Stevo’s Forcing Class Fall 2012 – Class 1

(In the fall of 2012 I will be taking Stevo Todorcevic’s class in Forcing at the University of Toronto. I will try to publish my notes here, although that won’t always be possible.)

This class is going to be held Wed 1-2 and Friday 3:30-5:30 in BA 6183.

The theme for today is “Things that are ccc, but whose products are not”. So first some examples:

Example 1. If $T$ is a Suslin tree, then $T^2$ is not a Suslin tree. (See Lemma III.2.18 in Newnen.)

Example 2. Let $\mathcal{A}(T) =$ finite antichains of $T$, where $T$ is a Suslin Tree. Then $\mathcal{A}(T)$ is powerfully ccc (its finite products are ccc), but $T \times \mathcal{A}(T)$ is not ccc.

“A Suslin tree is the skeleton of a Suslin continuum. If you take x-rays, that’s what you get.” – Stevo

This example was first studied by Baumgartner in his thesis:

Theorem (Baumgartner, 1969): TFAE for a tree $T$:

• $\mathcal{A}(T)$ is ccc;
• $T$ has no uncountable branches.

It is clear that NOT (2) imples NOT (1), as the singletons of a branch form an antichain.

“There are many proofs of this fact, but Baumgartner’s original proof is still the most useful.” – Stevo

proof of Theorem. Like most proofs that a poset is ccc, we start with an uncountable subset of the poset and refine, refine, refine.

Let $\{p_\xi : \xi \in \omega_1\} \sse \mathcal{A}(T)$, we want to find $\xi \neq \eta$ such that $p_\xi \cup p_\eta \in \mathcal{A}(T)$. We refine 3 times to an uncountable set $\Gamma \sse \omega_1$ so that:

1. (All conditions are the same length) $\vert p_\xi \vert = n$ for all $\xi \in \Gamma$,
2. ($\Gamma$ is a $\Delta$-sytem) There is an $r \in \mathcal{A}(T)$ such that for $\xi \neq \eta$ in $\Gamma$ we have $p_\xi \cap p_\eta = r$.
3. (Heights are increasing) The $ht(p_\xi) < ht(p_\eta)$ for $\xi < \eta$.

We want to assume that $r$ is actually empty, and we can do this by considering $q_\xi := p_\xi \setminus r \neq \emptyset$. Note that $q_\xi \not\perp q_\eta$ implies $p_\xi \not \perp p_\eta$.

WLOG, we get $r = \emptyset$.

We suppose that $p_\xi \perp p_\eta$ for all $\xi \neq \eta$ in $\Gamma$ (and look for a contradiction). This means that if $\xi < \eta$ then there is an $x \in p_\xi$ and a $y \in p_\eta$ such that $x <_T y$.

Now take a uniform ultrafilter $\mathcal{U}$ on $\omega_1$ such that $\Gamma \in \mathcal{U}$.

Fix $\xi \in \Gamma$. Write $p_\xi = \{p_\xi (0), \dots , p_\xi (n-1) \}$.

There is an $i_\xi < n$ such that $\Gamma_\xi := \{\eta \in \Gamma : \exists y \in p_\eta, p_\xi (i_\xi) <_T y \} \in \mathcal{U}$.

Define $X := \{p_\xi (i_\xi) : \xi \in \Gamma\}$.

Question: Is there a large antichain in X?

Now for each $\xi \in \Gamma$ there is a $j_\xi < n$ so that $\Gamma_\xi (j_\xi) \in \mathcal{U}$. Thus $\bigcap_{i \leq n+1} \Gamma_\xi (i_\xi) \in \mathcal{U}$ and we can choose an $\eta$ in this intersection so that $ht(p_\eta) > ht(p_{\xi_i})$ for all $i \leq n$.

Now if $j_\xi$ does not depend on $\xi$ then we are done. This follows from exercise 2. [QED]

Some Exercises:

Exercise 1: Suppose that $\mathbb{P}$ is a ccc poset and let $\{ p_\xi : \xi < \omega_1\} \sse \mathbb{P}$. Show that there is an infinite set $I \sse \omega_1$ such that $\{p_i : i \in I\}$ are pairwise compatible.

Exercise 2: Let $T$ be a tree with an uncountable subset that has no antichains of size $\geq N$. Show that $T$ has an uncountable chain.

Some chain conditions, listed from easiest to satisfy to hardest to satisfy:

• ccc
• powerfully ccc
• productively ccc
• $\sigma$-finite-cc
• $\sigma$-bounded-cc
• $\sigma$-2-linked
• $\sigma$-n-linked
• $\sigma$-n-linked ($\forall n$)
• $\sigma$-centred
• countable

Note that Borel posets can distinguish all of the blue and dark blue properties. By the following exercise, the dark blue posets are small.

Exercise 3. If $\mathbb{P}$ is an atomless $\sigma$-(2)-linked poset then $\vert \mathbb{P} \vert \leq \mathfrak{c}$.

hint. Use the Erdös-Rado theorem.

Question. Can a boolean algebra supporting a continuous submeasure algebra distinguish $\sigma$-finite-cc from $\sigma$-bounded-cc?