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A partially non-proper ordinal beyond \(L(V_{\lambda +1})\). (English) Zbl 1270.03099

Summary: In his recent work, Woodin has defined new axioms stronger than I0 (the existence of an elementary embedding \(j\) from \(L(V_{\lambda +1})\) to itself), that involve elementary embeddings between slightly larger models. There is a natural correspondence between I0 and determinacy, but to extend this correspondence in the new framework we must insist that these elementary embeddings are proper. Previous results validated the definition, showing that there exist elementary embeddings that are not proper, but it was still open whether properness was determined by the structure of the underlying model or not. This paper proves that this is not the case, defining a model that generates both proper and non-proper elementary embeddings, and compare this new model to the older ones.

MSC:

03E55 Large cardinals
03E45 Inner models, including constructibility, ordinal definability, and core models
91A44 Games involving topology, set theory, or logic
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References:

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