In mathematics, Ψ0(Ωω) 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 -CA0 of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999).
- , and for n > 0.
- is the smallest set of ordinals that contains for n finite, and contains all ordinals less than , and is closed under ordinal addition and exponentiation, and contains if j ≥ i and and .
- is the smallest ordinal not in
- 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