论文标题
强大的向下löwenheim-skolem定理用于固定逻辑，ii-反射到连续体
Strong downward Löwenheim-Skolem theorems for stationary logics, II -- reflection down to the continuum
论文作者
论文摘要
继续上一篇论文，我们研究了固定逻辑及其变化的强大向下löwenheim-skolems（SDLSS）。已经表明，具有较弱的二阶参数的普通固定逻辑的SDL降低到$ <\ aleph_2 $等于CH和Cox的对角线反射原理的内部俱乐部的连接。我们证明，固定逻辑的SDL无弱二阶参数降低至$ <2^{\ aleph_0} $，这意味着连续体的大小为$ \ aleph_2 $。相比之下，对固定逻辑的内部解释可以使SDL在连续体下的$> \ aleph_2 $下的$ <2^{\ aleph_0} $。该SDL被证明等于对角线反射原理的内部版本，直至内部固定的大小$ <2^{\ aleph_0} $。 We also consider a ${\cal P}_κλ$ version of the stationary logic and show that the SDLS for this logic in internal interpretation for reflection down to $<2^{\aleph_0}$ is consistent under the assumption of the consistency of ZFC $+$ "the existence of a supercompact cardinal" and this SDLS implies that the continuum is (at least) weakly Mahlo.就SDL而言，这三个“公理”是加强通用超紧凑性的三个实例的后果，我们称之为Laver-Generic-Generic SuperCompactness。在这三个实例中的每一个中，都存在Laver-Generic-Generic SuperCompact Cardinal，还将连续体的基数固定为$ \ aleph_1 $或$ \ Aleph_2 $或非常大。我们还表明，这些通用大型红衣主教之一的存在意味着相应的强迫型公理的“ $ ++ $”版本。
Continuing the previous paper, we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. It has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters down to $<\aleph_2$ is equivalent to the conjunction of CH and Cox's Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak second-order parameters down to $<2^{\aleph_0}$ implies that the size of the continuum is $\aleph_2$. In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to $<2^{\aleph_0}$ under the continuum being of size $>\aleph_2$. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size $<2^{\aleph_0}$. We also consider a ${\cal P}_κλ$ version of the stationary logic and show that the SDLS for this logic in internal interpretation for reflection down to $<2^{\aleph_0}$ is consistent under the assumption of the consistency of ZFC $+$ "the existence of a supercompact cardinal" and this SDLS implies that the continuum is (at least) weakly Mahlo. These three "axioms" in terms of SDLS are consequences of three instances of a strengthening of generic supercompactness which we call Laver-generic supercompactness. Existence of a Laver-generic supercompact cardinal in each of these three instances also fixes the cardinality of the continuum to be $\aleph_1$ or $\aleph_2$ or very large respectively. We also show that the existence of one of these generic large cardinals implies the "$++$" version of the corresponding forcing axiom.
