A note on the consistency operator

Walsh, James

doi:10.1090/proc/14948

A note on the consistency operator
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Proc. Amer. Math. Soc. 148 (2020), 2645-2654 Request permission

Abstract:

It is a well-known empirical observation that natural axiomatic theories are pre-well-ordered by proof-theoretic strength. For any natural theory $T$, the next strongest natural theory is $T+\mathsf {Con}_T$. We formulate and prove a statement to the effect that the consistency operator is the weakest natural way to uniformly extend axiomatic theories.

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