The Oscillation Mapping and the Square-bracket Operation

Abstract

In what follows, θ will be a fixed regular infinite cardinal.

$$ osc:\mathcal{P}\left( \theta \right)^2 \to Card $$

(8.1.1)

is defined by

$$ osc\left( {x,y} \right) = \left| {x\backslash \left( {\sup \left( {x \cap y} \right) + 1} \right)/ \sim } \right|, $$

(8.1.2)

where ∼ is the equivalence relation on x\ (sup(x ⋂ y)+1) defined by letting α ∼ β iff the closed interval determined by α and β contains no point from y. Hence, osc(x, y) is simply the number of convex pieces the set x \ (sup(x ⋂ y)+1) is split by the set y (see Figure 8.1). Note that this is slightly different from the way we have defined the oscillation between two subsets x and y of ω₁ in Section 2.3 above, where osc(x, y) was the number of convex pieces the set x is split by into the set y \ x. Since the variation is rather minor, we keep the same old notation as there is no danger of confusion. The oscillation mapping has proven to be a useful device in various schemes for coding information. Its usefulness in a given context depends very much on the corresponding ‘oscillation theory’, a set of definitions and lemmas that disclose when it is possible to achieve a given number as oscillation between two sets x and y in a given family X. The following definition reveals the notion of largeness relevant to the oscillation theory that we develop in this section.