Takeuti–Feferman–Buchholz ordinal: Difference between revisions

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The Takeuti-Feferman-Buchholz ordinal, sometimes abbreviated to TFBO^[1],^{[self-published source]} is a large countable ordinal, which acts as the limit of (largest number definable using) Buchholz's psi function and Feferman's theta function.^[2] It was named by David Madore^[3], after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as $\psi (\varepsilon _{\Omega _{\omega }+1})$ in Buchholz's psi function^[4], an OCF invented by Wilfried Buchholz^[5]^[6]^[7], and $\theta _{\varepsilon _{\Omega _{\omega }+1}}(0)$ in Feferman's theta function, an OCF invented by Solomon Feferman^[8]^[9]. It is the proof-theoretic ordinal of $\Pi _{1}^{1}-CA+BI$ ^[10], another subsystem of second-order arithmetic and $\Pi _{1}^{1}$ - comprehension + transfinite induction^[4].

Definition

Let $\Omega _{\alpha }$ represent an uncountable ordinal with cardinality $\aleph _{\alpha }$ .
Let $\varepsilon _{\beta }$ represent the $\beta$ th epsilon number, equal to $\varepsilon _{\beta -1}$ tetrated to itself.
Let $\psi$ represent Buchholz's psi function
The TFBO is equal to $\psi _{0}(\varepsilon _{\Omega _{\omega }+1})$ .

References

^ "Kyodaisuu/Abbreviation". Googology Wiki. Retrieved 2021-08-10.
^ "Buchholz's ψ functions". cantors-attic. Retrieved 2021-08-10.
^ "Ordinal collapsing function", Wikipedia, 2008-04-16, retrieved 2021-08-10
^ ^a ^b "A Zoo of Ordinals" (PDF). Madore. 2017-07-29. Retrieved 2021-08-10.{{cite web}}: CS1 maint: url-status (link)
^ "Collapsingfunktionen" (PDF). University of Munich. 1981. Retrieved 2021-08-10.{{cite web}}: CS1 maint: url-status (link)
^ "A new system of proof-theoretic ordinal functions". Annals of Pure and Applied Logic. 32: 195–207. 1986-01-01. doi:10.1016/0168-0072(86)90052-7. ISSN 0168-0072.
^ Buchholz, W.; Schütte, K. (1988). "Proof Theory of Impredicative Subsystems of Analysis". undefined. Retrieved 2021-08-10.
^ "[PDF] Proof Theory Second Edition by Gaisi Takeuti | Perlego". www.perlego.com. Retrieved 2021-08-10.
^ Buchholz, Wilfried (1975). "Normalfunktionen und Konstruktive Systeme von Ordinalzahlen". doi:10.1007/BFB0079544. {{cite journal}}: Cite journal requires |journal= (help)
^ Buchholz, Wilfried; Feferman, Solomon; Pohlers, Wolfram; Sieg, Wilfried (1981). "Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies". Lecture Notes in Mathematics. doi:10.1007/bfb0091894. ISSN 0075-8434.

[1] "Kyodaisuu/Abbreviation". Googology Wiki. Retrieved 2021-08-10.

[2] "Buchholz's ψ functions". cantors-attic. Retrieved 2021-08-10.

[3] "Ordinal collapsing function", Wikipedia, 2008-04-16, retrieved 2021-08-10

[:1-4] "A Zoo of Ordinals" (PDF). Madore. 2017-07-29. Retrieved 2021-08-10.{{cite web}}: CS1 maint: url-status (link)

[5] "Collapsingfunktionen" (PDF). University of Munich. 1981. Retrieved 2021-08-10.{{cite web}}: CS1 maint: url-status (link)

[6] "A new system of proof-theoretic ordinal functions". Annals of Pure and Applied Logic. 32: 195–207. 1986-01-01. doi:10.1016/0168-0072(86)90052-7. ISSN 0168-0072.

[7] Buchholz, W.; Schütte, K. (1988). "Proof Theory of Impredicative Subsystems of Analysis". undefined. Retrieved 2021-08-10.

[8] "[PDF] Proof Theory Second Edition by Gaisi Takeuti | Perlego". www.perlego.com. Retrieved 2021-08-10.

[9] Buchholz, Wilfried (1975). "Normalfunktionen und Konstruktive Systeme von Ordinalzahlen". doi:10.1007/BFB0079544. {{cite journal}}: Cite journal requires |journal= (help)

[10] Buchholz, Wilfried; Feferman, Solomon; Pohlers, Wolfram; Sieg, Wilfried (1981). "Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies". Lecture Notes in Mathematics. doi:10.1007/bfb0091894. ISSN 0075-8434.

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 {{refimprove|date=August 2021}}
+The Takeuti-Feferman-Buchholz ordinal, sometimes abbreviated to TFBO<ref>{{Cite web|title=Kyodaisuu/Abbreviation|url=https://googology.wikia.org/wiki/User:Kyodaisuu/Abbreviation|access-date=2021-08-10|website=Googology Wiki|language=en}}</ref>,{{self published source}} is a [[large countable ordinal]], which acts as the limit of (largest number definable using) [[Buchholz psi functions|Buchholz's psi function]] and Feferman's theta function.<ref>{{Cite web|title=Buchholz’s ψ functions|url=https://neugierde.github.io/cantors-attic/Buchholz%27s_%CF%88_functions#takeuti-feferman-buchholz-ordinal|access-date=2021-08-10|website=cantors-attic|language=en-US}}</ref> It was named by David Madore<ref>{{Citation|title=Ordinal collapsing function|date=2008-04-16|url=https://en.wikipedia.org/w/index.php?title=Ordinal_collapsing_function&oldid=206127084|work=Wikipedia|language=en|access-date=2021-08-10}}</ref>, after [[Gaisi Takeuti]], [[Solomon Feferman]] and Wilfried Buchholz. It is written as <math>\psi(\varepsilon_{\Omega_\omega + 1})</math> in Buchholz's psi function<ref name=":1">{{Cite web|date=2017-07-29|title=A Zoo of Ordinals|url=http://www.madore.org/~david/math/ordinal-zoo.pdf|url-status=live|access-date=2021-08-10|website=Madore}}</ref>, an [[Ordinal collapsing function|OCF]] invented by Wilfried Buchholz<ref>{{Cite web|date=1981|title=Collapsingfunktionen|url=https://www.mathematik.uni-muenchen.de/~buchholz/articles/Collapsing.pdf|url-status=live|access-date=2021-08-10|website=University of Munich}}</ref><ref>{{Cite journal|date=1986-01-01|title=A new system of proof-theoretic ordinal functions|url=https://www.sciencedirect.com/science/article/pii/0168007286900527|journal=Annals of Pure and Applied Logic|language=en|volume=32|pages=195–207|doi=10.1016/0168-0072(86)90052-7|issn=0168-0072}}</ref><ref>{{Cite web|last=Buchholz|first=W.|last2=Schütte|first2=K.|date=1988|title=Proof Theory of Impredicative Subsystems of Analysis|url=https://www.semanticscholar.org/paper/Proof-Theory-of-Impredicative-Subsystems-of-Buchholz-Sch%C3%BCtte/d98c02aebc65319f3653d1813ca7c4cf03e8bdea|access-date=2021-08-10|website=undefined|language=en}}</ref>, and <math>\theta_{\varepsilon_{\Omega_\omega + 1}}(0)</math> in Feferman's theta function, an OCF invented by Solomon Feferman<ref>{{Cite web|title=[PDF] Proof Theory Second Edition by Gaisi Takeuti {{!}} Perlego|url=https://www.perlego.com/book/112275/proof-theory-second-edition-pdf|access-date=2021-08-10|website=www.perlego.com}}</ref><ref>{{Cite journal|last=Buchholz|first=Wilfried|date=1975|title=Normalfunktionen und Konstruktive Systeme von Ordinalzahlen|url=https://www.semanticscholar.org/paper/Normalfunktionen-und-Konstruktive-Systeme-von-Buchholz/6668b830ca38e6458a3f2d8f1d7a964f86d63bad|doi=10.1007/BFB0079544}}</ref>. It is the proof-theoretic ordinal of <math>\Pi_1^1 -CA + BI</math><ref>{{Cite journal|last=Buchholz|first=Wilfried|last2=Feferman|first2=Solomon|last3=Pohlers|first3=Wolfram|last4=Sieg|first4=Wilfried|date=1981|title=Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies|url=https://link.springer.com/book/10.1007/BFb0091894|journal=Lecture Notes in Mathematics|language=en-gb|doi=10.1007/bfb0091894|issn=0075-8434}}</ref>, another subsystem of second-order arithmetic and  <math>\Pi_1^1</math>- comprehension + transfinite induction<ref name=":1" />.
-The Takeuti-Feferman-Buchholz ordinal, sometimes abbreviated to TFBO<ref>{{Cite web|title=Kyodaisuu/Abbreviation|url=https://googology.wikia.org/wiki/User:Kyodaisuu/Abbreviation|access-date=2021-08-10|website=Googology Wiki|language=en}}</ref>,{{self published source}} is a [[large countable ordinal]], which acts as the limit of [[Buchholz psi functions|Buchholz's psi function]] and Feferman's theta function. It was named by David Madore, after [[Gaisi Takeuti]], [[Solomon Feferman]] and Wilfried Buchholz. It is written as <math>\psi(\varepsilon_{\Omega_\omega + 1})</math> in Buchholz's psi function, an [[Ordinal collapsing function|OCF]] invented by Wilfried Buchholz.
 == Definition ==