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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-orderness phenomenon by studying a coarsening of the
consistency strength order, namely, the reflection strength order. We
prove that there are no descending sequences of sound extensions of
in this order. Accordingly, we can attach a rank in this
order, which we call reflection rank, to any sound extension of
. We prove that for any sound theory extending
, the reflection rank of equals the proof-theoretic
ordinal of . We also prove that the proof-theoretic ordinal of
iterated reflection is . Finally, we use our
results to provide straightforward well-foundedness proofs of ordinal notation
systems based on reflection principles