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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 $$\alpha$$ less than the successor $${\displaystyle \kappa ^{+}}$$ of some cardinal number $$\kappa$$ can be written as the union of sets X1,X2,... where Xn is of order type at most κn for n a positive integer.
Proof

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

Fix an increasing sequence $$\{\beta_\gamma\}_{\gamma<\mathrm{cf}\,(\alpha)}$$ cofinal in $$\alpha$$ with $$\beta _{0}=0$$.

Note $$\mathrm{cf}\,(\alpha)\le\kappa.$$

Define:

$$X^\alpha _0 = \{0\};\ \ X^\alpha_{n+1} = \bigcup_\gamma X^{\beta_{\gamma+1}}_n\setminus \beta_\gamma$$

Observe that:

$$\bigcup_{n>0}X^\alpha_n = \bigcup _n \bigcup _\gamma X^{\beta_{\gamma+1}}_n\setminus \beta_\gamma = \bigcup_\gamma \bigcup_n X^{\beta_{\gamma+1}}_n\setminus \beta_\gamma = \bigcup_\gamma \beta_{\gamma+1}\setminus \beta_\gamma = \alpha \setminus \beta_0$$

and so $$\bigcup_nX^\alpha_n = \alpha.$$

Let $$\mathrm{ot}\,(A)$$ be the order type of A. As for the order types, clearly $$\mathrm{ot}(X^\alpha_0) = 1 = \kappa^0.$$

Noting that the sets $$\beta_{\gamma+1}\setminus\beta_\gamma$$ form a consecutive sequence of ordinal intervals, and that each $$X^{\beta_{\gamma+1}}_n\setminus\beta_\gamma$$ is a tail segment of $$X^{\beta_{\gamma+1}}_n$$ we get that:

$$\mathrm{ot}(X^\alpha_{n+1}) = \sum_\gamma \mathrm{ot}(X^{\beta_{\gamma+1}}_n\setminus\beta_\gamma) \leq \sum_\gamma \kappa^n = \kappa^n \cdot \mathrm{cf}(\alpha) \leq \kappa^n\cdot\kappa = \kappa^{n+1}$$

References

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

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