# Ψ₀(Ωω)

(Redirected from Psi0(Omega omega))
In mathematics, Ψ0ω) 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).
• $\Omega_0 = 0$, 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 ji and $\xi \in C_i(\alpha)$ and $\xi < \alpha$.
• $\Psi_i(\alpha)$ is the smallest ordinal not in $C_i(\alpha)$