This question concerns proper forcings of size $\aleph_1$. In the context of $\rm ZFC+\neg CH$, I couldn't find any counter example to the following property. Suppose $\mathbb P$ is a proper forcing of size $\aleph_1$ in $H_\theta$, and suppose $p$ is $(M,\mathbb P)$-generic, where $M\prec H_\theta$ is countable and contains $\mathbb P$. Then if $M^*\prec H_\theta$ is an end extension of $M$ in the sense that $M\cap\omega_1=M^*\cap\omega_1$, $p$ is then $(M^*,\mathbb P)$-generic.
I may notice that I am not looking for a artificial construction of a single condition, I want every generic condition $p\in\mathbb P$ have(morally) the above property.
Let me give you the motivation, and then I state my question in a formal way.
Definition(Aspero-Mota): A forcing notion $\mathbb P$ is finitely proper if whenever $\mathcal M$ is a finite set of countable models and $p$ is a condition in $\bigcap \mathcal M$, then there is a condition $q\leq p$ which is $(M,\mathbb P)$-generic for every $M\in\mathcal M$.
The following is a particular case of their theorem..
Theorem The forcing axiom for meeting $\aleph_2$-dense sets by finitely proper forcings of size $\aleph_1$ is consistent with continuum large.
But they give an example of a proper forcing of size $\aleph_1$ which is not finitely proper. I have a relaxed version of the above property(they call it $\aleph_{1.5}-cc$), let us call it $\aleph_{\sqrt{\pi}}-cc$. I think this notion is iterable using virtual models and we get a forcing axiom with $2^{\aleph_0}=\aleph_3$. This is slightly stronger than Aspero-Mota's axiom(in the sense of the above particular case), but in fact this is not important. I wonder if we can prove the proper forcing axiom for posets of size $\aleph_1$ does not imply $2^{\aleph_0}=\aleph_2$.
What is the issue? The issue is that in our construction using two types models we don't have $\in$-chains, so we can't extend our conditions inductively to be generic for all models in the side condition. But if every proper poset of size $\aleph_1$ is $\aleph_{\sqrt{\pi}}-cc$, then it seems everything is fine and we can show that proper forcing axiom for poset of size $\aleph_1$(meeting $\aleph_2$ many dense sets) is consistent with $2^{\aleph_0}=\aleph_3$.
I think for a proper poset $\mathbb P$ of size $\aleph_1$, the $M$-genericity of a condition $p$ should only depend on $M\cap\omega_1$ when $\rm CH$ fails. The following is a formal attempt to find a structure theory for posets of size $\aleph_1$.
Question Assume $\rm CH$ fails. Let $\langle p_\alpha:\alpha<\omega_1\rangle$ be an enumeration of $\mathbb P$. Can one find a function $\rho:\omega_1\rightarrow\omega_1$ such that if $M\prec H_\theta$ is countable, then for every $p\in M$, there is some $\beta\leq \rho(M\cap\omega_1)$ such that $p_\beta\leq p$ is $(M,\mathbb P)$-generic?
Remarks:
If the answer is yes, then it means we don't have to go unboundedly may times for finding generics for various models with the same $\omega_1$, and by a suitable decoration of side condition, we get $\rm PFA_{\omega_2}[\aleph_1]$ is consistent with $2^{\aleph_0}=\aleph_3$
Maybe the answer has to do with Chang's conjecture!
We may assume $\mathbb P$ has the $\omega_1$-approximation property.
We are also interested in the following case. If $\mathbb P$ is a counterexample to the above question we may think about a forcing notion of size $\aleph_1$ which kills the properness of $\mathbb P$, though itself enjoys the above property.
finitely proper and $\aleph_{1.5}-cc$ are not necessarily the same, but for the forcing of size $\aleph_1$ they coincide.
