18. Large cardinals

Lemma 18.9. Let A be a set of infinite cardinals such that for every regular cardinal
κ, the set A ∩ κ is non-stationary in κ. Then there is a one-one regressive function with
domain A.
Proof. We proceed by induction on γ def
= ⋃ A. Note that γ is a cardinal; it is 0 if
A = ∅. The cases γ = 0 and γ = ω are trivial, since then A = ∅ or A = {ω} respectively.
Next, suppose that γ is a successor cardinal κ + . Then A = A ′ ∪ {κ + } for some set A ′
of infinite cardinals less than κ + . Then ⋃ A ′ < κ + , so by the inductive hypothesis there
is a one-one regressive function f on A ′ . We can extend f to A by setting f(κ + ) = κ, and
so we get a one-one regressive function defined on A.
Suppose that γ is singular. Let 〈µ ξ : ξ < cf(γ)〉 be a strictly increasing continuous
sequence of infinite cardinals with supremum γ, with cf(γ) < µ 0 . Note then that for every
cardinal λ < γ, either λ < µ 0 or else there is a unique ξ < cf(γ) such that µ ξ ≤ λ < µ ξ+1 .
For every ξ < cf(γ) we can apply the inductive hypothesis to A ∩ µ ξ to get a one-one
regressive function g ξ with domain A ∩ µ ξ . We now define f with domain A. In case
cf(γ) = ω we define, for each λ ∈ A,
⎧
g 0 (λ) + 2 if λ < µ 0 ,
⎪⎨ µ ξ + g ξ+1 (λ) + 1 if µ ξ < λ < µ ξ+1 ,
f(λ) = µ ξ if λ = µ ξ+1 ,
⎪⎩ 1 if λ = µ 0 ,
0 if λ = γ ∈ A.
Here the addition is ordinal addition. Clearly f is as desired in this case. If cf(γ) > ω, let
〈ν ξ : ξ < cf(γ)〉 be a strictly increasing sequence of limit ordinals with supremum cf(γ).
Then we define, for each λ ∈ A,
⎧
g ⎪⎨ 0 (λ) + 1 if λ < µ 0 ,
µ
f(λ) = ξ + g ξ+1 (λ) + 1 if µ ξ < λ < µ ξ+1 ,
⎪⎩ ν ξ if λ = µ ξ ,
0 if λ = γ ∈ A.
Clearly f works in this case too.
Finally, suppose that γ is a regular limit cardinal. By assumption, there is a club C
in γ such that C ∩ γ ∩ A = ∅. We may assume that C ∩ ω = ∅. Let 〈µ ξ : ξ < γ〉 be the
strictly increasing enumeration of C. Then we define, for each λ ∈ A,
⎧
⎨g 0 (λ) + 1 if λ < µ 0 ,
f(λ) = µ
⎩ ξ + g ξ+1 (λ) + 1 if µ ξ < λ < µ ξ+1 ,
0 if λ = γ ∈ A.
Clearly f works in this case too.
Lemma 18.10. Suppose that κ is weakly compact, and S is a stationary subset of κ. Then
there is a regular λ < κ such that S ∩ λ is stationary in λ.
Proof. Suppose not. Thus for all regular λ < κ, the set S ∩ λ is non-stationary in
λ. Let C be the collection of all infinite cardinals less than κ. Clearly C is club in κ, so
206