论文标题
在集合和功能的牢固链上
On strong chains of sets and functions
论文作者
论文摘要
谢拉（Shelah）表明，没有长度为$ω_3$增加模型有限的链条中的$ {}^{ω_2}ω________________________2$。我们将此结果提高到集合。也就是说，我们表明，$ [ω_2]^{\ aleph_2} $增加modulo有限的长度$ω_3$没有长度$ω_3$。这与Koszmider的结果形成鲜明对比的是，Koszmider的结果表明，始终存在$ω_2$的链条，以$ [ω_1]^{\ Aleph_1} $增加Modulo有限，以及$ {}^{}^{ω_1}ω____________________________________________________________________________________________________$。更笼统地，我们研究了功能空间的深度$ {}^κμ$，由理想的$ [κ]^{<θ} $ sotionaldience，其中$θ<κ$是无限的红衣主教。
Shelah has shown that there are no chains of length $ω_3$ increasing modulo finite in ${}^{ω_2}ω_2$. We improve this result to sets. That is, we show that there are no chains of length $ω_3$ in $[ω_2]^{\aleph_2}$ increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length $ω_2$ increasing modulo finite in $[ω_1]^{\aleph_1}$ as well as in ${}^{ω_1}ω_1$. More generally, we study the depth of function spaces ${}^κμ$ quotiented by the ideal $[κ]^{< θ}$ where $θ< κ$ are infinite cardinals.
