关于$Σ_1$ -和$Σ_2$ -归纳与单调枚举之间的原理
On principles between $Σ_1$- and $Σ_2$-induction, and monotone enumerations
摘要：
我们表明，许多原则的一阶算术，以前只知道严格之间的$Σ_1$ -诱导和$Σ_2$ -诱导，相当于良好的基础$ω^ω$。这些原理包括哈耶克和帕里斯的部分函数迭代原理（$PΣ_1$），钟、斯拉曼和杨的有界单调枚举原理（非迭代，BME $_1$），相对化的巴黎-哈林顿对原理，以及相对化的阿克曼-佩特函数的总体。有了这一点，我们表明，良好的基础$ω^ω$是一个更广泛的比通常怀疑。进一步，我们研究了有界单调迭代原理的$k$ -迭代版本（BME $_k$），并证明了它等价于$k+1$ -高度$ω$ -塔的适基性。
英文摘要：
We show that many principles of first-order arithmetic, previously only known to lie strictly between $Σ_1$-induction and $Σ_2$-induction, are equivalent to the well-foundedness of $ω^ω$. Among these principles are the iteration of partial functions ($PΣ_1$) of Hájek and Paris, the bounded monotone enumerations principle (non-iterated, BME$_1$) by Chong, Slaman, and Yang, the relativized Paris-Harrington principle for pairs, and the totality of the relativized Ackermann-Péter function. With this we show that the well-foundedness of $ω^ω$ is a far more widespread than usually suspected. Further, we investigate the $k$-iterated version of the bounded monotone iterations principle (BME$_k$), and show that it is equivalent to the well-foundedness of the $k+1$-height $ω$-tower.
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