  We show that the Proper Forcing Axiom for forcing notions of size $\aleph_1$
is consistent with the continuum being arbitrarily large. In fact, assuming
$GCH$ holds and $\kappa\geq\omega_2$ is a regular cardinal, we prove that there
is a proper and $\aleph_2$-c.c.\ forcing $\mathbb{P}$ giving rise to a model of
this forcing axiom together with $2^{\aleph_0}=\kappa$ and which, in addition,
satisfies all statements of the form $\mathcal{H}(\aleph_2)\models \forall
x\exists y\varphi(x, y)$, where $\varphi(x, y)$ is a restricted formula with
the property that for every $a\in \mathcal{H}(\aleph_2)$ and every inner model
$M$ of $CH$ with $a\in M$ there is, in $M$, a suitably nice proper poset adding
some $b$ such that $\varphi(a, b)$. In particular, $\mathbb{P}$ forces Moore's
Measuring principle, Baumgartner's Axiom for $\aleph_1$-dense sets of reals,
Todor\v{c}evi\'{c}'s Open Colouring Axiom for sets of size $\aleph_1$, and
Todor\v{e}vi\'{c}'s P-ideal Dichotomy for $\aleph_1$-generated ideals on
$\omega_1$, among other statements. In particular, all these statements are
simultaneously compatible with a large continuum.
