Takeuti-Feferman-Buchholz ordinal: Difference between revisions
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The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's original set of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the [[Buchholz ordinal]], although one could comparatively describe the difference in size as like that between [[Epsilon numbers|\( \varepsilon_0 \)]] and the [[Bachmann-Howard ordinal]]. It is equal to the proof-theoretic ordinal of |
The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's [[Buchholz's psi-functions|original set]] of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the [[Buchholz ordinal]], although one could comparatively describe the difference in size as like that between [[Epsilon numbers|\( \varepsilon_0 \)]] and the [[Bachmann-Howard ordinal]]. It is equal to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-CA_0} \) (of which the Buchholz ordinal is the proof-theoretic ordinal) with additional induction schemata. It also is the proof-theoretic ordinal of Peano arithmetic, augmented by iterated inductive definitions of length \( \omega \) (while the Buchholz ordinal has iterated definitions of arbitrary finite lengths). |
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Connection to Buchholz hydras |
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Latest revision as of 16:52, 25 March 2024
The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's original set of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".[1] It is not particularly vastly larger than the Buchholz ordinal, although one could comparatively describe the difference in size as like that between \( \varepsilon_0 \) and the Bachmann-Howard ordinal. It is equal to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-CA_0} \) (of which the Buchholz ordinal is the proof-theoretic ordinal) with additional induction schemata. It also is the proof-theoretic ordinal of Peano arithmetic, augmented by iterated inductive definitions of length \( \omega \) (while the Buchholz ordinal has iterated definitions of arbitrary finite lengths).
Connection to Buchholz hydras
- ↑ A Zoo of Ordinals, David A. Madore, July 29 2017