Axiomatic Set Theory 1: Exercises 4
April 3, 2017
Exercise 1. Let κ be an infinite cardinal and let C be any well-order of κ.
Show that there is an X ⊆ κ with |X| = κ such that C and < agree on X.
Hint: For regular κ, it’s easy to choose X by recursion in κ steps.
For singular κ show that there exist (λi |i < cf(κ)) increasing and <-cofinal in
κ and such that for every λi exists Xi with |Xi | = λi such that C and < agree
on Xi which is neither <-cofinal nor C-cofinal in κ and with the property that
for j < i holds that Xj ⊆ Xi .
Let Xi0 = {αρ |ρ ∈ λi } <-inceasing enumerated with Xi0 ⊆ λi with |Xi0 | = λi
such that C and < agree on Xi0 . Show that there exists Xi with Xi ⊆ λi+1 with
|Xi | = λi such that C and < agree on Xi and such that Xi is not C-cofinal in
κ.
S
Finally take X = Xi .
Exercise 2. If κ is anSinfinite cardinal and κ ≤ α < κ+ , then there are Xnα ⊆ α
for n < ω such that
Xnα = α and each type(Xnα ) ≤ κn (ordinal exponentian<ω
tion).
Hint: Induction on α.
α
For α limit let (βγ |γ < cf(α)) be increasing and cofinal in α. Let Xn+1
:=
S
βγ+1
\ βγ .
Xn
γ<cf(α)
Exercise 3 (Löwenheim-Skolem Downwards). Let L be a language an B and
L-structure. Fix κ such that max(|L|, ℵ0 ) ≤ κ ≤ |B|, and fix S ⊆ B with
|S| ≤ κ. Show that there is an A B such that S ⊆ A and |A| = κ.
Exercise 4. Let γ > ω1 be a limit ordinal. Prove that there is a countable
transitive M and ordinals α, β ∈ M such that M ≡ R(γ) and (α ≈ β)M is false
but (α ≈ β)R(γ) is true.
Hint: By Löwenheim-Skolem Downwards get a countable A with ω, ω1 ∈ A R(γ). Then, let M be the Mostowski collapse of A; let α = mos(ω) = ω
and β = mos(ω1 ). Then β will be a countable ordinal that M “thinks” is
uncountable.
Exercise 5. Show that the following are ∆0 definable:
1. x is a transitive set.
4. x is a limit ordinal.
2. x is an ordinal.
5. x is a natural number.
3. x is a successor ordinal.
6. x = ω.
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