Milner–Rado paradox explained

\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 ^{\beta} _{n\}} _{n} be a partition of \beta satisfying the requirements of the theorem.

Fix an increasing sequence

\{\beta _{\gamma\}} _{\gamma<cf(\alpha)} cofinal in \alpha with \beta _{0=0} .

Note

cf (\alpha)\le\kappa .

Define:

X ^{\alpha} _{0} = \{ 0 \}; X ^{\alpha} _{n +1} = c up _{\gamma} X ^{\beta _{\gamma+1}} _{n\setminus} \beta _{\gamma}

Observe that:

c up _{n>0} X ^{\alpha} _{n} = c up _{n} c up _{\gamma} X ^{\beta _{\gamma+1}} _{n\setminus} \beta _{\gamma} = c up _{\gamma} c up _{n} X ^{\beta _{\gamma+1}} _{n\setminus} \beta _{\gamma} = c up _{\gamma} \beta _{\gamma+1} \setminus \beta _{\gamma} = \alpha \setminus \beta _{0}

and so

c up ^{\alpha} _{n} = \alpha .

Let

ot (A) be the order type of A . As for the order types, clearly 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:ot (X ^{\alpha} _{n +1} ) = \sum _{\gamma} ot (X ^{\beta _{\gamma+1}} _{n\setminus\beta} _{\gamma)} \leq \sum _{\gamma} \kappa ^{n} = \kappa ^{n} ⋅ cf (\alpha) \leq \kappa ^{n ⋅ \kappa} = \kappa ^{n+1}

References

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