Epistemic set theory (Q1263587)

The modal provability operator is axiomatized in the context of set theory so that it provides a framework for consideration of the Post- Turing thesis. Following the ideas of Gödel and Kripke, the author first defines on ZFC a theory of concepts, on which the axioms for the modal provability operator are formulated. In this framework, then, the strong form of the Post-Turing thesis is formulated, not as a schema but as a sentence. The author also suggests that it is possible in this framework to deal with such a notion as ordinal notations that involve essentially both constructive and non-constructive notions.