A modal logic of consistency
Fraenkel-Mostowski permutation models of the set theory ZFA (it admits atoms) are proper classes of hereditarily symmetric sets, where symmetry is defined with respect to a topological group. If the construction of a permutation model is performed within another permutation model, then the resulting structure is isomorphic to a permutation model (Theorem 1) and the following result follows (Theorem 2): A modal propositional formula \(\varphi\) is in S5 if and only if \(\varphi\) is \(\text{FM}_{\text{FM}}\)-valid. This means that for all interpretations \((\cdot)^*\) the ZF-formula \(\varphi^*\) is true in all permutation models over a fixed ZFC universe \(V\). The key clause in the definition of \((\cdot)^*\) is the interpretation of the modal operator \(M\) of possibility: \((M\varphi)^*\) expresses that some permutation model satisfies \(\varphi^*\). It follows that in contrast to the case where the models are sets, the formalized second incompleteness theorem is not valid. Reviewer's remarks: (1) In the proof of Lemma 2.1 the embedding of \(V(X)(Y)\) is not into \(V(Y)\) but into \(V(Z_Y)\), where \(Z_Y\in V\) is the first coordinate of \(Y= (Z_Y, \alpha)\in V(X)\); otherwise the proof remains unaltered. (2) The reference 1 [\textit{M. Baaz}, \textit{N. Brunner} and \textit{K. Svozil}, ``Effective quantum observables'']\ may be completed as follows: Il Nuovo Cimento 110B, 1397-1413 (1995).
scientific article; zbMATH DE number 826762