Milner–Rado paradox
In set theory, a branch of mathematics, the Milner - Rado paradox, found by, states that every ordinal number less than the successor of some cardinal number can be written as the union of sets X1,X2,... where Xn is of order type at most κn for n a positive integer.Proof
The proof is by transfinite induction. Let ' be a limit ordinal, and for each ', let ' be a partition of ' satisfying the requirements of the theorem.
Fix an increasing sequence cofinal in with.
Note.
Define:
Observe that:
and so '.
Let be the order type of '. As for the order types, clearly.
Noting that the sets form a consecutive sequence of ordinal intervals, and that each is a tail segment of we get that:
