Is the consistency operator canonical?

James Walsh

It is a well-known empirical phenomenon that natural axiomatic theories are well-ordered by consistency strength. The restriction to natural theories is necessary; using ad-hoc techniques (such as self-reference and Rosser orderings) one can exhibit non-linearity and ill-foundedness in the consistency strength hierarchy. What explains the contrast between natural theories and axiomatic theories in general?

Our approach to this problem is inspired by work on an analogous problem in recursion theory. The natural Turing degrees (0,0′,…,Kleene’s O,…,0#,…) are well-ordered by Turing reducibility, yet the Turing degrees in general are neither linearly ordered nor well-founded, as ad-hoc techniques (such as the priority method) bear out. Martin’s Conjecture, which is still unresolved, is a proposed explanation for this phenomenon. In particular, Martin’s Conjecture specifies a way in which the Turing jump is canonical.

After discussing Martin’s Conjecture, we will formulate analogous proof-theoretic hypotheses according to which the consistency operator is canonical. We will then discuss results—both positive and negative—within this framework. Some of these results were obtained jointly with Antonio Montalbán.