Ψ₀(Ωω)

From Wikipedia, the free encyclopedia
  (Redirected from Psi0(Omega omega))
Jump to: navigation, search
The correct title of this article is Ψ0ω). It appears incorrectly here because of technical restrictions.

In mathematics, Ψ0ω) 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[edit]

  • \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 ji and \xi \in C_i(\alpha) and \xi < \alpha.
  • \Psi_i(\alpha) is the smallest ordinal not in C_i(\alpha)

References[edit]