The second incompleteness theorem and bounded interpretations
Let \(\mathrm{con}_i(U)\) be consistency of a theory \(U\) for provability involving axioms witnessed below \(i\). Let \(\mathrm{slb-con}(U)\) (super-logarithmically bounded consistency of \(U\)) mean \(\forall x( \mathrm{superexp}(x)\downarrow\iff \mathrm{con}_x(U))\). It is proved that \(C=\mathrm{slb-con}(S^1_2)\) is an Orey sentence for Kalmar-elementary arithmetic EA, i.e., EA interprets both \(\mathrm{EA}+\mathrm C\) and \(\mathrm{EA}+\neg \mathrm C\). Therefore C is independent of EA although it is weaker than \(\mathrm{con}(\mathrm{EA})\).
scientific article; zbMATH DE number 6088848
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