Take-home final exam for Math 771: Set Theory; Spring 2016
Due: May 16 at noon
1) Show that ZF proves the following fact: if (M, R) is a model such that
(M, R) ZF + V = L,
and also such that R is well-founded on M , then there is an ordinal α such that
(M, R) is isomorphic to (L(α), ∈).
2) In this problem, we assume that Martin’s Axiom is true, and that 2ℵ0 ≥ ℵ38 .
Let A, B ⊆ P(ω) both have size ℵ37 . Furthermore, let us assume that whenever
A0 ⊆ A and B 0 ⊆ B are such that A0 and B 0 both have size ℵ1 , then there is a
set x ⊆ ω satisfying that ∀a ∈ A0 (a ⊆∗ x) and ∀b ∈ B 0 (b ⊥ x) (see Definition
III.1.15). Show that there is an x ⊆ ω such that in fact ∀a ∈ A(a ⊆∗ x) and
∀b ∈ B(b ⊥ x).
Hint: consider
n
P = (X, Y ) | X, Y ⊆ ω, X ∩ Y = ∅, and there exist finite subsets U ⊆ A, V ⊆ B
[
[ o
such that X =∗
U and Y =∗
V.
3) Let M be a ctm for ZF. Let N = (L)M . Let P = Fn((ω2 )N × ω, ω). Let G be
P-generic over N . We consider the model N [G].
We will be showing, in steps, that N [G] is a model of the statement that
there is a sequence of ω2 many functions fγ : ω → ω such that for every g ∈ ω ω ,
{γ | fγ ≤∗ g} is countable.
In fact, we get these functions as follows: given γ < (ω 2 )N , let fγ = {(n, m) |
{((γ, n), m)} ∈ G}.
a) Show that N [G] satisfies the statement “every fγ is a total function, and the
set {fγ | γ < (ω2 )N } has size ω2 ”.
b) Let f ∈ N [G] be such that N [G] f : ω → ω. Show: there is set Γ ⊆ (ω2 )N
such that (Γ is countable)N , and there is an f˚ ∈ N P satisfying f˚G = f such
that f˚ is actually a Fn(Γ × ω, ω)-name.
c) Let f ∈ N [G] be such that N [G] f : ω → ω, and let Γ and f˚ be for f as in
the previous subproblem. Show: if γ 6∈ Γ, then
N [G] fγ 6≤ f.
Hint: let (pi )i<ω ∈ N be an enumeration of Fn(Γ × ω, ω). Working in N , for
every i ∈ ω, if there is a qi ≤ pi with qi ∈ Fn(Γ × ω, ω) and an ni ∈ ω such
that
qi f˚(i) = ňi ,
choose such a qi and ni . Now let Dγ be the set of those r ∈ P such that for
some i ∈ ω, both r ≤ qi and r ≤ {((γ, i), ni + 1)}. Argue that G ∩ Dγ 6= ∅.
d) Show that N [G] is indeed a model of the statement that there is a sequence
of ω2 many functions fγ : ω → ω such that for every g ∈ ω ω , {γ | fγ ≤∗ g}
is countable.