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Combinatorial principles in the core model for one Woodin cardinal. (English) Zbl 0834.03018
Summary: We study the fine structure of the core model for one Woodin cardinal, building on the work of Mitchell and Steel on inner models of the form \(L[\vec E]\). We generalize to \(L[\vec E]\) some combinatorial principles that were shown by Jensen to hold in \(L\). We show that \(L[\vec E]\) satisfies the statement: “\(\square_\kappa\) holds whenever \(\kappa\leq\) the least measurable cardinal \(\lambda\) of \(\vartriangleleft\) order \(\lambda^{++}\)”. We introduce a hierarchy of combinatorial principles \(\square_{\kappa, \lambda}\) for \(1\leq \lambda\leq \kappa\) such that \[ \square_\kappa \iff \square_{\kappa,1} \Rightarrow \square_{\kappa, \lambda} \Rightarrow \square_{\kappa, \kappa} \iff \square^*_\kappa. \] We prove that if \((\kappa^+ )^V= (\kappa^+ )^{L [\vec E]}\), then \(\square_{\kappa, \text{cf}(\kappa)}\) holds in \(V\). As an application, we show that \(\text{ZFC} + \text{PFA} \Rightarrow \text{Con(ZFC}+\)“there is a Woodin cardinal”). We also obtain one Woodin cardinal as a lower bound on the consistency strength of stationary reflection at \(\kappa^+\) for a singular, countably closed limit cardinal \(\kappa\) such that \((V_{\kappa^+} )^\#\) exists; likewise for the failure of \(\square^*_\kappa\) at such a \(\kappa\).

03E55 Large cardinals
03E35 Consistency and independence results
03E05 Other combinatorial set theory
Full Text: DOI
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