Descending sequences in reflection hierarchies

Abstract

There is no recursively enumerable sequence of sufficiently strong 2-consistent r.e. theories such that each proves the $2$-consistency of the next. Montalbán and Shavrukov independently asked whether this result generalizes to $0^{\prime }$-recursive sequences. We consider a general version of this problem: For arbitrary $n$, for which complexity classes $\hat{I}“$ are there $\hat{I}“$-definable sequences of $n$-consistent r.e. theories each of which proves the $n$-consistency of the next? The answer to this question depends not only on $n$ and $\hat{I}“$ but also on the manner in which sequences are encoded in arithmetic. We provide positive answers for certain encodings and negative answers for others.