Consider a normal function f on the ordinals (i. e. a function f that is strictly increasing and continuous at limit stages). By enumerating the fixed points of f we obtain a faster normal function $f^{\prime }$, called the derivative of f. The present paper investigates this important construction from the viewpoint of reverse mathematics. Within this framework we must restrict our attention to normal functions $f:{\mathrm{\aleph }}_1\to {\mathrm{\aleph }}_1$ that are represented by dilators (i. e. particularly uniform endofunctors on the category of well-orders, as introduced by J.-Y. Girard). Due to a categorical construction of P. Aczel, each normal dilator T has a derivative ∂T. We will give a new construction of the derivative, which shows that the existence and fundamental properties of ∂T can already be established in the theory ${\mathbf{RCA}}_0$. The latter does not prove, however, that ∂T preserves well-foundedness. Our main result shows that the statement “for every normal dilator T, its derivative ∂T preserves well-foundedness” is ${\mathbf{ACA}}_0$-provably equivalent to ${\mathrm{\Pi }}_1^1$-bar induction (and hence to ${\mathrm{\Sigma }}_1^1$-dependent choice and to ${\mathrm{\Pi }}_2^1$-reflection for ω-models).

考虑序数上的正态函数f（即严格增加并在极限阶段连续的函数f）。通过枚举f的不动点，我们可以获得更快的法线函数$F^{\prime }$，称为f的导数。本文从逆向数学的角度研究了这一重要构造。在此框架内，我们必须将注意力集中在正常功能上$F：{\mathrm{\aleph }}_{\mathrm{1个}}\to {\mathrm{\aleph }}_{\mathrm{1个}}$这些由扩张器表示（例如，由J.-Y. Girard提出的井号类别中的统一内爆函数）。由于P. Aczel的分类结构中，每一个正常扩张器Ť具有衍生物∂ Ť。我们将给它们的衍生物，这表明∂的存在和基本性能的新型建筑牛逼已经可以在理论建立${\mathbf{RCA}}_0$。后者并不能证明，但是，∂牛逼保持良好foundedness。我们的主要结果表明，该声明“为每一个正常的扩张牛逼，其衍生∂牛逼蜜饯精心foundedness”是${\mathbf{ACA}}_0$-等同于 ${\mathrm{\Pi }}_{\mathrm{1个}}^{\mathrm{1个}}$-bar感应（因此 ${\mathrm{\Sigma }}_{\mathrm{1个}}^{\mathrm{1个}}$依赖的选择和 ${\mathrm{\Pi }}_2^{\mathrm{1个}}$-反射的ω -models）。