Michael Rathjen and the present author have shown that $\Pi^1_1$-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 $f(\alpha)\leq 1+\alpha^2$ and $f'(\alpha)=\omega^{\omega^\alpha}$. We will also exhibit a normal function $g$ with $g(\alpha)\leq 1+\alpha\cdot 2$ and $g'(\alpha)=\omega^{1+\alpha}$.

中文翻译：

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关于正规函数的序数幂和导数的注记

Michael Rathjen 和现在的作者已经证明 $\Pi^1_1$-bar 归纳等价于（一个合适的形式化）每个正常函数都有导数的陈述，可以证明在 $\mathbf{ACA_0}$ 中。在本笔记中，我们表明基础理论可以弱化为 $\mathbf{RCA_0}$。我们的论证关键地使用了普通函数 $f$ 和 $f(\alpha)\leq 1+\alpha^2$ 和 $f'(\alpha)=\omega^{\omega^\alpha}$。我们还将展示具有 $g(\alpha)\leq 1+\alpha\cdot 2$ 和 $g'(\alpha)=\omega^{1+\alpha}$ 的普通函数 $g$。