The Newtonian gravitational potential is given by:
Which appears in the Schwarzschild metric tensor with a so-called singularity at $r=0$. Nonetheless, I can't get why is it necessarily the case that a singularity should form.
From Newtonian formulas, we would get that there is an infinite potential at the center of the Earth. But that is not the case because $M$ decreases along with $r$. When you get to the center of our planet, all the matter (approx.) is bordering you, so discounting the insane pressure in that place you are actually pulled towards the exterior in all directions (so that pull ends up cancelling).
My question is; doesn't it happen the same with black holes? In the same way the gravitational force is not infinite at $r=0$ because the mass $M$ becomes zero too, then why is this logic dismissed when studying the black holes case?