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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:
 3e+35 Consistency and independence results 3e+50 Continuum hypothesis and Martin’s axiom
##### Keywords:
forcing axioms; transfer principles
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