In this note we consider Gentzen's first ordinal notation, used in his first published proof of the consistency of Peano Arithmetic (1936). It is a decimal notation, quite different from our current notations. We give a rule to translate this notation into our usual set-theoretic notation and we show some of its peculiarities. Then we indicate how to decode Gentzen's assignment of ordinal notations to derivations and give some examples. Finally, we go through his proof of their decrease after the application of his reduction procedure, giving further examples.

中文翻译：

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解码 Gentzen 的符号

在这篇笔记中，我们考虑 Gentzen 的第一个序数符号，在他第一次发表的 Peano Arithmetic (1936) 一致性证明中使用。它是一种十进制表示法，与我们目前的表示法完全不同。我们给出了一个规则来将这个符号转换成我们通常的集合论符号，并展示它的一些特殊性。然后我们指出如何解码 Gentzen 将序数符号分配给导数并给出一些例子。最后，我们通过他的证明在应用他的归约程序后他们的减少，给出了进一步的例子。