If the gravity inside the BH appraches infinity, would this increase the density of the matter in the BH to infinity too ?
It's not exactly infinity. Calculating density or gravity requires dividing by a unit of linear measurement (to one power or another). When positive numbers get low, the result of dividing by them (or their squares or cubes) gets high... but the extreme end of that trend is NOT division by yielding infinity. The result is actually "undefined", because it's something you just can't do... not just you can't do it physically, but you can't even do it mathematically.
To illustrate the difference between infinity and undefinity (yes, I did just make that word up), consider that whenever
n/d=q, that means dq=n. Starting with any division problem, you can always multiply the answer by what you were dividing by, and get what you were dividing. So if division by zero yielded infinity, that would mean zero times infinity would equal whatever "n" was, whatever number you were dividing by zero in the first place. But does it? It can't; multiplying zero by infinity doesn't give you any particular answer. Multiplication by zero just yields zero, multiplication by infinity just yields another infinity, and even if those two facts somehow didn't get get in a fight with each other in this case, you'd still have no way to choose an answer among the infinite supply of finite non-zero numbers. And if you can't multiply two things together and get an answer, then there's nothing that you can divide by one of them that will give you the other as an answer.
(No "dq" means no "n".)
The rubber anology, but with the BH as a well in the rubber, still assists visualisation of it for me.
Then let me throw in a couple of things that will make it work better.
First, there's the matter of how far down the gravity well or cone pulls the surface, and how steep that means the sides of the well/cone are. Stronger gravity pulls it farther and creates a well/cone with steeper sides. A black hole is the answer to the question "What would you have if the sheet were pulled an infinite distance away and the well became a cylinder instead of a cone, with perfectly vertical sides?". That image presents us an interesting component: the lack of a bottom point for the gravity well's walls to converge to. Instead there's a tube, an area that would look like a circle from above, where the sheet does not cover. What could that represent? Something that is not within our universe. That's one way in which the name "black hole" fits pretty well.
Second, there's something we can add to the sheet to make the image more complete. In its original form, it's just a snapshot of space at some particular moment; it's got nothing to show the passage of time from one moment to the next. To show time, we'd need to draw the sheet multiple times, above and below each other, where each one represented where things are a moment after the one below and a moment before the one above. So up is the direction of the flow of time. But instead of representing that with lots of different layers in one image, we can just use one layer like we had before, with arrows added to show that time flows upward instead. If you're picturing the sheet with a grid drawn on it, then at each intersection, put a little arrow pointing up and away off of the sheet, perpendicular to it, like a nail. The look at what happens when you pull on the sheet to represent gravity. The rule about time and space is that time has to be perpendicular to space, so when you pull the sheet down, the time arrows tip in with it in order for each arrow to stay perpendicular to the point at which it's attached. The steeper the gravity well is, the farther they tip in. If the gravity well's walls go vertical, all of the time arrows then point in toward the central axis of that cylinder I described before. Whereas the flat sheet had space laid out horizontally and time vertically, in the black hole, you end up with space laid out vertically and time horizontally; they've switched places/roles. Each one acts like the other would normally have acted. When that happens, we get a whole new reason why you can't escape: the center, the singularity, is not a place you move toward or past or away from, but a moment in your future, and you can't not go to your future.
would that not mean that the dimensions of what ever matter there is in it would then approach NIL ? Is it still matter if this is the case ?
We don't know. The current theories don't know any force that can stop the mass from collapsing into a point.
We don't know of anything that could cause it either. It would require a bunch of particles to become smaller than a particle is, which is not an effect that any known force has. A field-based (non-quantum) view of gravity doesn't forbid it, but only because it doesn't pay any attention to particles, so that's like the fact that the rules of airplane design have nothing against owls living on a diet of lemons and shoelaces.
If the platform is inside the horizon, the chains will break.
And the reason why is important. When the answer to some kinds of questions in physics is that the proposed object would break, some people respond by just asking what if it's a really really strong substance. But any object's strength is based on the interactions of electrons and protons, which interact by emitting and absorbing photons... so if a photon can't get from one particle in the object to another, then there's no basis for the object having any strength at all, or even being an object at all. (And in this case, redshifting & blueshifting would have the same results before you got to the point of "can't get there at all anymore" anyway.)
