On the completenes principle: A study of provability in heyting's arithmetic and extensions

Annals of Mathematical Logic Pub Date : 1982-08-01 DOI:10.1016/0003-4843(82)90024-9

Albert Visser

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引用次数: 53

Abstract

In this paper extensions of HA are studied that prove their own completeness, i.e. they prove A → □ A, where □ is interpreted as provability in the theory itself. Motivation is three-fold: (1) these theories are thought to have some intrinsic interest, (2) they are a tool for producing and studying provability principles, (3) they can be used to proved independence results. Work done in the paper connected with these motivations is respectively:

1.
(i) A characterization is given of theories proving their own completeness, including an appropriate conservation result.
2.
(ii) Some new provability principles are produced. The provability logic of HA is not a sublogic of the of PA. A provability logic plus completeness theorem is given for a certain intuitionistic extension of HA. De Jongh's theorem for propositional logic is a corollary.
3.
(iii) FP-realizability in Beeson's proof that $∦_{HA}$ KLS is replaced by theories proving their own completeness. New consequences are $∦_{HA+−M_{PR}} KLS$ , $∦_{HA+DNS} KLS$ .

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完备原理:heyting算法及其扩展的可证明性研究

本文研究了证明自身完备性的HA的扩展，即证明了A→□A，其中□被解释为理论本身的可证明性。动机有三个方面:(1)这些理论被认为具有某种内在的利益，(2)它们是产生和研究可证明性原则的工具，(3)它们可用于证明独立性的结果。本文所做的与这些动机有关的工作分别是:1.(i)给出了证明其完备性的理论的一个表征，包括一个适当的守恒结果。2.(ii)给出了一些新的可证明性原理。HA的可证明性逻辑不是PA的子逻辑。对于HA的某一直觉扩展，给出了一个可证性逻辑加完备性定理。3.(iii) Beeson证明∦HA KLS被证明其自身完备性的理论所取代的fp -可实现性。新的结果是∦HA+−MPR KLS，∦HA+DNS KLS。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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