The Square-bracket Operation on Countable Ordinals

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


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.


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

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