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
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 nonempt
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
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 ordi
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
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
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
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 isomorph
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-
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
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.
D
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