\(\Pi^ 1_ 1\)-complete families of elementary sequences.

*(English)*Zbl 0646.03030In 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 |

##### Keywords:

models of set theory; well-foundedness; projective hierarchy; elementary embeddings of models of ZF; direct limits; Baire space
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\textit{P. Dehornoy}, Ann. Pure Appl. Logic 38, No. 3, 257--287 (1988; Zbl 0646.03030)

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##### References:

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