(This is the twelfth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the eleventh lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

**Summary**

- We show that PFA implies the Weak Baumgartner Axiom.
- With the weak BA is equivalent to BA.

## Baumgartner’s Axiom

Let be of cardinality . (We will see that -dense is sufficient.) We would like to construct a proper poset (cardinal-preserving under CH) which forces a strictly increasing . I.e. We wish to show the following theorem.

**Theorem**. PFA implies .

“This is one of those times where designing the dense sets can be a problem.“

**Conditions**. Let where:

- is a finite increasing map from into ; and
- ; and
- is a finite -chain of CESMs of which separates and arbitrary (???) .

Moreover,

- with and ;
- such that iff ;
- we have (No way you can cross the important model)

“This is the source of OCA.”

See the closed mapping lemma from partition problems in Topology. (REFERENCE!)

**Dense Sets**. is dense in .

**Claim**. is proper.

**
proof**.

Given a CESM, we need to find an -generic condition below .

Let .

(We want to use the previous lemma (from class 11), which needs everything on top of a model.)

Take such that and . We need to copy to a compatible condition .

Note . So we are after an end-extension of . “The bottom doesn’t interfere with the top. The previous lemma gives us what we want.”

Let be the collection of partial functions of the form where .

Of course , and .

By the previous lemma there is an such that is increasing. So in particular, there is such an .

(“By -density. I cannot define a function using as it is not in . But, I can use rational intervals.“)

Pick such that . Then . [**QED**]

“It is one thing to be silly, it is another to be wrong. This time we were lucky.”

“Shouldn’t the important models be limits of CESMs?” – David

“This quite correct. If you want to be really correct.”

Require: , where .

## Equivalence of Baumgartner’s Axiom and its weak form.

**Lemma**. If , then is equivalent to .

**
proof**.

Pick two -dense sets of reals and . We may assume that they are -dense in .

For each pair of open rational intervals fix two increasing injections:

;

(The indices suggest the domain and range.)

**Conditions**. Let be the poset of finite strictly increasing maps from to such that:

For each there exist a pair of open rational intervals such that either or .

Now we show that this is -centred and it does the job.

**Dense Sets**. and both and are dense open.

**Claim**. is -centred.

**proof**. Given we look at its invariants:

- Which rational intervals were used;
- Which directions they go.

[**QED**]

**Exercise**. Redefine this poset so it preserves . Do this with matrices (?).