This will be a talk for the Logic Seminar at the Mathematical Institute of the University of Oxford, 29 May 2025 5pm Andrew Wiles Building.

Abstract. For a given computably enumerable set $W$, consider the spectrum of assertions of the form $n\in W$. If $W$ is c.e. but not computably decidable, it is easy to see that many of these statements will be independent of PA, for otherwise we could decide $W$ by searching for proofs of $n\notin W$. 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 joint work with Atticus Stonestrom (Notre Dame).