MathDB

For an arbitrary ordinal number

\alpha

let

H(\alpha)

denote the set of functions

f\colon \alpha \rightarrow \{ -1,0,1\}

that map all but finitely many elements of

\alpha

0

. Order

H(\alpha)

according to the last difference, that is, for

f, g\in H(\alpha)

let

f\prec g

f(\beta) < g(\beta)

holds for the maximum ordinal number

\beta < \alpha

with

f(\beta) \neq g(\beta)

. Prove that the ordered set

(H(\alpha), \prec)

is scattered (i.e. it doesn't contain a subset isomorphic to the set of rational numbers with the usual order), and that any scattered order type can be embedded into some

(H(\alpha), \prec)

Problem Statement