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 α less than the successor κ⁺ of some cardinal number κ can be written as the union of sets X₁,X₂,... where X_n is of order type at most κⁿ for n a positive integer.

The proof is by transfinite induction. Let ${\displaystyle \alpha }$ be a limit ordinal (the induction is trivial for successor ordinals), and for each ${\displaystyle \beta <\alpha }$, let ${\displaystyle \{X_{n}^{\beta }\}_{n}}$ be a partition of ${\displaystyle \beta }$ satisfying the requirements of the theorem.

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