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.

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. (3), 15: 750–768, doi:10.1112/plms/s3-15.1.750, MR 0190003
• How to prove Milner-Rado Paradox? - Mathematics Stack Exchange