In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal[citation needed], 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;[1][2] this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999).
, the theory of finitely iterated inductive definitions, and of
,[3] a fragment of Kripke-Platek set theory extended by an axiom stating every set is contained in an admissible set.
Buchholz's ordinal is also the order type of the segment bounded by
in Buchholz's ordinal notation
[1] Lastly, it can be expressed as the limit of the sequence:
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