Article contents
REFLECTION RANKS AND ORDINAL ANALYSIS
Published online by Cambridge University Press: 10 July 2020
Abstract
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
$\Pi ^1_1$
reflection strength order. We prove that there are no descending sequences of
$\Pi ^1_1$
sound extensions of
$\mathsf {ACA}_0$
in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any
$\Pi ^1_1$
sound extension of
$\mathsf {ACA}_0$
. We prove that for any
$\Pi ^1_1$
sound theory T extending
$\mathsf {ACA}_0^+$
, the reflection rank of T equals the
$\Pi ^1_1$
proof-theoretic ordinal of T. We also prove that the
$\Pi ^1_1$
proof-theoretic ordinal of
$\alpha $
iterated
$\Pi ^1_1$
reflection is
$\varepsilon _\alpha $
. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles.
MSC classification
- Type
- Article
- Information
- Copyright
- © Association for Symbolic Logic 2020
References


- 5
- Cited by