In the context of reverse mathematics, effective transfinite recursion refers to a principle that allows us to construct sequences of sets by recursion along arbitrary well orders, provided that each set is ${\mathrm{\Delta }}_1^0$‐definable relative to the previous stages of the recursion. It is known that this principle is provable in ${\mathsf{ACA}}_0$. In the present note, we argue that a common formulation of effective transfinite recursion is too restrictive. We then propose a more liberal formulation, which appears very natural and is still provable in ${\mathsf{ACA}}_0$.

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什么是逆数学中的有效超限递归？

在逆数学的上下文中，有效的超限递归是指一种原理，该条件允许我们通过沿任意井阶的递归构造构造序列集，前提是每个集合都是 ${\mathrm{\Delta }}_{\mathrm{1个}}^0$相对于递归的先前阶段是可定义的。众所周知，这一原理在${\mathsf{ACA}}_0$。在本说明中，我们认为有效的超限递归的通用表述过于严格。然后，我们提出一种更为宽松的表述，该表述看起来很自然，并且在${\mathsf{ACA}}_0$。