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
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