Walks on Ordinals and Their Characteristics pp 133-160 | Cite as

# The Square-bracket Operation on Countable Ordinals

Chapter

## Abstract

Recall that a where the finite set of places visited in the minimal walk from

*walk*from a countable ordinal*β*to a smaller ordinal*α*along the fixed*C*-sequence*C*_{ξ}(*ξ*<*ω*_{1}) is a finite decreasing sequence$$
\beta = \beta _0 > \beta _1 > \cdots \beta _n = \alpha ,
$$

*β*_{i+1}= min(\( C_{\beta _i } \) \*α*) for all*i*<*n*. Recall also the notion of the*upper trace*of the minimal walk,$$
Tr\left( {\alpha ,\beta } \right) = \{ \beta _0 ,\beta _1 , \ldots ,\beta _n \} ,
$$

*β*to*α*. The following simple fact about the upper trace lies at the heart of all known definitions of square-bracket operations, not only on*ω*_{1}but also at higher cardinalities.## Keywords

Banach Space Force Notion Stationary Subset Geometrical Application Complete Binary Tree
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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© Birkhäuser Verlag AG 2007