Reflection ranks and ordinal analysis

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F21%3A00551636" target="_blank" >RIV/67985840:_____/21:00551636 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1017/jsl.2020.9" target="_blank" >https://doi.org/10.1017/jsl.2020.9</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1017/jsl.2020.9" target="_blank" >10.1017/jsl.2020.9</a>

Result language
angličtina
Original language name
Reflection ranks and ordinal analysis
Original language description
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 Π11 reflection strength order.We prove that there are no descending sequences ofΠ11 sound extensions of ACA0 in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any Π11 sound extension of ACA0.We prove that for any Π11 sound theory T extending ACA+0 , the reflection rank of T equals the Π11 proof-theoretic ordinal of T. We also prove that the Π11 proof-theoretic ordinal of α iterated Π11 reflection is εα. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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