# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Approachability and games onposets 68 pp 589–606– (2003)  (2005)  Mathematical Logic Quarterly 50 pp 297–302– (2004)  Archive for Mathematical Logic 43 pp 311–326– (2004)  The Higher Infinite (1997)  Annals of Mathematics 127 pp 1–47– (1988)  Games played on Boolean algebras 48 pp 714–723– (1983)  Journal of Mathematical Logic 1 pp 35–98– (2001)  Set theory (Curacao, 1995; Barcelona, 1996) pp 23–39– (1998)  Handbook of set-theoretic topology pp 913–959–  Archive for Mathematical Logic 39 pp 393–401– (2000)  Advances in Mathematics 94 pp 256–284– (1992)  Dimacs series 58 pp 135–148– (2002)  Proper and Improper Forcing (1998) · Zbl 0889.03041  Proper forcing 940 (1982)  Logic colloquium ’78 97 pp 357–380– (1979)  Journal of Mathematical Logic 5 pp 87–97– (2005)  Separating stationary reflection principles 65 pp 247–258– (2000)  Set theory, an introduction to independence proofs (1980) · Zbl 0443.03021  Saturated ideals 43 pp 65–76– (1978)  The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (1999) · Zbl 0954.03046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.