> QM is the only physical theory that has randomness as an inherent part
It's either inherent randomness or just a deep hole in the whole thing (similar to the alien chess thought experiment problem). Personally I choose to believe that the theory is just incomplete because nobody can even define what a "measurement" really is, meaning in which cases what we do is a "measurement" and in which cases it is not a "measurement". I also think that this is what people like Feynman refer to when they say things like "nobody understands QM", it's actually "nobody understands the wave function collapse", the rest is just maths.
My layman feeling wrt. QM, and Copenhagen school in particular, is that we're searching for too computationally simple mental models. Most other areas of physics - like GR, SR, thermodynamics - can get away with aggregating matter into points, perfect spheres, etc. because they're working in macro scale, but QM is trying to deal with the smallest bits of our reality. Now the boundary between QM and "classical physics" is one where your quantum system will interact with 10^{double digit} amount of other quantum-relevant bits. I have a feeling that searching for what constitutes "a measurement" in such scenario is missing the point, and even talking about the macro system being entangled with the test system is pretty much skipping over all the interesting bits.
It is, but that's in interpretations that also don't have the concept of "wave function collapse" in them. WFC is a feature of interpretations that considers measurement as something ontologically special.
If you don't consider measurement ontologically special, then you need to somehow derive a physically meaningful Born rule without reference to measurement, which so far is something that AFAIK has only been accomplished in theories with large amounts of nonlocality and extra assumptions. The idea that people cling to the obviously false projection postulate out of obstinance is really strange to me, there just aren't very good alternatives available (at least not with the math fully worked out).
I would love to consider measurement something ontologically special, but it's not possible because there is no well-defined definition for what a measurement is.
The definitions I have found always invoke the presence of a "classical system"/"observer".
But that just kicks the can down the road, because there is no well-defined definition of a "classical system" either.
Sure. Everyone agrees Copenhagen is just kicking the can down the road. I'm just saying, let's not act like we have a ton of viable theories to fill out the rest of the road; in the meanwhile, we still have to perform measurements and make predictions, and the projection postulate is handy for that.
(It would help tremendously if we ever measured quantum states that weren't "collapsed", but as we've never done this so far it makes most of the stochastic collapse stuff hard to justify, even if it seems intuitively like the right approach).
Turing machine as an abstract concept is known to not hold a solution to the halting problem. There's no question about that, the only question is how the abstract concept assumptions pertain to the real world (infinite tape? maybe a problem, maybe not?; is the execution speed bounded? or maybe we can somehow count to infinity by exponentially increasing the speed? things like that).
On the other hand nobody has any clue what the quantum measurement / wave function collapse actually is. There are theories/interpretations but no truly satisfying answers in the same way as for example Newton's equations were a satisfying answer to the elliptical movement of planets, even though we later found out in the 20th century that F ~ 1/r^2 was actually an approximation.
We simply don't know, and we have no idea when shall we know.
It's either inherent randomness or just a deep hole in the whole thing (similar to the alien chess thought experiment problem). Personally I choose to believe that the theory is just incomplete because nobody can even define what a "measurement" really is, meaning in which cases what we do is a "measurement" and in which cases it is not a "measurement". I also think that this is what people like Feynman refer to when they say things like "nobody understands QM", it's actually "nobody understands the wave function collapse", the rest is just maths.