Buchholz's ordinal
In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal[citation needed], is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem -CA0 of second-order arithmetic;[1][2] this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of , the theory of finitely iterated inductive definitions, and of ,[3] a fragment of Kripke-Platek set theory extended by an axiom stating every set is contained in an admissible set. Buchholz's ordinal is also the order type of the segment bounded by in Buchholz's ordinal notation .[1] Lastly, it can be expressed as the limit of the sequence: , , , ...
Definition
[edit]- , and for n > 0.
- is the closure of under addition and the function itself (the latter of which only for and ).
- is the smallest ordinal not in .
- Thus, ψ0(Ωω) is the smallest ordinal not in the closure of under addition and the function itself (the latter of which only for and ).
References
[edit]- ^ Jump up to: a b Buchholz, W. (1986-01-01). "A new system of proof-theoretic ordinal functions". Annals of Pure and Applied Logic. 32: 195–207. doi:10.1016/0168-0072(86)90052-7. ISSN 0168-0072.
- ^ Simpson, Stephen G. (2009). Subsystems of Second Order Arithmetic. Perspectives in Logic (2 ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-88439-6.
- ^ T. Carlson, "Elementary Patterns of Resemblance" (1999). Accessed 12 August 2022.
- G. Takeuti, Proof theory, 2nd edition 1987 ISBN 0-444-10492-5
- K. Schütte, Proof theory, Springer 1977 ISBN 0-387-07911-4