PFA implies the P-ideal dichotomy

Fix a large regular $\theta$. For every countable $\mathcal{X}\subseteq \mathcal{I}$, use F2 and A1 to fix $I_{\mathcal{X}}\in \mathcal{I}$ with $a{\subseteq }^{\ast }{I}_{\mathcal{X}}$ for all $a\in \mathcal{X}$, and put $I_N=I_{N\cap \mathcal{I}}$ for a countable $N\prec H_{\theta }$. Define $Q_{\mathcal{I}}$ as follows. A condition $p=(Z_p,{\mathcal{N}}_p)$ has finite $Z_p\subseteq S$ and a finite membership-chain ${\mathcal{N}}_p$ of countable elementary submodels containing $\mathcal{I}$; distinct points of $Z_p$ are separated by some $N\in {\mathcal{N}}_p$; and if $N\in {\mathcal{N}}_p$ and $X\in N\cap {\mathcal{I}}^{\perp }$, then $X\cap Z_p\subseteq N$. Put $p\le q$ when $Z_q\subseteq Z_p$, ${\mathcal{N}}_q\subseteq {\mathcal{N}}_p$, and $(Z_p\setminus Z_q)\cap N\subseteq I_N$ for every $N\in {\mathcal{N}}_q$. These clauses are preserved by extension and make the empty pair a greatest condition.

We verify properness, including the combinatorial compatibility step. Let $M$ be suitable, $p_0\in Q_{\mathcal{I}}\cap M$, and add $N=M\cap H_{\theta }$ to its side chain; the result $q$ is a condition below $p_0$. Fix once and for all the well-order of $H_{\theta }$ carried by the elementary structure. For a condition $u$ and a model $K\in {\mathcal{N}}_u$, define the condition trace $u\upharpoonright K=(Z_u\cap K,{\mathcal{N}}_u\cap K),$ and enumerate the finite set $Z_u\setminus K$ in the fixed well-order, writing $t_{u,K}\in S^{\mathrm{\mid }{Z}_u\setminus K\mathrm{\mid }}$ for the resulting tuple. It suffices by F3 to take $r\le q$ and dense $D\in M$, first strengthen $r$ into $D$, and then find a member of $D\cap M$ compatible with this strengthening; rename the strengthened condition $r$. Put $r_0=r\upharpoonright N$ and $n=\mathrm{\mid }{Z}_r\setminus N\mathrm{\mid }$. By elementarity, restrict $D$ to the conditions $s$ carrying a distinguished $N_s\in {\mathcal{N}}_s$ such that $s\upharpoonright N_s=r_0$ and $\mathrm{\mid }{Z}_s\setminus N_s\mathrm{\mid }=n$; the witnesses for $r$ are $N_r=N$ and $t_{r,N}$. Let $T=\{t_{s,N_s}:s\in D\}\subseteq S^n$ and let $\mathcal{J}$ be the sigma-ideal generated by ${\mathcal{I}}^{\perp }$. For $U\subseteq S^n$, let $\mathrm{\partial }U$ retain the tuples $u$ for which, at every coordinate $k<n$, the fibre of possible $k$th entries above $u\upharpoonright k$ is $\mathcal{J}$-positive. The derivative claim in Moore's cited tutorial says that $T^{\mathrm{\prime }}={\mathrm{\partial }}^{n}T$ is a nonempty ${\mathcal{J}}^{+}$-splitting member of $M$ and contains the external tuple $t_{r,N}$. Its finite induction uses the membership chain and clause 4 of the forcing: if $t_{r,N}$ first disappeared, the least bad fibre's countable ${\mathcal{I}}^{\perp }$ decomposition, coded in the relevant side model, would put one of the corresponding outside points of $Z_r$ in a member of ${\mathcal{I}}^{\perp }$ from that model, contrary to clause 4. In particular, this claim does not assume that $t_{r,N}$ itself belongs to $M$. Starting with the empty tuple, choose successively in $M$ an initial segment extendible in $T^{\mathrm{\prime }}$. Its next-coordinate set $C\in M$ is not in $\mathcal{J}$, hence is not orthogonal to $\mathcal{I}$; elementarity gives an infinite $H\in \mathcal{I}\cap M$ with $H\subseteq C$. For every one of the finitely many outer models $P\in {\mathcal{N}}_r\setminus M$, membership-chain coherence gives $H\in P\cap \mathcal{I}$, so $H{\subseteq }^{\ast }{I}_P$. Choose the next coordinate in $H\cap {\bigcap }_{P\in {\mathcal{N}}_r\setminus M}{I}_P$. After $n$ choices, elementarity supplies $s\in D\cap M$ whose tuple $t_{s,N_s}$ is the chosen one, and hence $Z_s\setminus N_s$ is contained in every $I_P$. Then $(Z_s\cup Z_r,{\mathcal{N}}_s\cup {\mathcal{N}}_r)$ is a common extension. Thus $q$ is an $(M,Q_{\mathcal{I}})$-master and $Q_{\mathcal{I}}$ is proper.

If $S$ is a countable union of members of ${\mathcal{I}}^{\perp }$, PID's second alternative holds. Otherwise choose a suitable countable $M$ and $x\in S\setminus \bigcup (M\cap {\mathcal{I}}^{\perp })$. Then $q=(\{x\},\{M\cap H_{\theta }\})$ is a condition and, by step 2.1, an $M$-master. For a $Q_{\mathcal{I}}$-generic $G$ containing $q$, set $\dot{Z}={\bigcup }_{p\in \dot{G}}{Z}_p$ and $\dot{\mathcal{N}}={\bigcup }_{p\in \dot{G}}{\mathcal{N}}_p$. The master condition forces $\dot{Z}$ uncountable: if an $M$-name enumerated it countably, F3 would put all of its ground points in $M$, contrary to $x\in \dot{Z}\setminus M$. Properness gives the ground-model countable-covering property by the same master-name argument, so every countable subset of $\dot{Z}$ is contained in some side model $N\in \dot{\mathcal{N}}$. The order clause gives $N\cap \dot{Z}{\subseteq }^{\ast }{I}_N$, and therefore $N\cap \dot{Z}\in \mathcal{I}$. Hence $q$ forces every countable subset of $\dot{Z}$ to lie in $\mathcal{I}$.

Work in the proper cone $Q_{\mathcal{I}}\upharpoonright q$. Choose names $\dot{f}:{\omega }_1\to \dot{Z}$ and $\dot{g}:{\omega }_1\to \mathcal{I}$ such that $q$ forces $\dot{f}$ injective and $\dot{f}\mathrm{‘}\mathrm{‘}\xi \subseteq \dot{g}(\xi )$ for every $\xi <{\omega }_1$; step 3.1 supplies them. For each $\xi$, the set $D_{\xi }$ of conditions deciding both values is dense. By F1 there is a filter meeting all $D_{\xi }$. Compatibility within the filter makes the decided values coherent, producing in the ground universe an injection $f:{\omega }_1\to S$ and $g:{\omega }_1\to \mathcal{I}$ with $f\mathrm{‘}\mathrm{‘}\xi \subseteq g(\xi )$. Put $Z=f\mathrm{‘}\mathrm{‘}{\omega }_1$. If $a\subseteq Z$ is countable, the set of its $f$-indices is bounded by some $\xi <{\omega }_1$, so $a\subseteq f\mathrm{‘}\mathrm{‘}\xi \subseteq g(\xi )\in \mathcal{I}$ and downward closure gives $a\in \mathcal{I}$. Thus $Z$ witnesses PID's first alternative. Together with the first sentence of step 3.1, this proves PID. AC is used exactly through A1 and the stated ZFC suppliers.