Veblen function

In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φα is the function enumerating the common fixed points of φβ for β0 is a natural number and each term after the first is less than or equal to the previous term, and each If a fundamental sequence can be provided for the last term, then that term can be replaced by such a sequence to get For any β, if γ is a limit with then let No such sequence can be provided for = ω0 = 1 because it does not have cofinality ω. For we choose For we use and i.e. 0, , , etc.. For , we use and Now suppose that β is a limit: If , then let For , use Otherwise, the ordinal cannot be described in terms of smaller ordinals using and this scheme does not apply to it. The function Γ enumerates the ordinals α such that φα(0) = α. Γ0 is the Feferman–Schütte ordinal, i.e. it is the smallest α such that φα(0) = α.