Stevo’s Forcing Class Fall 2012 – Class 6

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

Caliber Properties

Definition. $\PP$ has caliber $(\theta, \lambda, \kappa)$ if for every $\theta$-sized family $F \sse \PP$ can be refined to a $\lambda$-sized $G \sse F$ such that all families of $H \sse G$ size $\kappa$ have a lower bound in $\PP$.

Examples:

• $\PP$ has pre-caliber $\theta$ iff $\PP$ has caliber $(\theta, \theta, <\omega)$.
• $\PP$ has property $K_\theta$ iff $\PP$ has caliber $(\theta, \theta, 2)$.
• $\PP$ has property K(naster) iff $\PP$ has caliber $(\omega_1, \omega_1, 2)$.

Topological Definition (Šanin): A compact space $K$ is $\theta$-caliber iff the poset of non-empty open subsets of $K$ have pre-caliber $\theta$.

This definition was introduced because separability is not preserved by products, but caliber properties are preserved.

Exercise: Show that Cohen $\times$ Random has caliber $(\omega_1, \omega, \omega)$ while Cohen * Random does not.

The ccc Functor is ccc

Here is a summary of the notation from Class 5, where we defined this functor.

We are starting with a powerfully ccc poset $\PP$ of cardinality $<\theta_2$ and we are building its specializing poset $\mathcal{S}(\PP)$ and

• $d: [\kappa]^{<\omega} \rightarrow \PP$ such that $d[X]^{<\omega} = \PP$ for all $X \in [\kappa]^\kappa$;
• Fix a 1-1 sequence of reals $\{r_\xi : \xi < \kappa\} \sse \{0,1\}^\omega$;
• $c: \kappa \times \kappa \rightarrow \omega$ is not constant on any product of two infinite sets;
• $c^* (\alpha_0, …, \alpha_{k-1}) := \max \{c (\alpha_i, \alpha_j) : i \neq j < k\}$;
• $\mathcal{S}(\PP) \sse [\kappa]^{<\omega}$;
• For $F \in [\kappa]^{<\omega}, \vec s \in (2^{<\omega})^{<\omega}$ of some length $k$, define $\PP_F := \{d(\alpha_0, …, \alpha_{k-1}) : \forall i < k, \alpha_0 < … < \alpha_{k-1} \in F, r_{\alpha_i} \upharpoonright c^* (\alpha_0, …, \alpha_{k-1}) = s_i \}$;
• $F \in \mathcal{S}(\PP)$ iff $\forall \vec s \in (2^{<\omega})^{<\omega}$, $\PP_F$ is centred in $\PP$.

Claim: $\mathcal{S}(\PP)$ is a (powerfully) ccc poset.

proof. Let $\mathcal{F} \sse \mathcal{S}(\PP)$ be a given uncountable family forming a $\Delta$-system with some root $D$.

We may assume (by refining) that for some $m < \omega$ and all $f \in \mathcal{F}$ that

• $\Delta(r_\xi, r_\eta) < m$, for all $\xi \neq \eta$;
• $c(\xi, \eta) < m$, for all $\xi \neq \eta$

We may also assume that for some $n, \vert F \vert = n < \omega$ for all $F \in \mathcal{F}$, and

• $r_\xi \upharpoonright m \neq r_\eta \upharpoonright m$, for all $\xi \neq \eta$;
• $F \mapsto \{r_\xi \upharpoonright m : \xi \in F\}$, for $F \in \mathcal{F}$ is a constant mapping, and in fact that $\forall F,G \in \mathcal{F}$ we get $r_{F(i)} \upharpoonright m = r_{G(i)} \upharpoonright m$, for all $i < n$, (listing $F = \{F(0), …, F(n-1)\}$).

“Now we do something unusual.”

Using the fact that $\PP^k$ is ccc for $k = \vert (2^{\leq m})^{\leq n}\vert$, we may assume that for each $\vec s \in 2^{\leq m})^{\leq n}$ that $\bigcup \{\bigcup\PP_F (\vec s) : F \in \mathcal{F}\}$ is a centred family in $\PP$.

“This exists in a forcing extension by $\PP$, where $\PP$ becomes $\sigma$-centred. This looks like I’m cheating, going to a forcing extension. But I’m only looking for 2 compatible elements. Use absoluteness of the colouring’s property.“

Find $\overline{m} > m$ and two uncountable (“this is a separable metric space thing“) $\mathcal{F}_0, \mathcal{F}_1 \sse \mathcal{F}$ such that $\forall F \in \mathcal{F}_0, \forall G \in \mathcal{F}_1, \forall i < n$ we have $\Delta(r_{F(i)}, r_{G(i)}) < \overline{m}$.

“Here the key comes.”

By the property of $c$ given in a previous lemma, there are $F \in \mathcal{F}_0, G \in \mathcal{F}_1$ such that $c(\alpha, \beta) > \overline{m}$ for $\alpha \in F \setminus D$ and $\beta \in G \setminus D$ “because of the unbounded property of $c$.”

Claim: $F \cup G \in \mathcal{S}(\PP)$.

“What does it mean? You have to check the defining property of $\mathcal{S}(\PP)$.“

Take $\vec s \in (2^{<\omega})^{<\omega}$, where $\vec s = (s_0, …, s_k)$, and $\vert s_i \vert = \vert s_j \vert = l, \forall i,j$ for some particular $l$.

There are three cases:

1. $l < m$;
2. $m \leq l < \overline{m}$;
3. $\overline{m} \leq l$.

“The important thing is what these numbers represent.” – Ivan Khatchatourian

Case 1 [$l < m$]:

Claim: $\PP_{F\cup G} (\vec s) = \PP_{F} (\vec s) \cup \PP_{G} (\vec s)$

Let $d (\alpha_0, …, \alpha_k) \in \PP_{F\cup G} (\vec s)$, which means $\{\alpha_0, …, \alpha_k\} \sse F\cup G$, and $r_{\alpha_i}\upharpoonright l = s_i$ for all $i \leq k$.

“Okay, so let us see… Why you cannot start in one and go to another?“

[Picture argument to be filled in later]

Then $\{\alpha_0, …, \alpha_k\} \sse F$ or $\{\alpha_0, …, \alpha_k\} \sse G$.

Case 2 [$m \leq l < \overline{m}$]:

Claim: $\PP_{F \cup G} = \emptyset$.

“Again, the picture tells you much more than you can get by chasing definitions.”

Take $d (\alpha_0, …, \alpha_k) \in \PP_{F\cup G} (\vec s)$, and $r_{\alpha_i} \upharpoonright l = s_i$, for all $i \leq k$ where $l = c^* (\alpha_0, …, \alpha_k)$.

There are no $i,j$ such that $\alpha_i \in F \setminus D$ and $\alpha_i \in G \setminus D$. So $\{\alpha_i\} \sse F$ or $\{\alpha_i\} \sse G$.

“The main thing is now comes case 3.”

Case 3 [$\overline{m} \leq l$]:

Claim: $\vert \PP_F (\vec s) \vert \leq 1$.

“Suppose not, so you have two possibilities:”

$\{\alpha_0, …, \alpha_k\}, \{\alpha_0^\prime, …, \alpha_k^\prime\} \sse F \cup G$ and $c^* (\alpha_0, …, \alpha_k) = l = c^* (\alpha_0^\prime, …, \alpha_k^\prime)$, and $r_{\alpha_i}\upharpoonright l = s_i = r_{{\alpha_i}^\prime} \upharpoonright l$ for all $i \leq k$.

If $\alpha_i \neq \alpha_i^\prime$, then $\alpha_i \in F \setminus D, \alpha_i^\prime \in G \setminus D$.

“The projections” $r_\alpha \mapsto r_\alpha \upharpoonright m$ “are 1-1″ for $\alpha \in F$.

“The place below where they split is below $\overline{m}$, a contradiction.” [QED]

“It is a slippery argument but … *Shrug*.”

“Showing that $\mathcal{S}(\PP)$ is powerfully ccc is the same argument.“

Corollary. TFAE:

• MA$(\aleph_1)$;
• Every compact ccc scpace has caliber $\aleph_1$;
• Every ccc poset has precaliber $\aleph_1$.

“The proof of this is like the previous proof comparing $\mathfrak{m}_{SH}$ and $\mathfrak{m}$.“

Note that this functor also specializes trees:

Let $X \sse \mathcal{S}(\PP)$ be a centred subset of size $\kappa$. Then $\Gamma := \bigcup X \in [\kappa]^\kappa$ and $[\Gamma]^{<\omega} \in \mathcal{S}(\PP)$ and $d [X]^{<\omega} = \PP$ and $$\bigcup_{\vec s \in (2^{<\omega})^{<\omega}} \PP_X (\vec s) = \PP$$

Question: ATFE? (Are The Following Equivalent)

• MA$(\aleph_1)$;
• Every ccc poset has property K.

“This involves checking something for every pair. As the previous proof shows.“

Now we go to the CH

“Log of weight is density.“

Definition: $\cc_d := \sup \{ d(K) : K \textrm{ is compact, ccc, weight }\leq \cc\}$.

Note that it is comparable to replace weight with $\pi$-weight here.

Question: Is $\cc_d = \cc$? In combinatorial language, how many centred sets do you need to cover a poset?

Some Combinatorial CHs

CH$^\omega$: $\cc_d = \aleph_1$, i.e. every ccc poset of size $\leq \cc$ is $\aleph_1$-centred.
CH$^n$: Every ccc poset of size $\leq \cc$ is $\aleph_1$-n-linked.
CH$^2$: Every ccc poset of size $\leq \cc$ is $\aleph_1$-2-linked.

Open Question: Are they different?

“We will see that most consequences of CH are actually consequences of these forms of CH.”

Theorem: CH$^3$ implies $\theta_2 = \aleph_2$.

“These give you another way to create ccc posets. Here ccc is a quite weak condition.“

Theorem*: If every ccc poset has property $K_{\bb^+}$, then $\bb \leq \theta_2$.

“In this course we pretend $\cc = \theta_2$.“

Theorem**: There is a productively ccc poset $\PP$ without property $K_{\bb^+}$.

“So plugging in CH$^3$ gives $\bb \leq \aleph_1$, i.e. $\bb = \aleph_1$, so $\bb^+ = \aleph_2$.“

Corollary: CH$^2$ implies $\bb = \aleph_1$.

(Downward) Lemma: Suppose $(X, \leq_W)$ is a well-ordered separable metric space. “We know well orderedness usually has nothing to do with metric spaces.” Let $F: X \rightarrow \textrm{Exp}(X)$ be a (closed) set mapping which respects $\leq_W$, i.e. $F(X) \sse \{ y : y\leq x\}$. Let $\PP = \{ A \in [X]^{<\omega}: \forall x \neq y \in A, y \notin F(x)\}$ be the collection of finite $F$-free sets ordered by reverse inclusion. THEN, $\PP$ is productively ccc.

Recall that $\PP \times \dot \Q$ is ccc iff $\vdash_\PP $ “$\Q$ is ccc”.

proof. Let $A \sse \PP$ be a given uncountable family of finite $F$-free subsets of $X$. We need to find $a \neq b$ such that $a \cup b$ is $F$-free.

“You see, being free is a 2-dimensional property.“

As usual, by forming a $\Delta$-system and removing the root, we may assume that elements of $A$ are pairwise disjoint and of the same size $n$.

“Showing they’re free on a tail is enough.“

Think of $A$ as a subset of $X^n$. Let $A_0 \sse A$ be a countable dense subset of $A$.

Since $X$ is a separable metric space, we may assume that for some fixed sequence $V_0, …, V_n$ of pariwise disjoint open subsets of $X$ we have that

• $A \sse V_0 \times V_1 \times … \times V_{n-1}$; and
• for all $a \in A$, and $i < n$ we have $F(a(i)) \cap V_j = \emptyset$.

We only need to check that $a(i)$ and $b(i)$ are compatible, by separability. That is, we don’t need to check $a(i)$ against $b(j)$, for $i \neq j$.

“Now we use the well-ordering.”

“There is always this ‘moreover’ business.“

Moreover, we may assume that for all $i<n$ we have $\textrm{otp}\{a(i): a \in A\} = \omega_1$.

Find $c \in A$ such that for all $i<n$ and $a \in A_0$ we have $a(i) <_W c(i)$. (For this, use uncountability.)

Take $b \in A$ such that $b(i) >_W c(i), \forall i < n$.

Then $$b \in (X \setminus F(c_0)) \times (X \setminus F(c_1)) \times … \times (X \setminus F(c_{n-1})) =: \vec U $$

“Closed sets look downwards, they don’t hit $b(i)$’s.“

So there is $a \in A_0$ such that $a \in \vec U$. So $a \cup c$ is $F$-free. “$a$ cannot hit $c$ because it is above. $c$ cannot hit $a$ because I picked it out. $b$ was just for non-emptyness.” [QED]

Question: Do we need the well-ordering? It is unnatural for metric spaces.

Remark: Changing this lemma to arbitrary posets is equivalent to OCA.

“Now we come to this example there.”

Example: Let $X \sse \omega^{\uparrow \omega}$ be a $<^*$-well-ordered, $<^*$-unbounded family of cardinality $\bb$.

Define $F: X \rightarrow\textrm{Exp}(X)$ by $F(x) := \{y \in X : \forall n, y(n) \leq x(n)\} \sse \{y \in X : y \leq^* x\}$.

So the poset $\PP$ of finite $F$-free sets is ccc.

Recall from oscillation theory: $\{\{x\}: x \in X\} \sse \PP_X$ does not have a 2-linked subset of size $\bb$.

[Assume $\bb^+ < \theta_2$.] Apply the functor $\PP \mapsto \mathcal{S}(\PP)$ to the poset $\PP_X$ to get a poset $\mathcal{S}(\PP)$ of size $\bb^+$ with no 3-linked subset of size $\bb^+$.

“We are losing 1-dimesnsion.“

Aside: “CH$^3$ seems to be quite strong.”

Some consequences of CH$^3$.

• $\textrm{cf}(\cc) = \aleph_1$;
• Every $f: \R \rightarrow \R$ can be decomposed into $\aleph_1$ continuous sub-functions.

CH$^\omega$ implies the density of Lebesgue measure algebra is $\aleph_1$. This gives you Luzin sets, Sierpinksi functions, etc..