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The logic of arithmetical hierarchy. (English) Zbl 0804.03045
Formulas of the propositional modal language with the unary modal operators $\square$, $\Sigma\sb 1, \bbfB\sb 1, \Sigma\sb 2, \bbfB\sb 2$, etc. are considered as schemata of sentences of arithmetic (PA), where $\square A$ is interpreted as (a formalization of) “$A$ is PA- provable”, $\Sigma\sb nA$ as “$A$ is PA-equivalent to a $\Sigma\sb n$- sentence” and $\bbfB\sb n A$ as “$A$ is PA-equivalent to a Boolean combination of $\Sigma\sb n$-sentences”. We give an axiomatization and show decidability of the sets of the modal formulas which are schemata of: (1) PA-provable, (2) true arithmetical sentences.
Reviewer: G.Dzhaparidze
MSC:
03F30First-order arithmetic and fragments
03B45Modal logic, etc.
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References:
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