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 consistent with
ZFC, and use to construct a Suslin tree.
In this chapter we study innite trees. The main things we look at are Knigs tree
theorem, Aronszajn trees, and Suslin trees.
A tree is a partially ordered set (T, <) such that for each t T , the set cfw_s T : s < t
is well-ordered by the relat
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 isomorphic, the existence of sets, the existence of compl
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 |f | < , , ).
We rst show that CH is consistent. The ma
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 c.t.m. for
countable transitive model; see Theorem 10.
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.
We now need to expand on the foundations of set theory a
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 their relation to certain
A Boolean algebra (BA)
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 bijection between them. A cardinal, or cardinal number, i
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, and we give some references for this after proving the
6. Ordinals, II
August 9, 2013
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. Then every nonempty subclass of A has a
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 classes
Classes and sets
Although expressions like
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. This is a very important notion in the foundations of
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 of this chapter is
to establish common notation.
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 x is included or contained in y , or that x is a subset
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 use of the framework is
just to make the development rig