This will be a talk for the Logic Seminar at the University of Notre Dame, 14 April 2026, 2pm, Room 125 Hayes-Healey.

Abstract After establishing several  general features of the hierarchy of consistency strength, we shall consider the possible spectrum of assertions of the form $n\in W$, where $W$ is a given computably enumerable set. If $W$ is c.e. but not computably decidable, many of these statements must be independent of PA, as well as ZFC, and indeed any consistent c.e. theory extending these. What kind of consistency strengths can be exhibited by these statements? In this work, we investigate the possible hierarchies of consistency strengths that arise. For example, there is a c.e. set $Q$ for which the consistency strengths of the assertions $n\in Q$ are linearly ordered like the rational line. More generally, I shall prove that every computable preorder relation on the natural numbers is realized exactly as the hierarchy of consistency strength for the membership statements $n\in W$ of some computably enumerable set $W$. After this, we shall consider the c.e. preorder relations. This is in part joint work with Atticus Stonestrom (Notre Dame).