Buchholz's ordinal explained

In mathematics, ψ₀(Ω_ω), widely known as Buchholz's ordinal, 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

\Pi ^{1} _{1} -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 ID _{<\omega} , the theory of finitely iterated inductive definitions, and of KP\ell _{0} ,^[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 D _{0D} _{\omega0} in Buchholz's ordinal notation (OT,<) . Lastly, it can be expressed as the limit of the sequence: \varepsilon _{0} = \psi _{0(\Omega)} , BHO = \psi _{0(\Omega} _{2)} , \psi _{0(\Omega} _{3)} , ...

Definition

See main article: article.

\Omega _{0} = 1 , and \Omega _{n} = \aleph _{n} for n > 0.C _{i(\alpha)} is the closure of \Omega _{i} under addition and the \psi _{η(\mu)} function itself (the latter of which only for \mu < \alpha and η \leq \omega ).\psi _{i(\alpha)} is the smallest ordinal not in C _{i(\alpha)} .

• Thus, ψ₀(Ω_ω) is the smallest ordinal not in the closure of

1 under addition and the \psi _{η(\mu)} function itself (the latter of which only for \mu < \Omega _{\omega} and η \leq \omega ).

References

• G. Takeuti, Proof theory, 2nd edition 1987
• K. Schütte, Proof theory, Springer 1977

Notes and References

1. 1986-01-01. A new system of proof-theoretic ordinal functions. Annals of Pure and Applied Logic. en. 32. 195–207. 10.1016/0168-0072(86)90052-7. 0168-0072. Buchholz . W. . free.
2. Book: Simpson, Stephen G.. Subsystems of Second Order Arithmetic. 2009. Cambridge University Press. 978-0-521-88439-6. 2. Perspectives in Logic. Cambridge.
3. T. Carlson, "Elementary Patterns of Resemblance" (1999). Accessed 12 August 2022.