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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 a number 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. == Definition == Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols ''c''1, ''c''2, ... for each positive integer. Then 0# is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with ''c''''i'' interpreted as the uncountable cardinal ℵ''i''. (Here ℵ''i'' means ℵ''i'' 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 0# works provided that there is an uncountable set of indiscernibles for some ''L''α, and the phrase "0# exists" is used as a shorthand way of saying this. There are several minor variations of the definition of 0#, which make no significant difference to its properties. There are many different choices of Gödel numbering, and 0# depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode 0# as a subset of formulae of a language, or as a subset of the hereditarily finite sets, or as a real number. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Zero sharp」の詳細全文を読む スポンサード リンク
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