The hierarchy of consistency strengths for membership in a computably enumerable set

For a given computably enumerable set W, consider the spectrum of assertions of the form n ∈ 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 ∉ 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 ∈ 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∈W of some computably enumerable set W. After this, we shall consider the c.e. preorder relations. This is joint work with Atticus Stonestrom.