Ordinal arithmetic based on Skolem hulling

Taking up ordinal notations derived from Skolem hull operators familiar in the field of infinitary proof theory we develop a toolkit of ordinal arithmetic that generally applies whenever ordinal structures are analyzed whose combinatorial complexity does not exceed the strength of the system KP ℓ 0 of set theory. The original purpose of doing so was inspired by the analysis of ordinal structures based on elementarity invented by T.J. Carlson, see [T.J. Carlson, Elementary patterns of resemblance, Annals of Pure and Applied Logic 108 (2001) 19–77], [G. Wilken, Σ 1 -Elementarity and Skolem hull operators, Annals of Pure and Applied Logic 145 (2) (2007) 162–175], and [G. Wilken, Assignment of ordinals to patterns of resemblance, The Journal of Symbolic Logic (in press)]. Within the arithmetical context laid bare in this work, the “ KP ℓ 0 -numbers” play a role analogous to the role epsilon numbers play in the ordinal arithmetic based on the notion of Cantor normal form.