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The Square-bracket Operation on Countable Ordinals

Part of the Progress in Mathematics book series (PM, volume 263)

Abstract

Recall that a walk from a countable ordinal β to a smaller ordinal α along the fixed C-sequence Cξ (ξ < ω1) is a finite decreasing sequence
$$ \beta = \beta _0 > \beta _1 > \cdots \beta _n = \alpha , $$
where βi+1 = min(\( C_{\beta _i } \) \ α) for all i < n. Recall also the notion of the upper trace of the minimal walk,
$$ Tr\left( {\alpha ,\beta } \right) = \{ \beta _0 ,\beta _1 , \ldots ,\beta _n \} , $$
the finite set of places visited in the minimal walk from 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 ω1 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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Copyright information

© Birkhäuser Verlag AG 2007

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