Ψ₀(Ωω)

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
Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, ISBN 978-0-521-88439-6, MR 2517689

Large countable ordinals

First infinite ordinal ω Epsilon numbers ε₀ Feferman–Schütte ordinal Γ₀ Ackermann ordinal θ(Ω²) small Veblen ordinal θ(Ω^ω) large Veblen ordinal θ(Ω^Ω) Bachmann–Howard ordinal ψ(ε_Ω+1) Ψ₀(Ω_ω) Church–Kleene ordinal ω₁^CK