Descending sequences in reflection hierarchies

TL;DR

This paper investigates the existence of infinite descending sequences of recursively enumerable theories with certain consistency properties, exploring how encoding methods affect their existence across different complexity classes.

Contribution

It generalizes the problem of descending sequences in reflection hierarchies to arbitrary complexity classes and encodings, providing conditions for their existence or non-existence.

Findings

Positive existence results for certain encodings

Negative results for other encodings

Dependence on complexity class and encoding method

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. Montalb\'an and Shavrukov independently asked whether this result generalizes to $0'$-recursive sequences. We consider a general version of this problem: For arbitrary $n$, for which complexity classes $\Gamma$ are there $\Gamma$-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 $\Gamma$ but also on the manner in which sequences are encoded in arithmetic. We provide positive answers for certain encodings and negative answers for others.