ON THE INEVITABILITY OF THE CONSISTENCY OPERATOR
ON THE INEVITABILITY OF THE CONSISTENCY OPERATOR
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论一致性算子的必然性
DOI：
10.1017/jsl.2018.65
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
J. Walsh
中科院分区：
文献类型：
--
作者：
A. Montalbán;J. Walsh
摘要研究了$EA$的Lindenbaum代数上的递归单调函数。我们证明了没有这样的函数将每一个一致的φ严格地赋给一个在φ和$\left( {\varphi \wedge Con\left( \varphi \right)} \right)$之间具有演绎强度的句子。我们将这一结果推广到有效超限的一致性迭代。然后我们证明了对于任何递归单调函数f，如果存在一个$Con$的迭代，它在任何地方都有f的边界，那么f一定在某个地方等于$Con$的迭代。
Abstract We examine recursive monotonic functions on the Lindenbaum algebra of $EA$. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and $\left( {\varphi \wedge Con\left( \varphi \right)} \right)$. We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function f, if there is an iterate of $Con$ that bounds f everywhere, then f must be somewhere equal to an iterate of $Con$.