论文标题
某些紧凑性原理在PFA模型上的坚不可摧
Indestructibility of some compactness principles over models of PFA
论文作者
论文摘要
We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $ω$ will not add an $ω_2$-Aronszajn tree or a weak $ω_1$-Kurepa tree, and moreover no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree (a tree of height and size $ω_1$，至少$ω_2$ COFINAL分支）。这部分回答了一个开放问题，CCC强制是否可以添加$ω_2$ -ARONSZAJN或$ω_1$ -KUREPA树。 我们实际上证明了更多：我们证明了$ \ mathsf {pfa} $的后果，即猜测模型原理，$ \ mathsf {gmp} $，这与不可效率的细长树属性，$ \ mathsf {isp} $相当，可以保留任何COHENSESES $ $ COHENSESES $ω$。此外，$ \ mathsf {gmp} $意味着没有$σ$以上的强迫可以添加弱$ω_1$ -KUREPA树。 为了获得更多的普遍性，我们以任意的常规红衣主教$κ=κ=κ^{<κ} $研究原理$ \ mathsf {gmp} $（我们表示此原理$ \ m arthsf {gmp} _ {κ{κ^{++}} $） $ \ aleph_ {ω+1} $ - kurepa树，没有$ \ aleph_ {ω+2} $ - aronszajn树。
We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $ω$ will not add an $ω_2$-Aronszajn tree or a weak $ω_1$-Kurepa tree, and moreover no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree (a tree of height and size $ω_1$ with at least $ω_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add $ω_2$-Aronszajn or $ω_1$-Kurepa trees. We actually prove more: We show that a consequence of $\mathsf{PFA}$, namely the guessing model principle, $\mathsf{GMP}$, which is equivalent to the ineffable slender tree property, $\mathsf{ISP}$, is preserved by adding any number of Cohen subsets of $ω$. And moreover, $\mathsf{GMP}$ implies that no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree. For more generality, we study the principle $\mathsf{GMP}$ at an arbitrary regular cardinal $κ= κ^{<κ}$ (we denote this principle $\mathsf{GMP}_{κ^{++}}$), and as an application we show that there is a model in which there are no weak $\aleph_{ω+1}$-Kurepa trees and no $\aleph_{ω+2}$-Aronszajn trees.
