Ψ₀(Ωω)

The correct title of this article is Ψ₀(Ω_ω). It appears incorrectly here because of technical restrictions.

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

Main article: Ordinal collapsing function

• $\Omega_0 = 0$, 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 $C_i(\alpha)$

References

• G. Takeuti, Proof theory, 2nd edition 1987 ISBN 0-444-10492-5
• K. Schütte, Proof theory, Springer 1977 ISBN 0-387-07911-4