(In the fall of 2012 I will be taking Stevo Todorcevic’s class in Forcing at the University of Toronto. I will try to publish my notes here, although that won’t always be possible.)
Summary of class 1.
- Discuss examples of ccc posets whose product is not ccc.
- Prove a theorem of Baumgartner’s that relates the branches in a tree to its antichain structure.
- Display the differences in chain conditions.
This class is going to be held Wed 1-2 and Friday 3:30-5:30 in BA 6183.
The theme for today is “Things that are ccc, but whose products are not”. So first some examples:
Example 2. Let finite antichains of
, where
is a Suslin Tree. Then
is powerfully ccc (its finite products are ccc), but
is not ccc.
“A Suslin tree is the skeleton of a Suslin continuum. If you take x-rays, that’s what you get.” – Stevo
This example was first studied by Baumgartner in his thesis:
Theorem (Baumgartner, 1969): TFAE for a tree :
is ccc;
has no uncountable branches.
It is clear that NOT (2) imples NOT (1), as the singletons of a branch form an antichain.
“There are many proofs of this fact, but Baumgartner’s original proof is still the most useful.” – Stevo
Proof. Like most proofs that a poset is ccc, we start with an uncountable subset of the poset and refine, refine, refine.
Let , we want to find
such that
. We refine 3 times to an uncountable set
so that:
- (All conditions are the same length)
for all
,
- (
is a
-sytem) There is an
such that for
in
we have
.
- (Heights are increasing) The
for
.
We want to assume that is actually empty, and we can do this by considering
. Note that
implies
.
WLOG, we get .
We suppose that for all
in
(and look for a contradiction). This means that if
then there is an
and a
such that
.
Now take a uniform ultrafilter on
such that
.
Fix . Write
.
There is an such that
.
Define .
Question: Is there a large antichain in X?
Now for each there is a
so that
. Thus
and we can choose an
in this intersection so that
for all
.
Now if does not depend on
then we are done. This follows from exercise 2.
[QED]
Some Exercises:
Exercise 1: Suppose that
is a ccc poset and let
. Show that there is an infinite set
such that
are pairwise compatible.
Exercise 2: Let
be a tree with an uncountable subset that has no antichains of size
. Show that
has an uncountable chain.
Some chain conditions, listed from easiest to satisfy to hardest to satisfy:
- ccc
- powerfully ccc
- productively ccc
-finite-cc
-bounded-cc
-2-linked
-n-linked
-n-linked (
)
-centred
- countable
Note that Borel posets can distinguish all of the blue and dark blue properties. By the following exercise, the dark blue posets are small.
Exercise 3. If
is an atomless
-(2)-linked poset then
.
Hint. Use the Erdös-Rado theorem.
Question. Can a boolean algebra supporting a continuous submeasure algebra distinguish -finite-cc from
-bounded-cc?
Previous Class – Next Class
Skip to a class: 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10 – 11 – 12 – 13 – 14 – 15 – 16 – 17
Thanks for sharing these notes Mike!
Small typo: In “[…] such that
" the first P should be lower case.
I also have a very hard time seeing the difference between "blue" and "dark blue" on my laptop. Could you use more contrasting colors?
LikeLike
Thanks for the feedback. These kind of comments are very helpful. I’ve made the light blue lighter, so hopefully there is more “pop”.
LikeLike
Do we need to add some additional assumptions in exercise 3? (like having the poset be separative) As it stands we could just add a huge blob of mutually compatible guys at the top of some centered forcing, say Cohen forcing, and violate the conclusion of cardinality bounded by the continuum
LikeLike
What is the ordering of the A(T) posets?
LikeLike
It should just be inclusion. I.e. One antichain is compatible with another, if their union is an antichain.
LikeLike