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:
- Milner, E. C.; Rado, R. (1965), "The pigeon-hole principle for ordinal numbers", Proc. London Math. Soc. (3) 15: 750–768, doi:10.1112/plms/s3-15.1.750, MR 0190003
- How to prove Milner-Rado Paradox? - Mathematics Stack Exchange
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