Stevo’s Forcing Class Fall 2012 – Class 12 – Mike Pawliuk

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

Baumgartner’s Axiom

Let $X,Y \subseteq \mathbb{R}$ be of cardinality $\aleph_1$ . (We will see that $\aleph_1$ -dense is sufficient.) We would like to construct a proper poset (cardinal-preserving under CH) which forces a strictly increasing $f: X \rightarrow Y$ . I.e. We wish to show the following theorem.

Theorem. PFA implies $BA^W (\aleph_1)$ .

“This is one of those times where designing the dense sets can be a problem.“

Conditions. Let $p = (f_p, \mathcal{M}_p, \mathcal{N}_p)$ where:

$f_p$ is a finite increasing map from $X$ into $Y$ ; and
$\mathcal{M}_p \subseteq \mathcal{N}_p$ ; and
$\mathcal{N}_p$ is a finite $\epsilon$ -chain of CESMs of $H_{\cc^+}$ which separates $f_p$ and arbitrary (???) $\mathcal{M}_p \subseteq \mathcal{N}_p$ .

Moreover,

$\forall x \in \textrm{dom}(f_p), \exists N \in \mathcal{N}_p,$ with $x \in N$ and $f(x) \notin N$ ;
$\forall x \neq y \in \textrm{dom}(f_p), \exists N \in \mathcal{N}_p$ such that $f(x) \in N$ iff $f(y) \notin N$ ;
$\forall M \in \mathcal{M}_p, \forall x \in \textrm{dom}(f_p) \cap M$ we have $f_p(x) \in M$ (No way you can cross the important model)

“This is the source of OCA.”

See the closed mapping lemma from partition problems in Topology. (REFERENCE!)

Dense Sets. $\forall x \in X, \mathcal{D}_x := \{p \in \mathbb{P} : x \in \textrm{dom}(p)\}$ is dense in $\mathbb{P}$ .

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

proof.

Given $p, \mathbb{P} \in M \prec H_\theta$ a CESM, we need to find an $(M, \mathbb{P})$ -generic condition below $p$ .

Let $q := \{f_p, \mathcal{M}_p \cup \{M \cap H_{\cc^+}\}, \mathcal{N}_p \cup \{M \cap H_{\cc^+}\}\}$ .

(We want to use the previous lemma (from class 11), which needs everything on top of a model.)

Take $\mathfrak{X} \subseteq \mathbb{P}$ such that $\mathfrak{X} \in M, r \leq q$ and $r \in \mathfrak{X}$ . We need to copy $r$ to a compatible condition $\overline{r} \in \mathfrak{X} \cap M$ .

Click to Zoom — Brown models are the Ms, Grey models are the Ns

Note $r \cap M \in M$ . So we are after an end-extension $\overline{r}$ of $r \cap M$ . “The bottom doesn’t interfere with the top. The previous lemma gives us what we want.”

Let $\mathcal{F}$ be the collection of partial functions of the form $f_s \setminus (f_r \cap M)$ where $s \in M$ .

Of course $\mathcal{F} \in M$ , and $f_r \setminus (f_r \cap M) \in \mathcal{F}$ .

By the previous lemma there is an $h \in \mathcal{F}$ such that $h \cup f_r \setminus (f_r \cap M)$ is increasing. So in particular, there is such an $h \in \mathcal{M}^\prime$ .

(“By $\aleph_1$ -density. I cannot define a function using $f_r$ as it is not in $M$ . But, I can use rational intervals.“)

Pick $\overline{r} \in \mathfrak{X} \cap M$ such that $f_{\overline{r}} \setminus (f_r \cap M) = h$ . Then $\overline{r} \not \perp r$ . [QED]

“It is one thing to be silly, it is another to be wrong. This time we were lucky.”

“Shouldn’t the important models be limits of CESMs?” – David

“This quite correct. If you want to be really correct.”

Require: $\mathcal{M}_p \subseteq \{ M \cap H_{\cc^+} : M \prec_{\textrm{ctble}} H_\lambda\}$ , where $\lambda = (2^{2^\cc})^+$ .

Equivalence of Baumgartner’s Axiom and its weak form.

Lemma. If $\mathfrak{p} > \omega_1$ , then $BA^W (\aleph_1)$ is equivalent to $BA(\aleph_1)$ .

proof.

Pick two $\aleph_1$ -dense sets of reals $X$ and $Y$ . We may assume that they are $\aleph_1$ -dense in $\mathbb{R}$ .

For each pair $I_0, I_1$ of open rational intervals fix two increasing injections:

$f_{01} : X \cap I_0 \rightarrow Y \cap I_1$ ;
$f_{10} : Y \cap I_1 \rightarrow X \cap I_0$

(The indices suggest the domain and range.)

Conditions. Let $\mathbb{P}$ be the poset of finite strictly increasing maps from $X$ to $Y$ such that:
For each $x \in \textrm{dom}(p)$ there exist a pair $I_0, I_1$ of open rational intervals such that either $p(x) = f_{01}(x)$ or $x = f_{10} (p(x))$ .

Now we show that this is $\sigma$ -centred and it does the job.

Dense Sets. $\forall x \in X$ and $y \in Y$ both $\mathcal{D}_x^0 := \{p \in \mathbb{P} : x \in \textrm{dom}(p)\}$ and $\mathcal{D}_y^1 := \{p \in \mathbb{P} : y \in \textrm{ran}(p)\}$ are dense open.