Formula omitted]-elementarity and Skolem hull operators

Abstract

The exact correspondence between ordinal notations derived from Skolem hull operators, which are classical in ordinal analysis, and descriptions of ordinals in terms of Σ 1 -elementarity, an approach developed by T.J. Carlson, is analyzed in full detail. The ordinal arithmetical tools needed for this purpose were developed in [G. Wilken, Ordinal arithmetic based on Skolem hulling, Annals of Pure and Applied Logic 145 (2) (2007) 130–161]. We show that the least ordinal κ such that κ < 1 ∞ (as defined in [T.J. Carlson, Elementary patterns of resemblance, Annals of Pure and Applied Logic 108 (2001) 19–77] and described below) is the proof theoretic ordinal of the set-theoretic system KP ℓ 0 , confirming a claim of Carlson. Moreover, we characterize the class of all ordinals κ such that κ < 1 ∞ and provide an ordinal arithmetical analysis of Carlson’s entire structure R 1 in the style of [T.J. Carlson, Ordinal arithmetic and Σ 1 -elementarity, Archive for Mathematical Logic 38 (1999) 449–460].