Following the Crisis of Foundations Hilbert proposed to consider a finitistic form of arithmetic as mathematics’ safe core. This approach to finitism has often admitted primitive recursive function definitions as obviously finitistic, but some have advocated the inclusion of additional variants of recurrence, while others argued that, to the contrary, primitive recursion exceeds finitism. In a landmark essay, William Tait contested the finitistic nature of these extensions, due to their impredicativity, and advocated identifying finitism with primitive recursive arithmetic, a stance often referred to as Tait’s Thesis. However, a problem with Tait’s argument is that the recurrence schema has itself impredicative and non-finitistic facets, starting with an explicit reference to the functions being defined, which are after all infinite objects. It is therefore desirable to buttress Tait’s Thesis on grounds that avoid altogether any trace of concrete infinities or impredicativity. We propose here to do just that, building on the generic framework of [ 13]. We provide further evidence for Tait’s Thesis by outlining a proof of a purely finitistic version of Parsons’ theorem, whose intuitive gist is that finitistic reasoning is equivalent to finitistic computing.

中文翻译：

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有限主义、命令式程序和原始递归

在基础危机之后，希尔伯特提议将一种有限形式的算术视为数学的安全核心。这种有限主义的方法经常承认原始递归函数定义显然是有限主义的，但有些人主张包含额外的递归变体，而另一些人则认为，相反，原始递归超越了有限主义。在一篇具有里程碑意义的文章中，William Tait 对这些扩展的有限主义性质提出了质疑，因为它们具有不可预测性，并主张将有限主义与原始递归算术相结合，这种立场通常被称为 Tait 的论文。然而，Tait 的论点存在一个问题，即递归模式本身具有不可预测性和非限定性方面，首先是对正在定义的函数的显式引用，这些函数毕竟是无限对象。因此，最好在完全避免任何具体的无限性或不可预测性的痕迹的基础上支持泰特的论点。我们建议在 [13] 的通用框架的基础上做到这一点。我们通过概述帕森斯定理的纯有限版本的证明来为泰特的论文提供进一步的证据，其直观的要点是有限推理等同于有限计算。