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 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

  • Milner, E. C.; Rado, R. (1965), "The pigeon-hole principle for ordinal numbers", Proceedings of the London Mathematical Society, Series 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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