REFLECTION RANKS AND ORDINAL ANALYSIS

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the III reflection strength order. We prove that there are no descending sequences of III sound extensions of ACA0 in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any III sound extension of ACA0. We prove that for any III sound theory T extending ACAloP, the reflection rank of T equals the III proof-theoretic ordinal of T. We also prove that the III proof-theoretic ordinal of a iterated III reflection is Ea. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles