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Workshop on Proof Theory, Modal Logic and Reflection Principles
October 17, 2017 15:00–15:35, Moscow, Steklov Mathematical Institute

On the naturalness of the consistency operator

J. Walsh
 Video records: MP4 951.1 Mb MP4 260.7 Mb

Abstract: It is a well-known empirical phenomenon that natural axiomatic theories are well-ordered by their consistency strength. ​ ​To investigate this phenomenon, we examine recursive monotonic functions on the​ ​ Lindenbaum algebra of EA. We prove that no such function sends every​ ​ consistent $\varphi$ to a sentence with deductive strength strictly between​ ​ $\varphi$ and $(\varphi\land \mathrm{Con}(\varphi))$. We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive​ ​ monotonic function $f$, if there is an iterate of $\mathrm{Con}$ that bounds $f$ everywhere, then $f$ must be somewhere equal to an iterate of $\mathrm{Con}$.