欧米茄逻辑的多面性

摘要：

我们考虑了“公式$ϕ$是$ω$ -逻辑的一个定理”的二阶算术语言的几种形式化，包括一些文献中已经研究过的形式化和一个通过最小不动点定义的新变体.我们分析了这些不同的形式化之间的关系，在标准理论的反向数学的可证明性。有了这个，我们研究了各种反射原理的强度所产生的这些概念的可证明性，调查已知的结果，并建立一些新的等价，包括一个表征的$Π^1_1$ - ${\sf CA}_0$在我们的不动点形式化的$ω$ -逻辑。

英文摘要：

We consider several formalizations in the language of second-order arithmetic of "The formula $ϕ$ is a theorem of $ω$-logic", including some which have been studied in the literature and a new variant defined via a least fixed point. We analyze the provability of relations between these different formalizations in standard theories of reverse mathematics. With this, we study the strength of various reflection principles arising from these notions of provability, surveying known results and establishing some new equivalences, including a characterization of $Π^1_1$-${\sf CA}_0$ in terms of our fixed-point formalization of $ω$-logic.

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