Countable Tightness and the Grothendieck Property in $C_p$-Theory

The Grothendieck property has become important in research on the definability of pathological Banach spaces [CI], [HT], and especially [HT20]. We here answer a question of Arhangelskiu{i} by proving it undecidable whether countably tight spaces with Lindelof finite powers are Grothendieck. We answer another of his questions by proving that $mathrm{PFA}$ implies Lindelof countably tight spaces are Grothendieck. We also prove that various other consequences of $mathrm{MA}_{omega_1}$ and $mathrm{PFA}$ considered by Arhangelskiu{i}, Okunev, and Reznichenko are not theorems of $mathrm{ZFC}$.