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The \(\Sigma^*\) approach to the fine structure of \(L\). (English) Zbl 0887.03039

The standard definition of the constructible universe \(L\) is through use of the sequence \(\langle L_{\alpha}\mid\alpha\in\text{ORD}\rangle\) based on iterated first order definability, with \(L_{\alpha+1}= \mathbf{Def} (L_{\alpha})\). Now \(L_{\alpha+1}\) is not closed under pairing, and R. B. Jensen [Ann. Math. Logic 4, 229-308 (1972; Zbl 0257.02035)] provided an alternative definition of \(L\) based on his hierarchy \(\langle J_{\alpha}\mid\alpha\in\text{ORD}\rangle\), which is closed under pairing. The paper under review starts with a new treatment of this \(J_{\alpha}\)-hierarchy, ensuring that \(J_{\alpha+1}\cap P(J_{\alpha}) =\mathbf{Def} (J_{\alpha})\), \(J_{\alpha+1}\) is closed under pairing, and \(J_{\alpha+1}\) satisfies \(\Sigma_0\)-Comprehension.
The author then proceeds to develop the fine structure theory for the \(J_{\alpha}\)-hierarchy, introducing a notion of \(\Sigma^*_n\) formula with the properties: (a) There is a universal \(\Sigma^*_n\) predicate for each \(n\), (b) For any \(X\subseteq J_{\alpha}\) the \(\Sigma^*_n\)-hull of \(X\) in \(J_{\alpha}\) exists for each \(n\), (c) There is a \(\Sigma^*_n\)- Skolem function for \(J_{\alpha}\) for each \(n\), (d) Every formula is \(\Sigma^*_n\) for some \(n\). This fine structure is used to prove the Square and the Fine Scale principles, and to develop several Morass principles.

MSC:

03E45 Inner models, including constructibility, ordinal definability, and core models
03E05 Other combinatorial set theory

Citations:

Zbl 0257.02035
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