Milner–Rado paradox

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 X1,X2,... where Xn is of order type at most κn for n a positive integer.

ProofEdit

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   we get that:

 

ReferencesEdit

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