So... uh, what is the problem of consciousness?
I'm not sure, but it seems to relate to what Eliezer highlighted in how an algorithm feels from inside; the way brains track concepts as separate from the components that define them. If you can imagine consciousness as something that persists as a first-order object, something separate from the brain - because it is hard to recognize "thinking" when looking at your brain - if you can see "I" as a concept distinct from the brain that you are, it makes sense to imagine "I wake up as Eliezer"; you just take the "I" object and reassign it to Eliezer's brain. That's why the sequences are so big on dissolving the question and looking at what experiences the concept actually makes you anticipate.
Afaics, the problem is hard not because of some intrinsic difficulty but because it requires us to recognize "ourselves" in our brains, and consciousness is so central to our experience that it's hard to go up against the intuitions we have about it.
Although Elizier has dealt with personal identity questions (in terms of ruling out the body theory), he has not actually, as far as I know, "solved" the problem of Personal Identity as it is understood in philosophy. Nor, as far as I know, has any thinker (Robin Hanson, Yvain, etc) broadly in the same school of thought.
Why do I think it worth solving? One- Lesswrong has a tradition of trying to solve all of philosophy through thinking better than philosophers do. Even when I don't agree with it, the result is often enlightening. Two- What counts as 'same person' could easily have significant implications for large numbers of ethical dilemnas, and thus for Lesswrongian ethics.
Three- most importantly of all, the correct theory has practical implications for cryonics. I don't know enough to assert any theory as actually true, but if, say, Identity as Continuity of Form rather than of Matter were the true theory it would mean that preserving only the mental data would not be enough. What kind of preservation is necessary also varies somewhat- the difference in requirement based on a Continuity of Consciousness v.s a Continuity of Psyche theory, for example should be obvious.
I'm curious what people here think. What is the correct answer? No-self theory? Psyche theory? Derek Parfit's theory in some manner? Or if there is a correct way to dissolve the question, what is that correct way?