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 ''κ''^''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 ''$\_n$'' be a partition of ''$\beta$'' satisfying the requirements of the theorem.

Fix an increasing sequence $\_$ cofinal in $\alpha$ with $\beta_0=0$.

Note $\mathrm\,(\alpha)\le\kappa$.

Define:

:$X^\alpha _0 = \;\ \ X^\alpha_ = \bigcup_\gamma X^_n\setminus \beta_\gamma$

Observe that:

:$\bigcup_X^\alpha_n = \bigcup _n \bigcup _\gamma X^_n\setminus \beta_\gamma = \bigcup_\gamma \bigcup_n X^_n\setminus \beta_\gamma = \bigcup_\gamma \beta_\setminus \beta_\gamma = \alpha \setminus \beta_0$

and so ''$\bigcup_nX^\alpha_n = \alpha$''.

Let $\mathrm\,(A)$ be the order type of ''$A$''. As for the order types, clearly $\mathrm(X^\alpha_0) = 1 = \kappa^0$.

Noting that the sets $\beta_\setminus\beta_\gamma$ form a consecutive sequence of ordinal intervals, and that each $X^_n\setminus\beta_\gamma$ is a tail segment of $X^_n$, then:

:$\mathrm(X^\alpha_) = \sum_\gamma \mathrm(X^_n\setminus\beta_\gamma) \leq \sum_\gamma \kappa^n = \kappa^n \cdot \mathrm(\alpha) \leq \kappa^n\cdot\kappa = \kappa^$

References

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

Set theory
Paradoxes

{{mathlogic-stub