Ψ₀(Ωω) - Alchetron, The Free Social Encyclopedia

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 Π 1 1 -CA₀ of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999).

Definition

Ω 0 = 0 , and Ω n = ℵ n for n > 0.

C i ( α ) is the smallest set of ordinals that contains Ω n for n finite, and contains all ordinals less than Ω i , and is closed under ordinal addition and exponentiation, and contains Ψ j ( ξ ) if j ≥ i and ξ ∈ C i ( α ) and ξ < α .

Ψ i ( α ) is the smallest ordinal not in C i ( α )

References

Ψ₀(Ωω) Wikipedia

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