Lecture 7.
The Innite Pigeonhole Principle If you split an innite set into nitely
many pieces, then one of the pieces is innite.
Denition 0.1. A tree T is a partially ordered set (T, <) such that for any
x T , pred(T, x) is well-ordered by <.
Denition 0.2
Lecture 4.
Theorem 0.1 (Hartog). For each set A, there exists an ordinal such that
A.
Theorem 0.2. (AC) implies (WP).
Theorem 0.3. (CP) implies (WP).
Denition 0.4 (Zorns Lemma). If (P, ) is a nonempty poset and every chain
in P has an upper bound, then P
Lecture 3.
1
Ordinals
Denition 1.1. A set T is transitive if whenever a b T , a T . Equivalently, every element of T is a subset of T .
Denition 1.2. A set is an ordinal number (or just an ordinal) if it is transitive
and well-ordered by .
Theorem 1.3.
1.
Lecture 5.
1
Cardinals
From here on out, we will be assuming the Axiom of Choice, and wont really
refer to it when we use it.
Denition 1.1. For each set A, the cardinality of A is
|A| = the least ordinal such that A .
Denition 1.2. An ordinal is a cardina
These notes are not intended to be full presentations of the material. Proofs
are excluded, and most examples are as well. Consult your favorite reference,
or attend lecture. (Or both!)
Lecture 1.
In set theory, everything we look at is a set, but we rare
Lecture 6.
Denition 0.1. If A is any set, then A< = n An .
Theorem 0.2. If is an innite cardinal, then |< | = .
Denition 0.3. Let cfw_i iI be an indexed set of cardinal numbers. Then dene
i = | iI (i cfw_i)|
i I
i = |cfw_f | f is a function, dom f = I, i
Lecture 9.
We began by using ultralters to reprove the following:
Corollary 0.1. Every innite graph contains an innite clique or an innite
anti-clique (i.e. independent set).
Denition 0.2. Suppose A N. Then
A = cfw_ aAf a | Af A is nite .
(Note that there
Lecture 11.
Theorem 0.1 (Solovay). Every regular uncountable is the disjoint union of
stationary sets.
Theorem 0.2. Let be a regular uncountable cardinal. Then there are 2
DLOs of size up to isomorphism.
Note that this is the most there possibly could be
Lecture 8.
Theorem 0.1 (Ramsey). Let n, k , S [ ] . For every coloring F : [S ]n
k , there is an innite homogenous H S .
Corollary 0.2. Every innite graph contains an innite clique or an innite
anti-clique (i.e. independent set).
Corollary 0.3 (Finite Ra
Lecture 12.
Denition 0.1 (The Continuum Hypothesis (CH). 20 = 1 .
The Continuum Hypothesis does not hold in every model of the ZFC axioms
of set theory. Next quarter we will see a model of set theory in which it does
hold.
Corollary 0.2 (CH). |R| = 1 .
Th
Lecture 10.
Throughout, we will use the abbreviation DLO to stand for dense linear
orders without endpoints.
Denition 0.1. Let A be a DLO. The rational right extension of A is the DLO
Ar = A
cfw_
Q.
Denition 0.2. Let A be a DLO. The irrational right exten
Lecture 2.
Denition 0.1 (The Comparability Principle (CP). If A, B are sets, then
A B or B A.
This does not follow from anything weve done so far! In fact, it is equivalent
to:
Denition 0.2 (The Axiom of Choice (AC). For every family F of nonempty
sets, t