Estrada index

In set theory, a branch of mathematics, the Milner – Rado paradox, found by Template:Harvs, 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

Let $\alpha$ be a limit ordinal, and for each $\beta <\alpha$, let $\{X_{\beta }^{n}\}_{n}$ be the obvious thing.

Fix an increasing sequence $\{\beta _{\gamma }\}_{\gamma <\mathrm {cf} \,(\alpha )}$ cofinal in $\alpha$ with $\beta _{0}=1$.

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:

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

References

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• How to prove Milner-Rado Paradox? - Mathematics Stack Exchange

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