Reflection ranks and ordinal analysis

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Reflection ranks and ordinal analysis

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Reflection ranks and ordinal analysis

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Symbolic Logic

ISSN

0022-4812

e-ISSN

1943-5886

Svazek periodika

86

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

35

Strana od-do

1350-1384

Kód UT WoS článku

000741497000005

EID výsledku v databázi Scopus

2-s2.0-85123943148