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 where is of order type at most κn for n a positive integer. Proof 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 . 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 , then: References How to prove Milner-Rado Paradox? - Mathematics Stack Exchange Set theory Paradoxes