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

of some cardinal number

can be written as the union of sets

is of order type at most κn for n a positive integer.

The proof is by transfinite induction.

be a limit ordinal (the induction is trivial for successor ordinals), and for each

β < α

satisfying the requirements of the theorem.

Fix an increasing sequence

γ

γ <

( α ) ≤ κ

Define: Observe that: and so

be the order type of

As for the order types, clearly

Noting that the sets

γ + 1

γ

form a consecutive sequence of ordinal intervals, and that each

is a tail segment of

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