15. Clubs and stationary sets
Here we introduce the important notions of clubs and stationary sets. A basic result here
is Fodors theorem. We also give a combinatorial principle , later proved consist
14. Trees
In this chapter we study innite trees. The main things we look at are Knigs tree
o
theorem, Aronszajn trees, and Suslin trees.
A tree is a partially ordered set (T, <) such that for each t T
13. Linear orders
In this chapter we prove some results about linear orders which form a useful background
in much of set theory. Among these facts are: any two denumerable densely ordered sets
are is
12. Independence of CH
The forcing orders used in this chapter are special cases of the following. For sets I, J and
for an innite cardinal,
Fn(I, J, ) = (cfw_f : f is a function contained in I J and
11. Generic extensions and forcing
In this chapter we give the basic denitions and facts about generic extensions and forcing.
Uses of these things will occupy much of remainder of these notes. We use
10. Models of set theory
In this chapter we describe the basics concerning models of set theory, ending with a proof
that if ZFC is consistent, then so is ZFC+there are no uncountable inaccessibles.
W
9. Boolean algebras and forcing orders
To introduce the apparatus of generic extensions and forcing in a clear fashion, it is necessary to go into a special set theoretic topic: Boolean algebras and t
8. Cardinals
This chapter is concerned with the basics of cardinal arithmetic.
Denition and basic properties
To abbreviate longer expressions, we say that sets A and B are equipotent iff there
is a bi
7. The axiom of choice
We give a small number of equivalent forms of the axiom of choice; these forms should be
sucient for most mathematical purposes. The axiom of choice has been investigated a
lot,
6. Ordinals, II
August 9, 2013
Transnite induction
The transnite induction principles follow rather easily from the following generalization
of Theorem 4.13.
Theorem 6.1. Let A be an ordinal, or On. T
5. Recursion
August 4, 2013
In this chapter we prove a general recursion theorem which will be used many times in
these notes. The theorem involves classes, so we begin with a few remarks about classe
4. Ordinals, I
August 28, 2013
In this chapter we introduce the ordinals and give basic facts about them.
A set A is transitive iff x Ay x(y A); in other words, iff every element of
A is a subset of A
3. Elementary set theory
August 1, 2013
Here we will see how the axioms are used to develop very elementary set theory. The axiom
of choice is not used in this chapter. To some extent the main purpose
2. The axioms of set theory
Before introducing any set-theoretic axioms at all, we can introduce some more abbreviations.
x y abbreviates z (z x z y ).
x y abbreviates x y x = y .
For x y we say that
Set theory
1. First-order logic
July 29, 2013
Here we describe the rigorous logical framework for set theory. This logical framework is
important for all the notes. For the rst few chapters the main u