Английская Википедия:Buchholz's ordinal

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In mathematics, ψ₀(Ω_ω), widely known as Buchholz's ordinalШаблон: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 <math>\Pi_1^1</math>-CA₀ of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of <math>\mathsf{ID_{<\omega}}</math>, the theory of finitely iterated inductive definitions, and of <math>KP\ell_0</math>, 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 <math>D_0D_\omega0</math> in Buchholz's ordinal notation <math>\mathsf{(OT, <)}</math>. Lastly, it can be expressed as the limit of the sequence: <math>\varepsilon_0 = \psi_0(\Omega)</math>, <math>\mathsf{BHO} = \psi_0(\Omega_2)</math>, <math>\psi_0(\Omega_3)</math>, ...

Definition

• <math>\Omega_0 = 1</math>, and <math>\Omega_n = \aleph_n</math> for n > 0.

• <math>C_i(\alpha)</math> is the closure of <math>\Omega_i</math> under addition and the <math>\psi_\eta(\mu)</math> function itself (the latter of which only for <math>\mu < \alpha</math> and <math>\eta \leq \omega</math>).
• <math>\psi_i(\alpha)</math> is the smallest ordinal not in <math>C_i(\alpha)</math>.
• Thus, ψ₀(Ω_ω) is the smallest ordinal not in the closure of <math>1</math> under addition and the <math>\psi_\eta(\mu)</math> function itself (the latter of which only for <math>\mu < \Omega_\omega</math> and <math>\eta \leq \omega</math>).

References

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

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