Samiksha Jaiswal (Editor)

Milner–Rado paradox

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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 α less than the successor κ+ of some cardinal number κ can be written as the union of sets X1,X2,... where Xn is of order type at most κn for n a positive integer.

Proof

The proof is by transfinite induction. Let α be a limit ordinal (the induction is trivial for successor ordinals), and for each β < α , let { X n β } n be a partition of β satisfying the requirements of the theorem.

Fix an increasing sequence { β γ } γ < c f ( α ) cofinal in α with β 0 = 0 .

Note c f ( α ) ≤ κ .

Define:

X 0 α = { 0 } ;     X n + 1 α = ⋃ γ X n β γ + 1 ∖ β γ

Observe that:

⋃ n > 0 X n α = ⋃ n ⋃ γ X n β γ + 1 ∖ β γ = ⋃ γ ⋃ n X n β γ + 1 ∖ β γ = ⋃ γ β γ + 1 ∖ β γ = α ∖ β 0

and so ⋃ n X n α = α .

Let o t ( A ) be the order type of A . As for the order types, clearly o t ( X 0 α ) = 1 = κ 0 .

Noting that the sets β γ + 1 ∖ β γ form a consecutive sequence of ordinal intervals, and that each X n β γ + 1 ∖ β γ is a tail segment of X n β γ + 1 we get that:

o t ( X n + 1 α ) = ∑ γ o t ( X n β γ + 1 ∖ β γ ) ≤ ∑ γ κ n = κ n ⋅ c f ( α ) ≤ κ n ⋅ κ = κ n + 1

References

Milner–Rado paradox Wikipedia
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