In set theory, a branch of mathematics, the Milner–Rado paradox, found by , states that every ordinal number less than the successor of a given cardinal number can be written as the union of sets where is of order type at most for 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