Buchholz's ordinal

In mathematics, Ψ₀(Ω_ω) 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).

Definition

$\Omega _{0}=0$ , and $\Omega _{n}=\aleph _{n}$ for n > 0.
$C_{i}(\alpha )$ is the smallest set of ordinals that contains $\Omega _{n}$ for n finite, and contains all ordinals less than $\Omega _{i}$ , and is closed under ordinal addition and exponentiation, and contains $\Psi _{j}(\xi )$ if j ≥ i and $\xi \in C_{i}(\alpha )$ and $\xi <\alpha$ .
$\Psi _{i}(\alpha )$ is the smallest ordinal not in $C_{i}(\alpha )$