(This is the seventeenth (and final) lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the sixteenth 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.)

NOTE: This page is pretty raw as I have not edited it yet. Please tread carefully!

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“I want to go a little over this proof.“

**Theorem 1 (Carlson-Laver, 1989)**. Assuming PFA, .

Recall the notation:

- ;
- , the Amoeba;
- is an -name for a ccc poset, identified with all -names for elements of ;

In we order as follows: iff in the generic filter such that .

**Aside**: We can assume and is continuous such that .

“I would like to show this is a ccc poset.”

**Claim 2**: [In ] is a ccc poset.

We actualy need claim 0. (Remember claim 1 was about a caliber property.)

**Claim 0**: For every sequence of elements of and there is an uncountable and a family of pairwise compactible elements of below such that:

(“Double negation!“)

**proof of Claim 0**.

Apply PFA to which will force uncountably many generics into ‘the set'(???). [**End of proof**]

**proof of Claim 2**.

Let of elements of . By lemma 1, some uncountable subset of belongs to so we may work in .

So we need to produce (below any given ) and such that $(p,n)

\vDash_\mathcal{A} \tau_\alpha \not \perp_\mathcal{A} \tau_\beta$.

In fact, . Set:

- ; and
- . (“Perfect tree, clopen set.”)

Applying lemma 0, we get an uncountable and such hat

Apply lemma 0 again to find an uncountable set and such that

Now if , then for each [… SEE DANA]

“Last time I mentioned these names: Souslin, Sacks, Solovay (random).”

“I decided not to talk about this now as I will be giving a lecture in the Czech Republic in February.”

“[Here is] exposition about two results, actually 3 (one which unifies them) which are actually quite complete.“

**Theorem 2 (1985, Laver)**. implies , for any measure algebra .

“Assuming isn’t interesting as measure algebras preserve Souslin trees. You’d like to use CH, but the next result says there is no hope.“

**Theorem (Hirshurn, 2000)**. PID implies .

## Ultrapowers of Measure Algebra

Let be an index set, a non-principal ultrafilter on , and consider be the ultrapower where .

**Theorem (Todorcevic, 1995)**. If the character (i.e. density of the metric space) of is bigger than then for every sequence , there is a sequence such that:

- (“The ‘s reflect.“) for all ;
- (“More important:”)

“You should dare to do something. It is a Brave New World.“

## Proof of Theorem 2

**Proof of Theorem 2**. “Once you do the proof, you will see you do not want to check if PID implies PID.”

Let be an -name for an Aronszajn tree nicely put on . I.e. is its level for countable limit.

“This idea comes from the original source for PID.“

, where is strongly unbounded if every infinite subset of is unbounded.

is a P-ideal.

“This of course is wht PID implies SH, but not here, because we just have names.“

Let .

Claim: is a P-ideal.

“I strongly suggest you do this with other posets and see if it is P-ideal. e.g. Sacks. Might have something to do with weakly-distributive or -bounding.”

. “What are the pieces to chop off?”

for countable limit.

Let be the -name for .

For , choose a _finite_ set such that

(“i.e. is very close to 1“)

[PICTURE 1]

Let

Note . We need to show this is in .

For , .

[PICTURE 2]

(“You don’t need to worry about the ‘s.“)

By PID, we have that either:

- There is an uncountable such that . (So contains an uncountable antichain .)
- There is an uncountable such that .

() “Here is where we need this measure theory lemma about ultrapowers.”

“What are the functions?“

Pick a uniform ultrafilter on such that . For let be defined by .

Claim. is countable.

**proof**. Otherwise, let .

Apply the ultrapower theorem. We will get some for () that somehow reflects.

(“Here the is really just 2. Just pairs.“)

“If you decode what I wrote, that’s what it says.”

Find such that is countable.

So is a branch of .

“This [] is a good name, a fantastic name! I can’t believe I picked such a good name. It is a branch.”

The generic must pick up some for which [SOMETHING]. So [SOMETHING ELSE]. [End of Proof]

“At () I force to add to the density, so the theorem applies. If I broke the Suslin tree, great!“

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“Let us see that MA() is enough to push through SH.”

“People are interested in some Martin principles.”

Statement of SM. For every *S*et-*M*apping either:

- , where each is -Free.
- There is an uncountable such that for every finite , is uncountable. (“Very much not free.”)

Compare this with the Free-Set Lemma:

Free-Set Lemma. [STATEMENT]

**Theorem (Todorcevic, 1995)**: MA() implies SM, for every measure algebra.

**Exercise**. SM implies that trees on with no uncountable chains are special.

(Use pred mapping, . Use , so you kill Suslin trees.)

**Proposition**. SM implies that if a compact space contains an L-subspace,then contains an uncountable free sequence.

**Recall**. An L-space is a regular, hereditarily Lindelof, non-separable space. (These exist in ZFC.)

**Corollary**. SM implies that perfect compacta are separable.

**Sketch**
SM

, let

be the measurable subsets of

and

## Proof of Proposition.

Assume is an L-space. This means (by possibly passing to a subsequence) that

By regularity, pick open in such that . Define , by .

“Free set is a discrete space.” (??)

For in the second alternative we get , we have that the collection of and (for ) has the finite intersection property. [**End of proof**]

## Now We Try to Prove the Theorem.

**proof**. Fix an -name for a set-mapping as in SM. For , is countable, by measure algebra.

“I want to apply MA, so I need a poset. The idea seems to be to force [the first alternative], and if it isn’t ccc, then [the second alternative].”

“I will force one free set, but there is the usual trick to get many.“

Let be the subalgebra generated by , which is separable as a metric space.

**Poset**. Let be the collection of all finite partial maps such that:

- for .
- for .

“[1] is a tricky move. For measure algebras making the requirement rarely gives ccc. But I’m hopeful because I’veput it in the separable algebra .“

**Ordering**. iff

“If it is (powerfully) ccc, you are forcing alternative 1.”

“So we focus on alternative 2.“

**Claim**. If is not ccc, then there is an -name for an uncountable subset of satisfying the second alternative of SM.

“First a -system, etc.“

Let be a sequence of pairwise incompatible elements in .

“Now we have to be careful.”

Fix , for , sitting in a separable [SOMETHING SOMETHING].

So [SOMETHING], we can ignore the root.

“Now the u.p. thing.“

Suppose , for all , . ?

[SOMETHING].

Here are some references:

- Singapore Lecture Notes. [SEE DANA]
- Combinatorial Dichotomies of Set Theory. (BSL 2011).