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 [math]\displaystyle{ \Pi_1^1 }[/math]-CA₀ of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999).

Definition

• [math]\displaystyle{ \Omega_0 = 0 }[/math], and [math]\displaystyle{ \Omega_n = \aleph_n }[/math] for n > 0.
• [math]\displaystyle{ C_i(\alpha) }[/math] is the smallest set of ordinals that contains [math]\displaystyle{ \Omega_n }[/math] for n finite, and contains all ordinals less than [math]\displaystyle{ \Omega_i }[/math], and is closed under ordinal addition and exponentiation, and contains [math]\displaystyle{ \Psi_j(\xi) }[/math] if j ≥ i and [math]\displaystyle{ \xi \in C_i(\alpha) }[/math] and [math]\displaystyle{ \xi \lt \alpha }[/math].
• [math]\displaystyle{ \Psi_i(\alpha) }[/math] is the smallest ordinal not in [math]\displaystyle{ C_i(\alpha) }[/math]

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, http://www.math.psu.edu/simpson/sosoa/