Description:
We give the definitions for Form 230 (\( L^{1} = HOD\)) from Myhill/Scott [1971]
Content:
We give the definitions for Form 230 (\( L^{1} = HOD\)) from Myhill/Scott [1971].
Definition:
- \(DF^{0}(A)\) is the set of all subsets of \(A\) definable by first order formulas in \(\langle A,\in _{A}\rangle \) using parameters from \(A\), i.e., \(DF^{0}(A)\) is all sets of the form \(\{ x \in A :\Phi^{(A)}(x,a_{0},\ldots a_{n-1}) \}\) where \(\Phi \) is a formula of the type described above, \(a_{0}, \ldots ,a_{n-1}\) are in \(A\) and \(\Phi ^{(A)}\) means all variables range over \(A\).
- Gödel's constructible sets are defined by \(L^{0}_{\alpha }=\bigcup^{}_{\beta <\alpha }DF^{0}(L^{0}_{\beta })\) and\(L^{0} =\bigcup^{}_{\alpha \in \hbox{On}}L^{0}_{\alpha }\).
- \(DF^{1}(A)\) consists of all sets of the form \(\{x\in A:\Phi (x,a_{0},\ldots ,a_{n-1}) \}\) where \(a_{0}, \ldots , a_{n-1}\)are in \(A\) and the quantifiers are restricted to \(A\) or \({\cal P}(A)\).
- \(L^{1}_{\alpha } = \bigcup^{}_{\beta <\alpha }DF^{1}(L^{1}_{\beta})\) and \(L^{1} = \bigcup^{}_{\alpha\in\hbox{On}}L^{1}_{\alpha}\).
- \(HOD\) is the class of hereditarily ordinal definable sets.
Howard-Rubin number:
82
Type:
Definitions
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