TITLE: Diagonalizing against $\omega_1$-sequences of functions mod finite
QUESTION [16 upvotes]: The following statement is a direct consequence of the Continuum Hypothesis:

There exists a sequence $\langle f_\alpha:\omega_1\rightarrow\omega_1 ~ \vert ~ \alpha<\omega_1\rangle$ of functions such that there is no function $f:\omega_1\rightarrow\omega_1$ with the property that the sets $\{\xi<\omega_1 ~ \vert ~ f(\xi)=f_\alpha(\xi)\}$ are finite for all $\alpha<\omega_1$.

Moreover, since a failure of this statement can be used to obtain an $\omega_2$-sequence  of subsets of $\omega_1$ with pairwise finite intersection, results of Baumgartner in
Baumgartner, James E., Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 9, 401-439 (1976). ZBL0339.04003.
show that the statement is not equivalent to CH.
Question: Can the above statement consistently fail?

REPLY [6 votes]: The arguments in Section 6 of my paper "The nonstationary ideal in the $\mathbb{P}_{\mathrm{max}}$ extension" show that there is a proper forcing adding a function from $\omega_{1}$ to $\omega_{1}$ which agrees with each such ground model function in only finitely many places. I say there that Todorcevic had done something similar in his "A note on the Proper Forcing Axiom". It follows that under PFA, and in the $\mathbb{P}_{\mathrm{max}}$ extension, the statement above fails. You don't need any large cardinals, however.