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The nonstationary ideal in the $$\mathbb P_{\max}$$ extension. (English) Zbl 1128.03044
The following are shown to hold in any $$\mathbb{P}_{\max}$$ extension of any model $$W$$ of $$\text{ZF}+ \text{DC}_{\mathbb R}+\text{AD}^+$$: (a) For any $$A\in P(\omega_2)\setminus W$$ there is $$\gamma<\omega_2$$ with $$A\cap\gamma\notin W$$; (b) No dense subset of $$P(\omega_1)/ NS_{\omega_1}$$ is order-isomorphic to a set in $$W$$; (c) $$P(\omega_1)/NS_{\omega_1}$$ is not the union of $$\aleph_1$$ many pairwise compatible sets; (d) For any sequence $$\langle Q_\alpha : \alpha<\omega_1\rangle$$ of partitions of $$\omega_1$$ into $$\aleph_1$$ many pieces there is a stationary subset of $$\omega_1$$ having nonstationary intersection with at least one member of each $$Q_\alpha$$; (e) Let $$X$$ be a family of stationary subsets of $$\omega_1$$ that has no size $$\aleph_1$$ subfamily with stationary diagonal intersection, then for every stationary subset $$S$$ of $$\omega_1$$ there is a size $$\aleph_1$$ collection $$Y$$ of stationary subsets of $$S$$ such that $$|\{A\in X : \forall B\in Y (A\cap B\notin NS_{\omega_1})\}\not=\aleph_2.$$ It is also shown that if $$P(\omega_1)/NS_{\omega_1}$$ is $$\aleph_2$$-c.c., then Axiom $$\text{F}^{+}$$ is equivalent to the assertion that $$NS_{\omega_1}| A$$ has caliber $$(\aleph_2,\aleph_0)$$ for some $$A$$ (Axiom $$\text{F}^{+}$$ states that for any $$f : \omega_2\times \omega_1\rightarrow\omega$$ there is $$n<\omega$$ and an infinite $$a\subseteq\omega_2$$ such that $$\bigcap_{\alpha\in a}\{\beta<\omega_1 : f(\alpha,\beta)=n\}$$ is stationary).

##### MSC:
 3e+35 Consistency and independence results 3e+60 Determinacy principles 3e+65 Other set-theoretic hypotheses and axioms
##### Keywords:
nonstationary ideal
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