论文标题
适合$ \ aleph_1 $尺寸的posets的适当强迫，$ω_1$互联在对称上适当的强迫和连续体的大小
The proper forcing axiom for $\aleph_1$-sized posets, $ω_1$-linked symmetrically proper forcing, and the size of the continuum
论文作者
论文摘要
我们表明，用于强制强迫尺寸$ \ aleph_1 $的概念的适当强迫与连续体任意大。实际上，假设$ gch $持有和$κ\geqΩ_2$是常规的红衣主教，我们证明有一个适当的和$ \ aleph_2 $ -c.c。 $ \ nathcal {h}（\ aleph_2）\ Models \存在yφ（a，y）$，其中$ a \ in \ mathcal {h}（\ aleph_2）$ and $φ（x，y）$是$ c $ m $ m $ $ $ ch $ $ ch a $ a pos a $ a \ a pos a $ a \ a p pos in a $ a $ a p pos，in具体而言，POSET $ \ MATHBB {Q} \ subseteq \ Mathcal {h}（κ）^m $，是$ω_1$ - 链接且对称地正确 - 添加一些$ b $，例如$φ（a，b）$。 In particular, $\mathbb{P}$ forces Moore's Measuring principle, Baumgartner's Axiom for $\aleph_1$-dense sets of reals, Todorčević's Open Colouring Axiom for sets of size $\aleph_1$, the Abraham-Rubin-Shelah Open Colouring Axiom, and Todorčević's P-ideal Dichotomy for $ \ aleph_1 $ - 在$ω_1$上的创立理想等等。因此，所有这些陈述都与大连续体同时兼容。最后，我们表明，除了所有较早的结论外，我们的建筑的进一步变化还产生了一个满足模型，这是Martin尺寸$ \ aleph_1 $的最大值。
We show that the Proper Forcing Axiom for forcing notions of size $\aleph_1$ is consistent with the continuum being arbitrarily large. In fact, assuming $GCH$ holds and $κ\geqω_2$ is a regular cardinal, we prove that there is a proper and $\aleph_2$-c.c.\ forcing giving rise to a model of this forcing axiom together with $2^{\aleph_0}=κ$ and which, in addition, satisfies all statements of the form $\mathcal{H}(\aleph_2)\models \exists yφ(a, y)$, where $a\in \mathcal{H}(\aleph_2)$ and $φ(x, y)$ is a $Σ_0$ formula with the property that for every ground model $M$ of $CH$ with $a\in M$ there is, in $M$, a suitably nice poset -- specifically, a poset $\mathbb{Q}\subseteq\mathcal{H}(κ)^M$ which is $ω_1$-linked and symmetrically proper -- adding some $b$ such that $φ(a, b)$. In particular, $\mathbb{P}$ forces Moore's Measuring principle, Baumgartner's Axiom for $\aleph_1$-dense sets of reals, Todorčević's Open Colouring Axiom for sets of size $\aleph_1$, the Abraham-Rubin-Shelah Open Colouring Axiom, and Todorčević's P-ideal Dichotomy for $\aleph_1$-generated ideals on $ω_1$, among other statements. Hence, all these statements are simultaneously compatible with a large continuum. Finally, we show that a further small variation of our construction yields a model satisfying, in addition to all the earlier conclusions, Martin's Maximum for posets of size $\aleph_1$.
