Milner–Rado paradox
Introduction 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 ''κ''''n'' 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