Ah, but the gravity isn't infinite, since the object has a finite mass. Pressure is obviously always infinite, since you have infinite density object in a finite density environment.
Hm, how can a singularity form in the first place? As pressure approaches infinity, wouldn't it eventually match the gravity's pull and leave the matter hanging midway between black hole centre and event horizon?
But there is no pressure. The only characteristics the black hole retains of it's parent star, are mass, angular momentum, electric charge.(it's called the No Hair principle)
The collapsing star undergoes a number of steps - the gravity needs to be strong enough to overcome radiation pressure first, which happens when too much of heavy elements accumulated in the stellar core; then there's the electron degenerancy pressure, when atoms are squeezed so close together that the Pauli's exclusion principle(that no two identical fermions, e.g. electrons, protons, neutrons; but not e.g. photons; can occupy the same "spot") acting on free electrons becomes the main factor preventing matter from moving any closer. At this point, the classical gas pressure stops being accountable for.
With high enough mass, the electrons in the degenerate matter are so energetic, that they can combine with protons to create neutrons, and the result is a neutron star, where the pressure is provided by Pauli's exclusion principle acting on neutrons. The neutron star is much smaller, as there are no more energetic electrons filling the space between the nuclei and keeping them farther apart. The electron degenerancy pressure dissapears.
If the mass is high enough, the collapse isn't stopped by neutron degenerancy pressure, and the star collapses further, the surface gravity becoming high enough to prevent light from escaping, and we have a black hole.
The point is, once the consecutively appearing "pressures" are overcame, these stop acting any further. Once the last line of defence is broken, there is no more resistance preventing the collapse towards a singularity.
The density of a singularity is infinite, but there is no infinite, or any pressure at all.
I don't know much about physics, but with enough resources I should be able to math it out... I also happen to have a book about the physics of black holes that I've been ignoring for the past two years. Can you tell me what, precisely, the questions you're trying to answer are?
I don't even know myself. It was about understanding how does a naked singularity appears.