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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 $$\kappa$$.

##### MSC:
 3e+55 Large cardinals 3e+35 Consistency and independence results 300000 Other combinatorial set theory
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##### References:
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