## Abstract

This book is devoted to a particular recursive method of constructing mathematical structures that live on a given ordinal which we call a ‘

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

*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*<*ε*_{0}, 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*=*β*_{0}>*β*_{1}> ... >*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*.## Keywords

Oscillation Mapping Force Notion Stationary Subset Singular Cardinal Unbounded Subset
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Birkhäuser Verlag AG 2007