The Oscillation Mapping and the Square-bracket Operation

In what follows,

θ

will be a fixed regular infinite cardinal.

8.1.1

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

is defined by

8.1.2

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

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

ω

1

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.