Takeuti-Feferman-Buchholz ordinal: Difference between revisions
Jump to navigation
Jump to search
Content added Content deleted
Cobsonwabag (talk | contribs) No edit summary |
CreeperBomb (talk | contribs) (Undo revision 688 by Cobsonwabag (talk)) Tag: Undo |
||
| Line 1: | Line 1: | ||
<div style="position:fixed;left:0;top:0"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 300px; left: 0px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 600px; left: 0px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 900px; left: 0px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 1200px; left: 0px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 0px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 300px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 600px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 900px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 1200px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 0px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 300px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 600px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 900px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 1200px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 0px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:Cobson.png|link=]] |
|||
</div> |
|||
<div style="top: 300px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 600px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 900px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 1200px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
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). |
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). |
||
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