16. Innite combinatorics
In this chapter we survey the most useful theorems of innite combinatorics; the best
known of them is the innite Ramsey theorem. We derive from it the nite Ramsey
theorem.
Two sets A, B are almost disjoint iff |A| = |B | while |A
28. Main conality theorems
The sets H
We will shortly give several proofs involving the important general idea of making elementary chains inside the sets H . Recall that H , for an innite cardinal , is the collection
of all sets hereditarily of size less
27. Basic properties of PCF
For any set A of regular cardinals dene
pcf(A) = cf
A/D : D is an ultralter on A .
By denition, pcf() = . We begin with a very easy proposition which will be used a lot
in what follows.
Proposition 27.1. Let A and B be sets of
26. Conality of posets
We begin the study of possible conalities of partially ordered setsthe PCF theory. In
this chapter we develop some combinatorial principles needed for the main results.
Ordinal-valued functions and their orderings
A lter on a set A
25. More examples of iterated forcing
We give some more examples of iterated forcing. These are concerned with a certain partial
order of functions. For any regular cardinal we dene
f < g
iff
f, g and there is an < such that f ( ) < g ( ) for all [, ).
Th
24. Proper forcing
The notion of proper forcing is dened in terms of clubs and stationary sets of the form
[]<1 , where is an uncountable cardinal. We write [] in place of []<1 .
Let A be uncountable. Let KA be the collection of all algebras with universe
23. Various forcing orders
In this section we briey survey various forcing orders which have been used. Many of
them give rise to new real numbers, i.e., new subsets of . (It is customary to identify
real numbers with subsets of , since these are simpler
22. Embeddings, iterated forcing, and Martins axiom
In this chapter we mainly develop iterated forcing. The idea of iterated forcing is to
construct in succession M [G0 ], M [G0 ][G1 ], etc., continuing transnitely, but stopping at
some stage M [G0 ][G1 ]
21. Isomorphisms and AC
In this chapter we prove that if ZF is consistent, then so is ZF + AC. First, however, we
go into the relationship of isomorphisms of forcing orders to forcing and generic sets; this
is needed for the consistency proof, and is inde
20. Powers of regular cardinals
In this chapter we continue Chapter 12, and describe in more detail the possibilities for
2 when is regular, where the results are fairly complete. The case of singular is more
involved, and there are still important open p
19. Constructible sets
This chapter is devoted to the exposition of Gdels constructible sets. We will dene a
o
proper class L, the class of all constructible sets. The development culminates in the proof
of consistency of AC and GCH relative to the consis
18. Large cardinals
The study, or use, of large cardinals is one of the most active areas of research in set theory
currently. There are many provably dierent kinds of large cardinals whose descriptions
are dierent from one another. We restrict ourselves
17. Martins axiom
Martins axiom is not an axiom of ZFC, but it can be added to those axioms. It has many
important consequences. Actually, the continuum hypothesis implies Martins axiom, so
it is of most interest when combined with the negation of the con
Algebra 2 Cheat Sheets!
(shhhhhh.)
Graphing Absolute Value equations (Cheat Sheet)
Steps:
1: set inside = to zero.
2: solve for x.
3: create a table with the found x value in the middle.
4: Plug x back into the equation to find y. (This is the vertex coor