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 ${\displaystyle \alpha }$ less than the successor ${\displaystyle \kappa ^{+}}$ of some cardinal number ${\displaystyle \kappa }$ can be written as the union of sets ${\displaystyle X_{1},X_{2},...}$ where ${\displaystyle X_{n}}$ is of order type at most κⁿ for n a positive integer.

Proof

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

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

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

Define:

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

Observe that:

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

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

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

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

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

References

• Milner, E. C.; Rado, R. (1965), "The pigeon-hole principle for ordinal numbers", Proceedings of the London Mathematical Society, Series 3, 15: 750–768, doi:10.1112/plms/s3-15.1.750, MR 0190003
• How to prove Milner-Rado Paradox? - Mathematics Stack Exchange