Introduction

Abstract

This book is devoted to a particular recursive method of constructing mathematical structures that live on a given ordinal θ, using a single transformation ξ ↦ C_ξ which assigns to every ordinal ξ < θ a set C_ξ of smaller ordinals that is closed and unbounded in the set of ordinals < ξ. The transfinite sequence

$$ C_\xi \left( {\xi < \theta } \right) $$

which we call a ‘C-sequence’ and on which we base our recursive constructions may have a number of ‘coherence properties’ and we shall give a detailed study of them and the way they influence these constructions. Here, ‘coherence’ usually means that the C_ξ’s are chosen in some canonical way, beyond the already mentioned and natural requirement that C_ξ is closed and unbounded in ξ for all ξ. For example, choosing a canonical ‘fundamental sequence’ of sets C_ξ ⊆ ξ for ξ < ε₀, relying on the specific properties of the Cantor normal form for ordinals below the first ordinal satisfying the equation x = ω^x, is a basis for a number of important results in proof theory. In set theory, one is interested in longer sequences as well and usually has a different perspective in applications, so one is naturally led to use some other tools besides the Cantor normal form. It turns out that the sets C_ξ can not only be used as ‘ladders’ for climbing up in recursive constructions but also as tools for ‘walking’ from an ordinal β to a smaller one α, β = β₀ > β₁ > ... > β_n−1 > β_n = α where the ‘step’β_i → β_i+1 is defined by letting β_i+1 be the minimal point of C_βi that is bigger than or equal to α.