The logic of arithmetical hierarchy (Q1315831)

Formulas of the propositional modal language with the unary modal operators \(\square\), \(\Sigma_ 1, \mathbb{B}_ 1, \Sigma_ 2, \mathbb{B}_ 2\), etc. are considered as schemata of sentences of arithmetic (PA), where \(\square A\) is interpreted as (a formalization of) ``\(A\) is PA- provable'', \(\Sigma_ nA\) as ``\(A\) is PA-equivalent to a \(\Sigma_ n\)- sentence'' and \(\mathbb{B}_ n A\) as ``\(A\) is PA-equivalent to a Boolean combination of \(\Sigma_ n\)-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.