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 $X_{1},X_{2},...$ 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 $\{X_{n}^{\beta }\}_{n}$ 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 $X_{n}^{\beta _{\gamma +1}}$ , then:

\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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.