Stevo’s Forcing Class Fall 2012 – Class 4

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

Measure Algebras

When talking about how Cohen Forcing $\mathcal{C}$ does not collapse $\omega_1$ it is better to think about how the regular open algebra $\B = \textrm{ro}(\mathcal{C})$ does not collapse $\omega_1$. This we can tell from some facts about the distributivity of the algebra:

Fact. In $\B$, $\forall \{a_{n_\xi} : \xi < \omega_1, n < \omega\} \sse \B^+$ and $\forall b \in \B^+$ there are $\alpha_n < \omega_1$ (for $n<\omega$) such that $$(b \cap \bigwedge_{n<\omega} \bigvee_{\xi < \alpha_n} a_{n_\xi} )> 0$$.

Recall the analogous distributive law with $\omega_1$ replaced by $\omega$.

Weak Distributive Law. $\forall \{a_{n_k} : k < \omega, n < \omega\} \sse \B^+$ and $\forall b \in \B^+$ there are $l_k < \omega$ (for $k<\omega$) such that $$(b \cap \bigwedge_{n<\omega} \bigvee_{k \leq l_k} a_{n_k} ) > 0$$.
Fact. If $\B$ is ccc, then $\B$ has the weakly-$(\omega_1, \omega_1)$-distributive.

Examples of ccc and weakly-$(\omega, \omega)$-distributive algebras.

Example. Try $\B = \textrm{ro}(\PP)$ where $\PP = \{K \sse [0,1] : K \textrm{ is compact and has positive measure}\}$.

“Weakly Distributive and $\sigma$-centred do not go together.”

Question. Is there a $\sigma$-centred, weakly distributive algebra? (Note that measure algebras do not work.)

For a Measure Algebra, $\mu: \B \rightarrow [0,1]$ strictly positive, $d_\mu (a,b) = \mu (a \Delta b)$ is countably additive or, it is finitely additive and weakly distributive.

Question. Are there nonseparable (non trivial) Maharam algebras? (You need to “make submeasure far away from measure” and “find $\sigma$-finite-cc, not $\sigma$-bounded-cc“.)

Back to the “Suslin Family”.

Theorem 2. There is a $\sigma$-linked, non $\sigma$-centred poset of size $\pp$.

“Smaller families are more interesting.”

Compare this with the following adjustments:

Theorem 1. There is a $\sigma$-linked, non $\sigma$-centred poset of size $\tt$

Theorem 0. There is a non $\sigma$-linked poset of size $\bb$ with no 2-linked subset of size $\bb$.

The proof is similar to Theorem 1, which we did in Class 2.

proof. We may assume that $\pp < \bb$, by Theorem 1.

Let $\{a_\xi : \xi < \pp\} \sse [\omega]^\omega$ have the finite intersection property, that is non-extendible. That is, there is no $b \in [\omega]^\omega$ such that $b \sse^* a_\xi$ for all $\xi < \pp$.

We build (“if possible, or in other words we simply try“), $C$, a tower $\{b_\alpha : \alpha <\pp\}$ such that

• $b_\alpha \neq b_\beta$ for $\alpha \neq \beta$
• $b_{\xi + 1} \sse a_\xi$
• $\vert b_\alpha \cap a_\xi \vert = \aleph_0$, for all $\alpha, \xi < \pp$.

“Of course it is going to be a recursive construction. This tells you you have hope. Otherwise, there is no future.”

Given $b_\xi$, let $b_\xi = b_\xi \cap a_\xi$.

“Everybody intersects everybody.”

Suppose $\xi$ is a limit $<\pp$.

Case 1, where $\cof(\xi) = \omega$.

Pick a sequence $\xi_n$ increasing to $\xi$. Define $c_n := \bigcap_{i \leq n} b_{\xi_i}$. And for $\alpha < \pp$, define $f_\alpha \in \omega^\omega$ by $f_\alpha (n) = \min (a_\alpha \cap c_n) \setminus (f_\alpha (n-1) + 1)$.

Since we are assuming that $\pp < \bb$, fix $g \in \omega^\omega$ such that $f_\alpha <^* g$ for all $\alpha < \pp$. Let $b_\xi := \bigcup_{n < \omega} c_n \cap g(n)$. (Remember we were trying to define, recursively, $b_\xi$.)

Then $b_\xi \sse^* b_\eta$ for $\eta < \xi$, and $b_\xi \cap a_\alpha$ is infinite for all $\alpha < \pp$.

Case 2, where $\cof(\xi) > \omega$.

“We force $b_\xi$, but we are not allowed to force in the obvious ($\sigma$-centred) way.”

Let $<F, G> \in \PP$ iff $F \in [\xi]^{<\omega}$, $G \in [\pp]^{<\omega}$ and $\forall k < \omega, \forall \alpha \in G, \vert b_F \cap a_\alpha \cap k \vert \geq \vert \Delta_F \cap k \vert$. The ordering is coordinate-wise inclusion.

(See Class 2 notes, for this notation, and intuition.)

Claim: $\PP$ is ccc (and in fact is $\sigma$-k-linked for all $k$)

“If the claim is true, if $\PP$ is not $\sigma$-centred we are done. So assume $\PP$ is $\sigma$-centred.“

Let $\PP = \bigcup_{n<\omega} \PP_n$ be such that $\PP_n$ is centred in $\PP$, for all $n$. Let $A_n := \bigcup\{ F : \exists G, <F,G> \in \PP_n\}$. We may assume that each $A_n$ is cofinal in $\xi$, so that $\xi = \bigcup A_n$.

“The next move is a dangerous move in general. Because we are asking for infinitely many requirements.“

Let $c_n := \bigcap \{ b_\eta : \eta \in A_n\}$.

Claim: These $c_n$ are infinite.

Since $A_n$ is cofinal in $\xi$, $c_n \sse^* b_\eta$, for all $\eta < \xi$. They are infinite since $\Delta_{A_n}$ is infinite and $\vert b_F \cap a_\alpha \cap k \vert \geq \vert \Delta_F \cap k \vert$.

“Now what do we know?“

If $\alpha \in G$, such that for some $n$ and $F$, $<F,G> \in \PP_n$, then $a_\alpha \cap c_n$ is infinite.

“Now $\bb$ is going to help us. You do the usual thing. What is the usual thing?“

For $\alpha < \pp$, define $f_\alpha \in \omega^\omega$ such that …

$f_\alpha (n) =$ minimal $k > f_\alpha (n-1)$ such that for every $m \leq n$ if $a_\alpha \cap c_n$ has some points above $f_\alpha (n-1)$ then it has some point below $k$.

Then, as before, $b_\xi := \bigcup_{n < \omega} c_n \cap g(n)$.

“We have that $\{b_\eta : \eta < \xi\}$ and $\{c_n : n<\omega\}$ form a gap, but we know that $\bb$ kills $\omega, *$-gaps.“

So we can continue on and either get that $\pp < \tt$, (so we have the last theorem), or we can continue to what we want. [QED]

Theorem. The following are equivalent:

1. MA($\aleph_1$) (i.e. $\mathfrak{m} > \aleph_1$);
2. Every first-countable ccc compact space is separable.

Some Preliminaries about the Cardinal $\theta_2$

A useful cardinal for us is $\theta_2 := \min \{\theta : {{\theta}\choose{\theta}} \rightarrow {{\omega}\choose{\omega}}^{1,1}_\omega\}$, where the arrow notation means $\forall c: \theta \times \theta \rightarrow \omega$ there are $A,B \in [\theta]^{\aleph_0}$ such that $c \upharpoonright A \times B$ is constant.

Exercise. $\theta_2 \geq \aleph_2$

Theorem ($\mathfrak{m} > \omega_2$). $\theta_2 = \aleph_2$ iff Chang’s Conjecture.

Theorem *. If Chang’s Conjecture fails, there is a ccc poset which forces ${{\omega_2}\choose{\omega_2}} \not\rightarrow {{\omega}\choose{\omega}}^{1,1}_\omega$.

Exercise. Chang’s Conjecture implies $\theta_2 = \aleph_2$.

[Chang's Conjecture]. For every structure of the form $\mathfrak{A} = (\omega_2, \omega_1, <f_n : n<\omega >)$ there is an elementary substructure $\mathfrak{B} = (B, B\cap \omega_1, <f_n \upharpoonright (B)^n : n < \omega>)$ of $\mathfrak{A}$ such that $\vert B \vert = \aleph_1$ and $\vert b \cap \omega_1 \vert \leq \aleph_0$.

Problem. Is it true (in ZFC) that $\theta_2 \leq \aleph_3$? “Any $\aleph_\alpha$ would be interesting.”

Now Amoebas and Functors.

We are going to produce a functor that goes from a ccc poset $\PP$ to another ccc poset $\mathcal{S}(\PP)$. The important thing is that we will be going “from local to global control“.

Theorem. Suppose that ${{\kappa}\choose{\kappa}} \not\rightarrow {{\omega}\choose{\omega}}^{1,1}_\omega$ Let $\PP$ be a powerfully ccc poset of size $\leq \kappa$. Then there is a (powerfully) ccc poset $\mathcal{S}(\PP)$ of size $\kappa$ such that if $\mathcal{S}(\PP)$ contains a centred subset of size $\kappa$, then $\PP$ is centred.

“We are getting away from meeting dense sets.”

Sketch. ${{\kappa}\choose{\kappa}} \not\rightarrow {{\omega}\choose{\omega}}^{1,1}_\omega$ implies that for every algebra $\mathfrak{A} = (\kappa, (f_n)_{n<\omega})$ there is a subalgebra $\mathfrak{B} = (B, (f_n \upharpoonright B^{k_n})_{n < \omega})$ such that $\vert B \vert = \kappa$ but $B \neq \kappa$.

This means that we have a map $f: [\kappa]^{<\omega} \rightarrow \PP$ such that $f”[\Gamma]^{<\omega} = \PP$ (this is global information), for all $\Gamma \sse \kappa$ of cardinality $\kappa$, (where we think of allowing $\Gamma$ to be any “thin” set.)