mattt said:
There are several mathematical theorems about that subject. "Willing to accept" seems a very odd way of saying "they know some mathematical theorems".
When you accept even simple mathematics (2+2=4) , you get Godel's proof along with it. Most of the time you can kick it into a corner and ignore it. Other times, it is in your face. I remember a time when studying the strong force meant dealing with infinities. Fortunately renormalization came along and pushed those infinities back into the dusty corners. But with general relativity, there are still infinities, and they are in the black holes that are all over the place. (On a stellar scale at least.) You may not be able to see into a black hole, but they are sort of tough to ignore.
Obviously you are using the word "inconsistent" in a different way than it is used in Mathematics or even in Physics. General Relativity is, basically, Semi-Riemannian Geometry with a given interpretation of some of its mathematical concepts. So it is mathematically consistent ( assuming consistency of ZFC ).
No it is not consistent, and at a very fundamental level. We talk about the Schwarzschild radius of a black hole, while ignoring the fact that GR says it is infinite. Any radius beginning at the center of a black hole is infinite in GR. That rubber sheet gets stretched to infinity. It is possible to (approximately) measure the circumference at the event horizon and divide by 2π, but that radius is not physical. Similarly, if I measure the potential energy of two particles relative to a black hole and subtract, I get garbage. The distributive law doesn't work. To be blunt, anywhere in a GR universe with at least one black hole, I can prove that 2+2=5, or 42, or whatever. Do I believe that there is a version of GR that gets rid of the infinities? Sure. It may or may not be physically correct, it is hard to take measurements inside a black hole, but it could be consistent. Hawking has recently done some work which says that the event horizon may be impossible to measure accurately. So the best we can do for now is to say that we have a (mostly) consistent physics except near black holes.
Another different thing (and I think this is what you are trying to point out) is that it is more useful or less useful, depending on what you want to do with it.
No, what I said was: "That it is useless to try merging GR and QM." Right now, absent black holes, GR is consistent, and so is QM. But wishing black holes away doesn't work. And the really nasty piece of work in GR is that it is possible for things, particles, information, anything to violate causality and do it outside the event horizon (in the ergosphere). So the problems with GR are not localized to the region of black holes like we might hope, but can spread to anywhere within the event cone of the black hole.
I would like for someone to "fix" that part of GR, but it seems impossible. You may have heard the joke: "QM, GR, and causality. Pick any two." If you try to rigorously merge QM and GR, causality seems to go out the window. Way out the window. Maybe someone can come up with a logic that works without causality, but I am not holding my breath. Until then the best choice seems to be some version of GR where the distortion of space in the area of a rotating black eliminates time-travel. Hawking seemed to be headed this way with his closed timelike loops (CTL). A true CTL outside an event horizon would contain immeasurable energy. (See Heisenberg.) If every route around a black hole that violates causality has to cross a CTL then causality is preserved. (Unfortunately, this may be true for a static rotating black hole, but throw enough mass in and you may be able to sneak through.)
If you are trying to say that there will always exist things that we can not predict or calculate with "total" accuracy, or even that there will still exist things that we can not even "explain"...well, it is probably true.
It is true, it was proved a century ago by Godel. Douglas Hofstetter's book "Godel, Escher, Bach, an Eternal Golden Braid" is probably the best book on the subject for non-specialists. The writing is excellent, but the material is very hard. What you should get out of the book is that there are true statements that cannot be proven true in any consistent formal system. (You will also understand emergent properties very well.)
But what the heck does it have to do with us creating better and better models with larger range of validity?
I mean, it seems to me you were kind of criticizing our efforts of trying to create models of quantum gravity (or whatever we think will be the next improvement in our quest for consistent models with larger range of validity), basing your opinion in that there will always be things we, possibly, could not explain or predict.
It is that what you're trying to say?
The point that (current) GR and QM cannot be usefully merged is very worth knowing. (AKA the renormalization issues are impossible.) Loop quantum gravity (LQG) and spin foams may result in a version of gravity that is consistent even inside event horizons. Then a merger with QM becomes useful.
Finally, the press may like the name theory of everything (ToE) for a unified field theory(UFT). But the distance between a UFT and a ToE is not just immense. It may be impossible to cross. (A real ToE at least has to deal with dark matter and dark energy.)