2阶纯模式
Pure patterns of order 2
摘要：
我们提供相互之间的经典序数记号的基本递归序同构，Skolem脱壳的基础上，和符号从纯粹的基本模式的相似顺序$2$，表明后者的特点证明理论的顺序片段$Π^1_1$ - $\mathrm{CA}_0$的二阶数论，或等价的集合论$\mathrm{KPl}_0$。作为推论，我们证明了Carlson关于$2$阶尊重森林的良拟序性的结果蕴涵了直到$\mathrm{KPl}_0$阶的超穷归纳法.我们希望我们的方法将有助于分析更强大的模式系统。
英文摘要：
We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order $2$, showing that the latter characterize the proof-theoretic ordinal of the fragment $Π^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order $2$ implies transfinite induction up to the ordinal of $\mathrm{KPl}_0$. We expect that our approach will facilitate analysis of more powerful systems of patterns.
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