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Forcing indestructibility of set-theoretic axioms. (English) Zbl 1128.03043
It is shown that strongly \((\omega_1+1)\)-game-closed forcings preserve MM(\(\Gamma)\), where \(\Gamma\) denotes the class of all posets \(\mathbb{Q}\) such that (a) \(\mathbb{Q}\) preserves stationary subsets of \(\omega_1,\) and (b) \(\mathbb{Q} = \mathbb{Q}_0\ast\mathbb{Q}_1,\) where \(\mathbb Q_0\) is \(\aleph_1\)-distributive and \(\Vdash_{\mathbb Q_0}| \mathbb Q_1| \leq\aleph_1.\) This and other preservation results are then used to prove the (relative) consistency of the following statements: (A) \(\text{MM}(\aleph_{\omega}) + \text{MM}(\Gamma) + \text{AP}_{\aleph_{\omega}}\), (B) \(\text{BPFA} + \text{PFA}(\Gamma)+\text{AP}_{\aleph_1},\) (C) “\(\omega_2\) is generically supercompact by \(\sigma\)-closed forcing”\( + \text{AP}_{\aleph_\omega},\) (D) \(\text{MM} + \text{``}\omega_\omega\) is not Jónsson”, and (E) \(\text{MM}+ \text{``}(\omega_{m+1},\omega_m) \twoheadrightarrow (\omega_{n+1},\omega_n)\) for all \(1<n<m\)”.

MSC:
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
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