Milner–Rado paradox

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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 .

Note .


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: