(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.)
- We show that PFA implies the Weak Baumgartner Axiom.
- With the weak BA is equivalent to BA.
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 (???) .
- 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.
Given a CESM, we need to find an -generic condition below .
(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 .
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.
Exercise. Redefine this poset so it preserves . Do this with matrices (?).