In set theory, a branch of mathematics, the Milner – Rado paradox, found by Eric Charles Milner and Richard Rado (1965), states that every ordinal number α less than the successor κ+ of some cardinal number κ can be written as the union of sets X1,X2,... where Xn is of order type at most κn for n a positive integer.
The proof is by transfinite induction. Let be a limit ordinal (the induction is trivial for successor ordinals), and for each , let be a partition of satisfying the requirements of the theorem.
Fix an increasing sequence cofinal in with .
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: