Ordinal Definable Set (Paperback)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first order formula. Ordinal definable sets were introduced by G del (1965). A drawback to this informal definition is that requires quantification over all first order formulas, which cannot be formalized in the language of set theory. However there is a different way of stating this which can be so formalized. By this way, a set S is formally defined to be ordinal definable if there some collection of ordinals 1... n such that S isin V_{alpha_1} and can be defined there by a first-order formula taking 1... n as parameters. Here V_{alpha_1} denotes the set indexed by the ordinal 1 in the von Neumann hierarchy of sets. In other words, S is the unique object such that (S, 1... n) holds with its quantifiers ranging over V_{alpha_1}.