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For the first question, think about the electric field of an electron, assuming the electron is infinitely small. The field scales as 1/r^2 (r is the distance away from the electron), and it is finite everywhere except at r=0, where it becomes infinite. This is an example of the finite becoming infinite at a singularity -- much easier to think about than the other way around (the infinite becoming finite). The same thing happens at a black hole. The event horizon is not infinite -- the horizon surrounds the singularity at the center of the black hole, and at the singularity "r goes to zero" and everything blows up to infinity!

Your second question, concerning the quantum information paradox is a very interesting topic. You ask why information should be conserved -- although be aware that some people (Hawking) don't think that it IS always conserved (see the above link). But most people think it should be conserved for a variety of reasons, and it all boils down to reversibility.

So what is information? Let's just say that information is conserved if I can reconstruct absolutely everything about the universe at some previous time, so long as I know everything about the universe right now.

If the laws of physics are time-reversible, which is currently believed to be true (actually, to be precise, you also have to take a mirror-image and charge-reverse everything -- this is known as "CPT" symmetry, C=Charge, P= Parity, T=Time) then information cannot be lost. It's impossible to completely "destroy" anything if you could look at the location of every particle in the universe and "run the universe backwards" to reconstruct what HAD been there at an earlier time. In practice, of course, I can't even put a broken teacup back together, but in principle it's possible to reconstruct just how the teacup broke apart.

This is true in quantum mechanics as well -- if you take a large enough system. One of the great paradoxes is modern physics which often gets swept under the rug is the "collapse" postulate in quantum mechanics, where a quantum wavefunction "collapses" when it is observed by an outside observer. But the collapse postulate is not reversible like all the rest of physics! This is one clue that this particular interpretation of Quantum needs an overhaul, but cosmologists often avoid thinking about it by taking the entire universe as their quantum wavefunction. That way, they figure, there's no "outside" observer to collapse anything, and everything stays reversible. (This is somewhat sketchy, because I think it's evident that something HAS collapsed the universe or it would look very different than it does. But this is getting off-subject...)

To summarize: if all the laws of physics are reversible, then information can't ever be destroyed. With the exception of the problematic "collapse postulate", the laws of physics all seem to be reversible. But the reason there's a quantum information paradox in the first place, is that Hawking seems to have shown that any information that gets dumped in a black hole ISN'T conserved! So is it conserved or isn't it?

My opinion on the matter is that information is NOT conserved when there is an interaction with a final-boundary condition. You can think of the Big Bang as an "initial-boundary condition" of the universe, and a black hole as a "final boundary condition" for whatever falls into it. If the universe ever collapses into a Big Crunch, that will be a "final-boundary condition" for the whole universe as well.

Currently quantum mechanics only assumes one boundary condition; it's not equipped for the two-boundary problem where it's constrained both in the past and the future. Murray Gell-Mann and James Hartle have constructed a new theory where wavefunctions have TWO boundary conditions, but this particular theory doesn't seem to correspond to the universe we live in. Another overlooked-possibility is that there are two components to the universe -- one set of wavefunctions with an initial boundary (but no final boundary), and another set with a final boundary (but no initial boundary). This theory keeps CPT symmetry, and also predicts that information would be lost whenever the two components interacted (i.e. at the event horizon of a black hole). Here is a not-too-dense paper which discusses this theory in detail.

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