Buchholz's ordinal

In mathematics, ψ0(Ωω), 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; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of $$\mathsf{ID_{<\omega}}$$, the theory of finitely iterated inductive definitions, and of $$KP\ell_0$$, 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 $$\mathsf{(OT, <)}$$. Lastly, it can be expressed as the limit of the sequence: $$\varepsilon_0 = \psi_0(\Omega)$$, $$\mathsf{BHO} = \psi_0(\Omega_2)$$, $$\psi_0(\Omega_3)$$, ...

Definition

 * $$\Omega_0 = 1$$, and $$\Omega_n = \aleph_n$$ for n > 0.


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