Stevo’s Forcing Class Fall 2012 – Class 9

(This is the ninth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the eighth 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.)

Proper Forcing

The most important notion in forcing. It is about generic conditions.”

You have $p, \mathbb{P} \in M \prec H_\theta$. You want to find $q \leq p$ which is $(M,\PP)$-generic.

Definition. $q \in \mathbb{P}$ is $(M, \mathbb{P})$-generic if $\forall \mathcal{D} \in M$, dense open and $\forall r \leq q$ in $\mathcal{D}$ there is an $\overline{r} \in \mathcal{D}\cap M$ such that $\overline{r} \not \perp r$. A $q$ with this property is also sometimes called a master condition.

Definition. A poset $\mathbb{P}$ is proper if for all large enough $\kappa$, for every condition $p \in \mathbb{P}$, and all CESMs (countable elementary submodels) $M \prec H_\kappa$ containing $\mathbb{P}$ and $p$, the condition $p$ can be extended to an $(M, \mathbb{P})$-generic condition.

This is a kind of generalization of ccc. In ccc every $r$ can be copied.”

(For a refresher on CESMs, you may wish to check out my FAQ. -Mike)

Examples

ccc is an example, but it is not the right example.”

Let $\epsilon$-Col$(\kappa)$, called “the epsilon collapse of $H_\kappa$“, be the collection of all finite $\in$-chains of CESMs of $H_\kappa$, ordered by inclusion.

Well, okay, so we have to check this.”

Claim: $\epsilon$-Col$(\kappa)$ is a proper poset.

proof. Take some $\theta$ much larger than $\kappa$ and $p,\PP \in \epsilon$-Col$(\kappa) \in M \prec H_\theta$. Take $q = p \cup \{M \cap H_\kappa\}$.

You don’t need to think. Let us check whether this is $(M, \epsilon$-Col$(\kappa))$-generic.

You have to do this ‘copying’.

Let $\mathcal{D} \in M$ be a dense open subset of $\epsilon$-Col$(\kappa)$ and $r \leq q$ in $\mathcal{D}$ be given.

By elementarity of $M$, there is an $\overline{r} \in \mathcal{D}$, $\overline{r} \in M$ such that $\overline{r}$ end-extends $r \cap M \in M$.

Now notice that $r \cup \overline{r}$ is an $\in$-chain, therefore is a member of $\epsilon$-Col$(\kappa)$. [QED]

This of course collapses. Next time we will see the one that preserves chain conditions. There is not much choice.”

Note: If $\mathfrak{G}$ is a generic filter of $\epsilon$-Col$(\kappa)$, then $\bigcup \mathfrak{G}$ is an $\in$-chain (of type $\omega_1$) of CESM of $(H_\kappa)^V$ which covers $H_\kappa$. Why order-type $\omega_1$? This is because if $M \in N \in K$ are all CESM of $H_\kappa$ then $M \in K$. This will tell us that there cannot be a CESM with uncountably many predecessors. The fact that it is at least $\omega_1$ is by a recursive construction.

Exercise: Proper posets preserve $\omega_1$.

Exercise (suggested by Justin Moore): This adds reals, and only cohen reals.

How much did we use that $\mathcal{D} \in M$?

This Suggests a General method Approach for Producing Proper Posets

Method is too strong. It just tells you in general what your proper poset should look like. It doesn’t tell you how to arrange the working parts.

Typically a proper poset will look like a collection of pairs $p = (H_p, \mathcal{N}_p)$ where $H_p$ is a finite subset of your structure and $\mathcal{N} \in \epsilon$-Col$(\kappa)$.

1. This is the poset you use if you want to show PFA implies something.”
2. If you want to show PFA implies something try finite approximations (the $H_p$). If it is ccc, great. If not, you add the CESMs (\mathcal{N}_p) and try again.
3. There are other equivalencies for a condition $q$ to be a master condition. For example, in the definition of master condition, we can replace the dense open sets $\mathcal{D}$ by all sets $X \in M \cap \mathcal{P}(\PP)$ containing $q$.
4. Genericity tells you every antichain below a condition looks countable.

I want now to go back a little to the original. It is up to you to decide what example you like. If you don’t choose, I will choose for you.

Most of the time you want to extend by $q = (H_p, \mathcal{N}_p \cup \{M \cap H_\kappa\})$, but it is not always to your advantage.

A Dichotomy about Open Graphs on Separable Metric Spaces.

That is a good example!” – Oswaldo

See where I use separability. See if you can relax it. Open is quite natural.”

Fix $(X,E)$, $X$ a separable metric space, $E \subseteq X^2 \setminus \Delta$ which is symmetric open.

Dichotomy:

1. $(X,E)$ embeds into a countable discrete subset; or
2. A complete uncountable graph embeds (injectively) into $(X,E)$.

Graph homomorphism preserves edges in one direction.”

We will start with an open graph $(X,E)$ that does not embed into a countable discrete subset, (and $X \notin J(X,E)$), and proceed to force a complete uncountable graph into $(X,E)$.

Definition. $J(X,E)$ is $\sigma$-generated by the ideal $\{Y \subseteq X : (Y,E)\textrm{ is discrete}, Y^2 \cap E = \emptyset\}$.

(Assume your metric space $X$ has some reasonable chain condition (like $\mathfrak{c}^+$-cc).) Let $\mathbb{P}$ be the collection of all maps $p : \mathcal{N}_p \rightarrow X$, where $\mathcal{N}_p \in \epsilon$-Col$(\cc^+)$ and $(X,E) \in N$ for all $N \mathcal{N}_p$ such that

1. range$(p)$ is a complete subgraph of $(X,E)$; and
2. $H_p$ is strongly separated by $\mathcal{N}_p$. “It is far away from all complete subgraphs.” i.e. for all $N \in \mathcal{N}_p, \forall Y \in J(X,E) \cap N, p(N) \in Y$ and $M \in N \in \mathcal{N}_p$ implies $p(m) \in M$.

You are trying to avoid the obvious disaster. Life is so simple, that this is enough.”

Claim: $\mathbb{P}$ is proper.

Let us do this because this is simple. We have 10 minutes, just enough.”

This is really Fubini combinatorics.”

Definition. $\mathcal{H}(X,E) := \mathcal{P}(X) \setminus J(X,E)$, and $\mathcal{H}^n(X,E)$ is called the $n$th Fubini power.

Why is this relevant? We forced it!

Lemma. If $p \in P$, $\vert \mathcal{N}_p \vert = n$ and $Z \subseteq X^n$, with $Z \in \min (\mathcal{N}_p)$, the minimal model that $Z$ is in, $\mathfrak{ran}(p) \in Z$ THEN $Z \in \mathcal{H}^n(X,E)$. (And vice versa.)

Copying Lemma. For $Z \in \mathcal{H}^n(X,E)$, its derivative is $\delta Z := \{\vec z \in Z : (\forall \epsilon)(\exists \vec y \in Z)[d(\vec y, \vec z) < \epsilon]\textrm{ and }\forall i, (y_i, z_i) \in E\}$. Observe, $Z \setminus \delta Z \notin \mathcal{H}^n(X,E)$.

This tells us that $\delta Z$, the set of all things you can copy, is almost all of $Z$.

This lemma is proved by induction. It is where separability is used. Use the countable base!

Go through this. Or Alan will be angry!

This lemma will give me $Y$ somewhere, but why in the model? Hereditary separability will give us something in the model close.

This density is actually interesting. You need this density to force an uncountable thing.

Start with a single point and a single model and this will force the generic to be uncountable. But, it is a cheap trick. I gave this example in Partition Problems in Topology, but some people didn’t like it.

Claim: If $p \in \mathbb{P}$ and $p \in M \prec H_{\cc^+}$ there is $x \in X \setminus M$ such that

1. $x \notin Y$, for every $Y \in M, Y \in J(X,E)$
2. range$(p) \cup \{x\}$ is $E$-complete.

If $p$ is empty, you’re done. Otherwise the last model is going to help us. Double it! If you find a point compatible anywhere, you can find positive many.”

6 thoughts on “Stevo’s Forcing Class Fall 2012 – Class 9”

1. Ari Brodsky says:

In the definition of “proper”, the $\kappa$ should be $\theta$. Also, the sentence should begin “A poset $\mathbb P$ is proper…”, since the $\mathbb P$ is used later in the sentence.

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2. Ari Brodsky says:

There’s a mistake in the definition of proper. “every condition in $\mathbb P$ can be extended to an $(M, \mathbb P)$-generic condition” should be “every condition in $\mathbb P \cap M$….”

This could have been a good example of “being relaxed when stating the theorems [or in this case, definitions], because we’ll figure it out when we do the proofs” (see the quote in my comment here ), except that apparently he didn’t notice where the extra condition was used in the proof. When you show that the epsilon-collapse is proper, the first step is to define
$q = p \cup \left\{ M \cap H_k \right\},$
and then we show that $q$ is $(M, \mathbb P)$-generic. But how do we know that $q$ is actually an element of the partial order? We need to show it’s an $\in$-chain of ESMs. This is slightly non-trivial and should be checked once as an exercise. In particular, in order for each model in $p$ to be compatible with $M \cap H_\kappa$, we need to assume that $p \in M$, otherwise it won’t work.

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1. saf says:

I second that. Here is a rephrase:
For every condition $p\in\mathbb P$ and every CESM containing $\mathbb P$ and $p$, admits a master condition…

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1. Micheal Pawliuk says:

Thanks you two! I’ve fixed that. Also, Assaf, do you mind explaining what your objection was in the “This Suggests a General method Approach for Producing Proper Posets” section?

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2. saf says:

My objection: the $\epsilon-\text{Col}(\kappa)$ does not do justice to $\sigma$-closed forcing, or any other proper posets that do not add reals).
Nevertheless, if one is only interested in producing proper posets in order to derive consequences of PFA, then Miyamoto has recently shown that the so-called general approach is indeed correct.

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3. Chris Brown says:

Wow. I’ve really enjoyed this series of posts! I gave a presentation on Baumgartner’s result on ℵ_1-dense sets of reals, but I still haven’t gotten as deep into forcing as I would like.

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