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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:
03E35 Consistency and independence results
03E60 Determinacy principles
03E65 Other set-theoretic hypotheses and axioms
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