平稳逻辑的强向下Löwenheim-Skolem定理，II --向下映射到连续统
Strong downward Löwenheim-Skolem theorems for stationary logics, II -- reflection down to the continuum
摘要：
在前一篇文章的基础上，我们研究了平稳逻辑的强向下Löwenheim-Skolem定理（SDLS）及其变体。本文证明了弱二阶参数下至$<\aleph_2$的一般平稳逻辑的SDLS等价于CH与Cox的对角反射原理的结合。我们证明了无弱二阶参数的平稳逻辑的SDLS下降到$<2^{\aleph_0}$意味着连续统的大小是$\aleph_2$。相反，静态逻辑的内部解释可以在连续体的大小为$>\aleph_2$的情况下满足SDLS下至$<2^{\aleph_0}$。这SDLS被证明是相当于一个内部版本的对角反射原理下降到一个内部固定的一套大小$<2^{\aleph_0}$。我们还考虑了一个${\cal P}_κλ$版本的静态逻辑，并表明，SDLS为这个逻辑在内部解释反射下降到$<2^{\aleph_0}$是一致的假设下的ZFC $+$的一致性“的超紧基数的存在”，这SDLS意味着连续（至少）弱Mahlo。这三个“公理”的SDLS是后果的三个实例加强一般超紧性，我们称之为拉弗一般超紧性。在这三种情况下，如果存在拉弗-泛属超紧基数，则连续统的基数将分别固定为$\aleph_1$或$\aleph_2$或非常大。我们还表明，这些通用的大基数之一的存在意味着“$++$“版本的相应的强制公理。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
