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Buchholz's ordinal

In mathematics, ψ<sub>0</sub>(Ω<sub>ω</sub>), widely known as Buchholz's ordinal, 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 -CA<sub>0</sub> of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of , the theory of finitely iterated inductive definitions, and of , 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 . Lastly, it can be expressed as the limit of the sequence: , , , ...

Definition

  • , and for n > 0.
  • is the closure of under addition and the function itself (the latter of which only for and ).
  • is the smallest ordinal not in .
  • Thus, ψ<sub>0</sub>(Ω<sub>ω</sub>) is the smallest ordinal not in the closure of under addition and the function itself (the latter of which only for and ).

References

  • G. Takeuti, Proof theory, 2nd edition 1987
  • K. Schütte, Proof theory, Springer 1977