Generalizations of Martin’s Axiom and the well-met condition

Recall that Martin’s Axiom asserts that for every partial order $\mathbb{P}$ satisfying c.c.c., and for any family $\mathcal{D}$ of $<2^{{\mathrm{\aleph }}_0}$ many dense subsets of $\mathbb{P}$, there exists a directed subset $G$ of $\mathbb{P}$ such that $G\cap D\ne \mathrm{\varnothing }$ for all $D\in \mathcal{D}$.

Solovay and Tennenbaum proved that Martin’s Axiom is consistent with $2^{{\mathrm{\aleph }}_0}>{\mathrm{\aleph }}_1$.

In [Sh:80], Shelah proved that the following weak generalization of Martin’s Axiom is consistent with $2^{{\mathrm{\aleph }}_1}>{\mathrm{\aleph }}_2$ and CH. If $\mathbb{P}$ is a partial order satisfying:

1. $|\mathbb{P}|<2^{{\mathrm{\aleph }}_1}$;
2. $\mathbb{P}$ is $\sigma$-closed;
3. $\mathbb{P}$ is ${\mathrm{\aleph }}_2$-stationary-c.c.: if $\{p_{\alpha }\mid \alpha <{\mathrm{\aleph }}_2\}$ is an injective enumeration of conditions, then there exists a regressive map $f:{\omega }_2\to {\omega }_2$ and a club $D\subseteq {\omega }_2$, so that $\{p_{\alpha }\mid \alpha \in D\cap E_{{\omega }_1}^{{\omega }_2},f(\alpha )=i\}$ is directed for all $i<{\omega }_2$;
4. well-met condition: for every $p,q\in \mathbb{P}$, if $\{r\in \mathbb{P}\mid p\le r\mathrm{\&}q\le r\}$ is nonempty (i.e., $p$ and $q$ are compatible), then the set has a least element,

then for any family $\mathcal{D}$ of $<2^{{\mathrm{\aleph }}_1}$ many dense subsets of $\mathbb{P}$, there exists a directed subset $G$ of $\mathbb{P}$ such that $G\cap D\ne \mathrm{\varnothing }$ for all $D\in \mathcal{D}$.

Let us look at requirements (1)–(4).

Requirement (1) is a rather natural restriction (though, this restriction may sometime be waived via tail-catching, or by reflection arguments in the presence of large cardinals).

Requirement (2) has to do with the fact that any forcing axiom for meeting more than ${\mathrm{\aleph }}_1$ many dense sets which will be compatible with CH, will have to be applicable only to forcing notions that do not add reals.

Requirement (3): Recall that Martin’s Axiom implies that the product of any two c.c.c. posets is again c.c.c., whereas there are ZFC examples of two ${\mathrm{\aleph }}_2$-c.c. posets whose product is not anymore ${\mathrm{\aleph }}_2$-c.c. (see here). So, plain ${\mathrm{\aleph }}_2$-c.c. must be replaced by a stronger variation.

Today, I learned from Ashutosh Kumar that [Sh:1036] offers a nice and relatively simple justification of Requirement (4). Luckily, the example builds upon a theorem of Shelah which I already presented in a previous blog post (see also Section 2 of this survey paper, for motivation):

Theorem (Shelah). If $⟨C_{\delta }\mid \delta \in E_{{\omega }_1}^{{\omega }_2}⟩$ is a sequence such that $C_{\delta }$ is a club in $\delta$ of order-type ${\omega }_1$ for each $\delta$, then there exists a sequence of local colorings $⟨f_{\delta }:C_{\delta }\to 2\mid \delta \in E_{{\omega }_1}^{{\omega }_2}⟩$, such that for every global coloring $f:{\omega }_2\to {\omega }_2$, there exists $\delta \in E_{{\omega }_1}^{{\omega }_2}$ for which $\{\alpha \in C_{\delta }\mid f(\alpha )\ne f_{\delta }(\alpha )\}$ is not bounded in $\delta$.

The very same proof shows that “not bounded” may be strengthened to “is stationary”. Thus, fix a sequence $⟨f_{\delta }\mid \delta \in E_{{\omega }_1}^{{\omega }_2}⟩$ witnessing the latter. For simplicity, we may assume that $min(C_{\delta })=1$ for all $\delta \in E_{{\omega }_1}^{{\omega }_2}$. Define a notion of forcing $\mathbb{P}$, as follows. $p\in \mathbb{P}$ iff it is a partial function from $E_{{\omega }_1}^{{\omega }_2}$ to ${\mathcal{P}}_{{\omega }_1}({\omega }_2)$, such that:

1. $\text{dom}(p)$ is a countable subset of $E_{{\omega }_1}^{{\omega }_2}$;
2. $p(\delta )$ is a closed bounded subset of $C_{\delta }$;
3. if ${\delta }_1<{\delta }_2$ are in $\text{dom}(p)$ and $\alpha \in p({\delta }_1)\cap p({\delta }_2)$, then $f_{{\delta }_1}(\alpha )=f_{{\delta }_2}(\alpha )$;
4. if ${\delta }_1\ne {\delta }_2$ are in $\text{dom}(p)$, then $\beta <max(p({\delta }_1))$ for all $\beta \in C_{{\delta }_1}\cap C_{{\delta }_2}$.

A condition $q$ extends $p$, iff

1. $\text{dom}(q)\supseteq \text{dom}(p)$;
2. $q(\delta )$ is an end-extension of $p(\delta )$ for all $\delta \in \text{dom}(p)$.

Let $\mathcal{D}:=\{D_{\alpha ,\delta }\mid \alpha <\delta ,\delta \in E_{{\omega }_1}^{{\omega }_2}\}$, where $$D_{\alpha ,\delta }:=\{p\in \mathbb{P}\mid \delta \in \text{dom}(p)\mathrm{\&}max(p(\delta ))>\alpha \}.$$

Then $|\mathcal{D}|={\mathrm{\aleph }}_2$ and if $G$ happens to be a directed subset of $\mathbb{P}$ such that $G\cap D$ for all $D\in \mathcal{D}$, then for all $\delta \in E_{{\omega }_1}^{{\omega }_2}$, $c_{\delta }:=\bigcup \{p(\delta )\mid p\in G\}$ is a club subset of $C_{\delta }$, and $f:=\bigcup \{f_{\delta }\upharpoonright c_{\delta }\mid \delta \in E_{{\omega }_1}^{{\omega }_2}\}$ is a function! In particular, $\{\alpha \in C_{\delta }\mid f(\alpha )\ne f_{\delta }(\alpha )\text{ or }\alpha \notin \text{dom}(f)\}$ is nonstationary for all $\delta \in E_{{\omega }_1}^{{\omega }_2}$.

Claim 1. $D_{\alpha ,\delta }$ is dense for every $\delta \in E_{{\omega }_1}^{{\omega }_2}$ and $\alpha <\delta$.
Proof. Given $q\in \mathbb{P}\setminus D_{\alpha ,\delta }$, we distinguish two cases.
If $\delta \in \text{dom}(q)$, then let ${\alpha }^{\prime }:=min(C_{\delta }\setminus (max(q(\delta )\cup \{\alpha \})+1))$, and set $$p:=(q\upharpoonright (\text{dom}(p)\setminus \{\delta \}))\cup \{(\delta ,q(\delta )\cup \{{\alpha }^{\prime }\})\}.$$If $\delta \notin \text{dom}(q)$, let $t:=\bigcup \{C_{\beta }\cap C_{\gamma }\mid \beta \ne \gamma \text{ in dom}(q)\cup \{\delta \}\}$. As $t$ and $\text{dom}(q)$ are countable, it is easy to construct an injective sequence $$⟨{\alpha }_{\beta }\mid \beta \in \text{dom}(q)\cup \{\delta \}⟩\in \prod _{\beta \in \text{dom}(q)\cup \{\delta \}}{C}_{\beta }$$ such that ${\alpha }_{\beta }>sup(q(\beta )\cup (t\cap \beta ))$ for all $\beta \in \text{dom}(q)$, and ${\alpha }_{\delta }>sup(\alpha \cup (t\cap \delta ))$.
Finally, let $p\in \mathbb{P}$ be such that $\text{dom}(p)=\text{dom}(q)\cup \{\delta \}$ and, $$p(\beta )=\{\begin{array}{ll}q(\beta )\cup \{{\alpha }_{\beta }\},& \beta \in \text{dom}(q)\\ \{{\alpha }_{\delta }\},& \text{otherwise}\end{array}.\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$$

Claim 2. $\mathbb{P}$ is $\sigma$-closed.
Proof. Given an increasing sequence $⟨p_n\mid n<\omega ⟩$ in $\mathbb{P}$, define $p\in \mathbb{P}$ such that $\text{dom}(p):=\bigcup _{n<\omega }\text{dom}(p_n)$ and $p(\delta )$ is the closure of $\bigcup \{p_n(\delta )\mid n<\omega ,\delta \in \text{dom}(p_n)\}$ for all $\delta \in \text{dom}(p)$. To see that $p\in \mathbb{P}$, we only need to verify that if ${\delta }_1<{\delta }_2$ are in $\text{dom}(p)$ and $\alpha \in p({\delta }_1)\cap p({\delta }_2)$, then $f_{{\delta }_1}(\alpha )=f_{{\delta }_2}(\alpha )$. Fix a large enough $N<\omega$ such that ${\delta }_1$ and ${\delta }_2$ are in $\text{dom}(p_N)$. Put $\beta :=sup(C_{{\delta }_1}\cap C_{{\delta }_2})$, then $\beta <{\delta }_1<{\delta }_2$, and by clause (4) above, $\beta <max(p_N({\delta }_i))$ for $i<2$. By $\alpha \in C_{{\delta }_1}\cap C_{{\delta }_2}$, we have $\alpha \le \beta <max(p_N({\delta }_i))$ for $i<2$. As $p_n({\delta }_i)$ end-extends  $p_N({\delta }_i)$ whenever $N\le n<\omega$ and $i<2$, we infer that $\alpha \in p_N({\delta }_1)\cap p_N({\delta }_2)$, and so by Clause (3), $f_{{\delta }_1}(\alpha )=f_{{\delta }_2}(\alpha )$. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$

Notation. For every $q\in \mathbb{P}$, denote $$g_q:=\bigcup \{f_{\delta }\upharpoonright q(\delta )\mid \delta \in \text{dom}(q)\}.$$

Claim 3. CH entails that $\mathbb{P}$ has size ${\mathrm{\aleph }}_2$ and satisfies the ${\mathrm{\aleph }}_2$-stationary-c.c.
Proof. It is a basic counting argument that shows that CH entails a bijection $\psi :{\omega }_2\leftrightarrow \mathbb{P}$. Next, given a sequence of conditions $⟨p_{\alpha }\mid \alpha \in E_{{\omega }_1}^{{\omega }_2}⟩$, we first extend each $p_{\alpha }$ to a condition $q_{\alpha }$ satisfying $\alpha \in \text{dom}(q_{\alpha })$, and $max(q_{\alpha }(\delta ))>\alpha$ for all $\delta \in \text{dom}(q_{\alpha })$ above $\alpha$.  As $g_{q_{\alpha }}$ is a countable partial function from ${\omega }_2$ to $2$, we infer from CH and the Engelking-Karlowicz theorem the exisetnce of a sequence $⟨{\phi }_i:{\omega }_2\to 2\mid i<{\omega }_1⟩$ such that for each $\alpha \in E_{{\omega }_1}^{{\omega }_2}$, there is $i_{\alpha }<{\omega }_1$ with $g_{q_{\alpha }}\subseteq {\phi }_{i_{\alpha }}$. Let $D$ be the set of all $\delta <{\omega }_2$ such that for all $\alpha <\delta$, $\{p\in \mathbb{P}\mid \text{dom}(p)\subseteq \alpha \}\subseteq \psi [\delta ]$. Then $D$ is a club, and for all $\delta \in D\cap E_{{\omega }_1}^{{\omega }_2}$, $\{p\in \mathbb{P}\mid \text{dom}(p)\subseteq \delta \}\subseteq \psi [\delta ]$. Define a function $h:E_{{\omega }_1}^{{\omega }_2}\to {\omega }_2$ by stipulating $h(\delta ):=sup(\bigcup \{\text{dom}(q_{\alpha })\mid \alpha \in E_{{\omega }_1}^{\delta }\})$, and let $E:=\{\delta \in D\mid h[\delta ]\subseteq \delta \}$. Finally, define a regressive function $f:E\cap E_{{\omega }_1}^{{\omega }_2}\to {\omega }_1\times {\omega }_2$ by letting $f(\delta ):=(i_{\delta },{\psi }^{-1}(q_{\delta }\upharpoonright \delta ))$.

To see that $\{p_{\alpha }\mid \alpha \in E\cap E_{{\omega }_1}^{{\omega }_2},f(\alpha )=i\}$ is directed for all $i\in {\omega }_1\times {\omega }_2$, fix arbitrary ${\delta }_1<{\delta }_2$ in $E\cap E_{{\omega }_1}^{{\omega }_2}$ with $f({\delta }_1)=f({\delta }_2)$.  As $q_{{\delta }_1}\upharpoonright {\delta }_1=q_{{\delta }_2}\upharpoonright {\delta }_2$ and ${\delta }_1\in {\delta }_2\in E$, we have $$q_{{\delta }_1}\upharpoonright (\text{dom}(q_{{\delta }_1})\cap \text{dom}(q_{{\delta }_2}))=q_{{\delta }_2}\upharpoonright (\text{dom}(q_{{\delta }_1})\cap \text{dom}(q_{{\delta }_2})).$$

Set $p:=q_{{\delta }_1}\cup q_{{\delta }_2}$. Then $p(\alpha )$ is a closed bounded subset of $C_{\alpha }$ for all $\alpha \in \text{dom}(p)$. Also, $g_p$ is a function, since $g_p\subseteq {\phi }_{i_{{\delta }_1}}={\phi }_{i_{{\delta }_2}}$. So, $p$ satisfies Clauses (1)–(3) above. Now, as in the end of the proof of Claim 1, it is easy to find $p^{\prime }\in \mathbb{P}$ with $\text{dom}(p^{\prime })=\text{dom}(p)$ such that $p^{\prime }(\alpha )$ is an end-extension of $p(\alpha )$ by a singleton for all $\alpha \in \text{dom}(p^{\prime })$. Clearly, $p^{\prime }$ is stronger than $q_{{\delta }_1}$ and $q_{{\delta }_2}$, let alone $p_{{\delta }_1}$ and $p_{{\delta }_2}$. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$

Note that any condition $p\in \mathbb{P}$ admits $\mathfrak{c}$ many pairwise incompatible extensions, hence, CH is also necessary for $\mathbb{P}$ to satisfy the ${\mathrm{\aleph }}_2$-c.c.

Claim 4. $\mathbb{P}$ does not satisfy the well-met condition.
Proof. By iterating the pressing down lemma, we can find a stationary subset $S$ of $E_{{\omega }_1}^{{\omega }_2}$ along with ordinals ${\alpha }_0<{\alpha }_1<{\alpha }_2$ such that for all $\delta \in S$, the first three elements of $C_{\delta }$ are $\{{\alpha }_0,{\alpha }_1,{\alpha }_2\}$. Pick ${\delta }_0<{\delta }_1$ from $S$. Let $p_i:=\{({\delta }_i,\{{\alpha }_i\})\}$ for $i<2$. Then $p_0$ and $p_1$ are compatible elements of $\mathbb{P}$, but due to Clause (4) above, $p_0\cup p_1$ is not in $\mathbb{P}$, and so the set $\{r\in \mathbb{P}\mid p_0\le r\mathrm{\&}{p}_1\le r\}$ does not admit a least element. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$