The size of Lindelof space Question. Suppose that $X$ is a Lindelof space such
that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?
 A: Unfortunately, the construction of a Lindelöf point $G_\delta$ space of size $\aleph_\omega$ in "A topological application of flat morasses" contains an error which seems to be  incorrigible. 
A: The consistency of this statement is Problem 1 in these slides by F. Tall, where the problem is attributed to Arhangelski. Some partial results already mentioned in Mohammad Golshani´s answer are given. The following is also mentioned:

Theorem (Tall-Usuba): Lévy-collapse a weak compact and then add $\aleph_3$ Cohen subsets of $\omega_1$. Then there are no counterexamples of size $\aleph_2$, even relaxing "points $G_\delta$" to "pseudocharacter $\leq \aleph_1$".

A: There are  consistency results due to Shelah and Gorelic for the negative answer. Shelah has proved the consistency of $GCH+$there exists a Lindelof space with points $G_\delta$ of size $\aleph_2.$ See Shelah's paper ''On  some  problems  in  general  topology''. Gorelic has extended this result to get $CH+2^{\aleph_1}$ arbitrary large+ there is such a space of size $2^{\aleph_1}$.
It is also consistent with $GCH$ that there exists a Lindelof space with points $G_\delta$ of size $\aleph_\omega.$ See R. Knight's paper ''A topological application of flat morasses'', Fundamenta Mathematicae, 194 (2007) pp. 45-66.
On the other hand Shelah has proved the consistency of $CH+$ there is no Lindelof space with points $G_\delta$  of size $\aleph_2.$ See Shelah's paper ''On  some  problems  in  general  topology''.
