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How can the density of a region of space go from finite density to infinite when there are no numbers larger than any Aleph0 number but smaller than any Aleph1 number (no decimal point in front of it, of course)? Aren't Planck volumes and strings designed to sidestep infinities?

My point there, stated differently, is how can the density go from finite to infinite when there is a 'no-number gap' between finite and infinite, with Cantor losing his mind contemplating that gap? And did he not prove that none was constructible/possible?

The entropy of the visible universe is ~(10^122)^2, if I'm not too far off the mark. This is a stupendous number but no closer to infinity than any other integer or real. And there is no room in it to tuck a singularity away.

In simpler terms. Our BH has a finite mass. If it has a region of infinite density, that region must be infinitesimal. But the Planck length is the lower limit on the size of regions of space. There are, therefore, no points in space, no infinitesimals, only punctoids, my term of convenience, the utility of which may become obvious in future posts.And no infinities.

Most physicists believe that the prediction of an infinite-density singularity (though note that for a Schwarzschild spacetime, the singularity is a moment in time, NOT a point in space) is a flaw in general relativity rather than a real physical thing that happens, and that at some density roughly around $m_p/ell_p^3$, where $m_p$ is the planck mass, and $ell_p$ is the planck length, quantum gravitational effects will take over and prevent a true singularity from forming. Obviously, without a working quantum theory of gravity, no one can know exactly how this happens, but this is the expectation.

Physics uses mathematical models of the real world. The best models are simpler, more accurate, and/or have broader scope that other models. General relativity is currently the best (simplest, most accurate, broadest scope) well-tested model that we have for describing gravitational phenomena. However, any time any model predicts a "singularity" (or "infinite density", etc), that's a sign that we have exceeded the limits of that model's validity. That's just as true for general relativity as it is for any other model.

Even ignoring singularities, we have other good reasons to think that general relativity is only approximately valid and that it breaks down in some extreme circumstances that are currently beyond our ability to explore experimentally. The black hole information paradox is a famous example. The basic problem is that general relativity doesn't account for quantum physics, and one of the lessons of the Black Hole Information Paradox (when analyzed carefully) is that any way of reconciling general relativity with quantum physics will necessarily require some radical change(s) in our current understanding of nature.

General relativity is not expected to be a good approximation under the extreme conditions where it would predict a singularity, and the black hole information paradox (and the related firewall paradox) gives us reasons to suspect that it might break down under even less extreme conditions. According to [1],

The black hole information paradox forces us into a strange situation: we must find a way to break the semiclassical approximation in a domain where no quantum gravity effects would normally be expected.

(In this excerpt, the "semiclassical approximation" is an approximation in which gravity is treated as a non-quantum thing and everything else is treated as quantum. That's the approximation we use today to describe everyday situations involving both gravity and quantum effects, like when individual atoms fall under the influence of earth's gravity. We don't need a theory of "quantum gravity" for that kind of thing, because the gravitational field itself is not exhibiting significant quantum behavior in that case.)

Page 2 in same paper summarizes the black hole information paradox like this:

...in any theory of gravity, it is hard to prevent the formation of black holes. Once we have a black hole, an explicit computation shows that the hole slowly radiates energy by a quantum mechanical process. But the details of this process are such that when the hole disappears, the radiation it leaves behind cannot be attributed any quantum state at all. This is a violation of quantum mechanics. Many years of effort could provide no clear resolution of this problem. The robustness of the paradox stems from the fact that it uses no details of the actual theory of quantum gravity. Thus one of our assumptions about low energy physics must be in error. This, in turn, implies that resolving the paradox should teach us something fundamentally new about the way that physics works.

Reference:

[1] Mathur (2012), "Black Holes and Beyond," http://arxiv.org/abs/1205.0776

There is no singularity associated with a Black-hole. At the final nanosecond in the formation of a black hole, essentially all of the matter confined within the envelope of a developing Schwarzschild sphere (whatever its size) can no longer be transformed into particle kinetic-energy, i.e., particle momentum.

Special relativity does NOT allow an inertial system to exceed the speed of light; and in the interior of a collapsing star, particles colliding at speeds approaching that of light cannot absorb a further increase in the momentum produced by gravitational forces. At this point, gravitational energy is transformed directly into radiation (entropy) rather than particle momentum.

This critical event, representing a change-in-state from matter to radiation, is preceded by an exponential increase in the momentum of particles confined within the volume of a contracting star (or any object). As particle velocities approach the speed of light, and as distance and time between particle collisions approach zero, the energy-density of a collapsing object will reach a limit where the transformation of gravitational force can no longer be defined in terms of particle collisions.

As the distance between colliding particles gets smaller, quantum mechanical factors require that the uncertainty in particle momentum correspondingly gets larger. At a critical point in this combination of events, the collision-distance between particles has decreased to a nanometer range that corresponds to the frequency of particle collisions; and particle interactions can be expressed in terms of a series of discrete quantum-mechanical wave-functions (distance and time between particle collisions) that approach the restricting value of 'h' (the Planck constant), producing a potential, quantum-mechanical catastrophe.

In order to preserve thermodynamic continuity, the thermodynamics of the system must change; consequently, particle matter is transformed into radiant energy by means of quantum mechanical processes. Gravitational energy is now expressed as a function of the total radiation-energy distributed over the surface of the ensuing Schwarzschild sphere, and any additional energy impacting the Black-hole produces an expansion of the Schwarzschild radius, while maintaining a constant energy-density, and a constant boundary acceleration, corresponding to a constant (Unruh) temperature.

A clue to the transformation of kinetic energy to radiation is seen in the function: e^hf/KT (from the Planck "key" to the ultraviolet paradox). In this function, particle kinetic-energy, "KT", increases due to an increase in particle velocity and effective particle temperature; the frequency of particle collisions, represented by "hf", increases with particle density due to increased gravitational confinement within a developing Schwarzschild object (Black Hole). But temperature and frequency do not rise to infinity, as one might expect from the Planck relationship.

The formation of a Schwarzschild boundary is coincidental with a maximum particle acceleration and a maximum temperature. This critical event represents the maximum energy-density (rather than the maximum energy) permitted by nature. Temperatures cannot rise beyond this critical point. Instead, these variables now become Black-hole constants and are conserved in all Black-Holes, regardless of their size and total energy. Subsequent to the formation of a Black-hole, temperature, acceleration, gravity and energy-density remain constant at their maximum values...even as more energy is added and the Schwarzschild envelope grows correspondingly larger.