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$$\Pi^ 1_ 1$$-complete families of elementary sequences. (English) Zbl 0646.03030
In this paper the author investigates elementary embeddings of models of ZF set theory, and their direct limits. A sequence of elementary embeddings is called elementary if their direct limit is well-founded. Let $$\Gamma$$ be a point class in the Baire space $$\omega^{\omega}$$. A family $$(I_{\alpha})_{\alpha \in \omega^{\omega}}$$ of elementary embeddings is $$\Gamma$$-complete iff the set of all $$\alpha$$-s such that $$I_{\alpha}$$ has a well-founded limit is $$\Gamma$$-complete. Let j be an elementary embedding of a model of ZF. Then this j is used to define for each $$\gamma \in \omega^{\omega}$$ two sequences of elementary embeddings in a natural way. It is shown that the corresponding families $$(I_{\alpha})_{\alpha \in \omega^{\omega}}$$ are $$\Pi^ 1_ 1$$- complete.
Reviewer: M.Weese

##### MSC:
 03C62 Models of arithmetic and set theory
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##### References:
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