# 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 $\alpha$ less than the successor $\kappa ^{+}$ 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 $\{X_{n}^{\beta }\}_{n}$  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 $\mathrm {cf} \,(\alpha )\leq \kappa$ .

Define:

$X_{0}^{\alpha }=\{0\};\ \ X_{n+1}^{\alpha }=\bigcup _{\gamma }X_{n}^{\beta _{\gamma +1}}\setminus \beta _{\gamma }$

Observe that:

$\bigcup _{n>0}X_{n}^{\alpha }=\bigcup _{n}\bigcup _{\gamma }X_{n}^{\beta _{\gamma +1}}\setminus \beta _{\gamma }=\bigcup _{\gamma }\bigcup _{n}X_{n}^{\beta _{\gamma +1}}\setminus \beta _{\gamma }=\bigcup _{\gamma }\beta _{\gamma +1}\setminus \beta _{\gamma }=\alpha \setminus \beta _{0}$

and so $\bigcup _{n}X_{n}^{\alpha }=\alpha$ .

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

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

$\mathrm {ot} (X_{n+1}^{\alpha })=\sum _{\gamma }\mathrm {ot} (X_{n}^{\beta _{\gamma +1}}\setminus \beta _{\gamma })\leq \sum _{\gamma }\kappa ^{n}=\kappa ^{n}\cdot \mathrm {cf} (\alpha )\leq \kappa ^{n}\cdot \kappa =\kappa ^{n+1}$