Abstract

CAICEDO, ANDRÉS EDUARDO. Goodstein's function. Rev.colomb.mat. [online]. 2007, vol.41, n.2, pp.381-391. ISSN 0034-7426.

Goodsteins function Ģ:N → N is an example of a fast growing recursive function. Introduced in 1944 by R. L. Goodstein [9], Kirby and Paris [12] showed in 1982, using model theoretic techniques, that Goodsteins result that Ģ is total, i.e., that Ģ(n) is defined for all n Є N, is not a theorem of first order Peano Arithmetic. We compute Goodsteins function in terms of the Löb-Wainer fast growing hierarchy of functions; from this and standard proof theoretic results about this hierarchy, the Kirby-Paris result follows immediately. We also compute the functions of the Hardy hierarchy in terms of the Löb-Wainer functions, which allows us to provide a new proof of a similar result, due to Cichon [2].

Keywords : Goodstein function; Hardy hierarchy; fast growing hierarchy; Peano Arithmetic.

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