Milner–Rado paradox

In set theory, a branch of mathematics, the Milner – Rado paradox, found by, states that every ordinal number $$\alpha$$ less than the successor $$\kappa^{+}$$ of some cardinal number $$\kappa$$ can be written as the union of sets $$X_1, X_2,...$$ where $$X_n$$ is of order type at most &kappa;n 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 $$X^{\beta_{\gamma+1}}_n$$, then:


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