关于正规函数的序数幂和导数的一个注记
A note on ordinal exponentiation and derivatives of normal functions
摘要：
Michael Rathjen和本作者已经证明了$Π^1_1$ -bar归纳等价于（一个合适的形式化）每个正规函数都有导数的陈述，在$\mathbf{ACA_0}$中是可证明的。本文证明了基理论可以弱化为$\mathbf{RCA_0}$。我们的论证关键地使用了一个普通函数$f$和$f(α)\leq 1+α^2$和$f'(α)=ω^{ω^α}$。我们还将展示具有$g(α)\leq 1+α\cdot 2$和$g'(α)=ω^{1+α}$的正常功能$g$。
英文摘要：
Michael Rathjen and the present author have shown that $Π^1_1$-bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in $\mathbf{ACA_0}$. In this note we show that the base theory can be weakened to $\mathbf{RCA_0}$. Our argument makes crucial use of a normal function $f$ with $f(α)\leq 1+α^2$ and $f'(α)=ω^{ω^α}$. We will also exhibit a normal function $g$ with $g(α)\leq 1+α\cdot 2$ and $g'(α)=ω^{1+α}$.
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