建筑方案中的树木和空隙
Trees and gaps from a construction scheme
摘要：
我们提出了一个相当普遍的建设方案的树木和差距的自然建设。特别地，我们解决了一个关于$(ω_1, ω_1)$ -间隙的自然问题。众所周知，$(ω_1, ω_1)$ -间隙有时可以填充在$ω_1$ -保持力迫扩张的集合论域。有两种自然条件，称为$S$和$T$，保证了这种力迫扩张的存在。条件$T$是条件$S$的自然加强，并且是由$(ω_1,ω_1)$ -缺口和高度$ω_1.$的某些树之间的许多类比激发的。事实证明，条件$S$实际上等价于这种力迫扩张的存在，但是我们通过证明存在可填充的$(ω_1, ω_1)$ -缺口（即，S-间隙），但没有T-间隙。
英文摘要：
We present natural constructions of trees and gaps using a quite general construction scheme. In particular, we solve a natural problem about $(ω_1, ω_1)$-gaps. As it is well known $(ω_1, ω_1)$-gaps can sometimes be filled in $ω_1$-preserving forcing extensions of the set-theoretic universe. There are two natural conditions, dubbed $S$ and $T$ below, that guarantee the existence of such forcing extensions. The condition $T$ is a natural strengthening of the condition $S$ and was motivated by the numerous analogies between $(ω_1,ω_1)$-gaps and certain trees of height $ω_1.$ It turns out that the condition $S$ is in fact equivalent to the existence of such forcing extensions but we show that the condition $T$ is strictly stronger by proving that it is consistent that there are fillable $(ω_1, ω_1)$-gaps (i.e., S-gaps) but no T-gaps.
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。
