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**](http://dx.doi.org/10.1016/0003-4843(76)90018-8), Ann. Math. Logic 9, 401-439 (1976). [ZBL0339.04003](https://zbmath.org/?q=an:0339.04003).


show that the statement is not equivalent to CH.


**Question:** Can the above statement consistently fail?



Diagonalizing against $\omega_1$-sequences of functions mod finite
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.


