The Square-bracket Operation on Countable Ordinals

Abstract

Recall that a walk from a countable ordinal β to a smaller ordinal α along the fixed C-sequence C_ξ (ξ < ω₁) is a finite decreasing sequence

$$ \beta = \beta _0 > \beta _1 > \cdots \beta _n = \alpha , $$

where β_i+1 = min($C_{\beta _i }$ \ α) for all i < n. Recall also the notion of the upper trace of the minimal walk,

$$ Tr\left( {\alpha ,\beta } \right) = \{ \beta _0 ,\beta _1 , \ldots ,\beta _n \} , $$

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.