A theorem on the second-order arithmetic with the $\omega$ -rule*)
A theorem on the second-order arithmetic with the $\omega$ -rule*)
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具有 $omega$ -规则*的二阶算术定理*)
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
M. Takahashi
中科院分区：
文献类型：
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作者：
Moto;M. Takahashi
$\omega$ -rule是递归限制的。另一方面，具有$\omega$ -规则的二阶算术是不完备的。因此，Shoenfield提出了一个问题，即是否可以用$\omega$-规则证明的二阶算术的每个句子都可以用递归限制的$\omega$-规则证明。针对这一问题，H. Tanaka [4]证明了用$\omega$ -规则可证明的二阶算术的每个句子都能用超算术限制的$\omega$-规则证明。本文的目的是对上述Shoenfield问题给予肯定的回答。这个结果可以推广到对应于任何高阶算术的结果。我们将在[2]中使用符号和术语。大致说来，证明的大纲如下.对于给定的公式
$\omega$ -rule is recursively restricted. On the other hand, the second-order arithmetic with the $\omega$ -rule is not complete. So Shoenfield has raised a question whether every sentence of the second-order arithmetic provable with the $\omega$-rule is provable with the recursively restricted $\omega$-rule. Concerning this problem, H. Tanaka [4] has shown that every sentence of the second-order arithmetic provable with the $\omega$ -rule is provable with the hyper-arithmetically restricted $\omega$-rule. The purpose of this paper is to give an affirmative answer to Shoenfield’s problem stated above. This result can be extended to one corresponding to any higher order arithmetic. We shall use notations and terminologies in [2]. Roughly speaking, the outline of proof is as follows. For a given formula