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arXiv · 1804.09451

Provability Logic and the Completeness Principle

Albert Visser·Jetze Zoethout
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Abstract

In this paper, we study the provability logic of intuitionistic theories of arithmetic that prove their own completeness. We prove a completeness theorem for theories equipped with two provability predicates $\Box$ and $\triangle$ that prove the schemes $A\to\triangle A$ and $\Box\triangle S\to\Box S$ for $S\in\Sigma_1$. Using this theorem, we determine the logic of fast provability for a number of intuitionistic theories. Furthermore, we reprove a theorem previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the $\Sigma_1$-provability logic of Heyting Arithmetic.

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Albert Visser, Jetze Zoethout. 2018-04-25. Provability Logic and the Completeness Principle. https://arxiv.org/abs/1804.09451

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