What is the problem that the Many Worlds Interpretation aims to solve?

[PeterDonis]

Physics has always had to deal with that.

Agreed, but I think the MWI is a particularly difficult case.

Not sure about 'not in superposition'. If dead is a valid solution to the cat's wave function, and so is alive, then dead and alive is also a solution.

But the total system is not just the cat. The total system (if we leave out the observer) is atom plus cat. And the atom plus cat system is entangled; there is no way of expressing its state as the product of an atom state and a cat state. So neither the atom nor the cat is in any definite state at all.

Whether or not the atom plus cat system is in a superposition depends on your choice of basis. Yes, in the "atom decayed or not decayed" plus "cat dead or alive" basis, the entangled state is a superposition. But one can always change basis to one in which the particular state of atom plus cat is one of the basis vectors, and in this basis, the atom plus cat state is not in a superposition.

Saying that the cat itself is in a "superposition" is therefore misleading, since the cat is entangled with the atom.

Similar remarks apply when we include you, the observer, in the system and describe the interaction between you and the atom plus cat system when you open the box to look at the cat.

you're declaring the atom-cat-you to be a system

I'm not "declaring" that; that's simply a fact of the scenario. The atom, cat, and you all interact during the scenario; that means you have to consider them as one complete system. Interaction causes entanglement, and you cannot treat an entangled system as multiple isolated systems.

I think in relational terms, so I have a hard time conveying any concept of and objective 'definite state'.

If you want to say that, relationally speaking, no subsystem in an entangled, interacting system is in a definite state, that's fine. If you want to draw a distinction between "objective definite state", which is what no subsystem in an entangled, interacting system is in, and "relational definite state", which has some other meaning, then you should at least attach the "relational" qualifier to the term" definite state", since the ordinary language meaning of "definite state" is "objective definite state". With that choice of language, you would say that none of the atom, cat, and you are in an "objective definite state" after the experiment, but all of them are in "relational definite states" in each branch.

I understand the "relational" viewpoint; I'm just not sure it actually makes sense. In fact that would be a way of rephrasing my previous objections that might make it more evident to you what I am objecting to.

I do have a definition of identity that works, something like "'I' am an event, coupled my worldline leading up to that event". The definition seems to work with MWI and RQM.

No, this definition does not work with MWI and RQM, since in those interpretations, nothing has a single definite worldline. The only single definite trajectory that exists in MWI and RQM is the single definite trajectory of the whole system (atom plus cat plus you, in the scenario we are discussing) in the state space of the system, which is not ordinary spacetime. The "you" in a particular branch of the overall wave function cannot point to a trajectory in spacetime that distinguishes that "you" from the other "yous" in other branches.

If dead-cat+observer-of-dead-cat is not a well defined state, then neither can be the cat+observer state, pre-measurement.

I didn't say dead cat + observer-of-dead-cat is not a well-defined state. It is. But the total cat + observer system being in that state does not mean either the cat or the observer, by themselves, is in a well-defined state.

The cat + observer state pre-measurement is not entangled (but note that you are leaving out the atom that either decayed or didn't decay, so what you are calling "cat" is actually an entangled state of atom + cat), so the cat (entangled atom + cat) and the observer are in well-defined states by themselves pre-measurement, and the cat + observer state is just a product state. The interaction involved in you observing the cat entangles the cat and observer subsystems, so the cat + observer system evolves into an entangled state.

(Almost) all pre-measurement states are post-some-other-measurement, and thus not an 'obviously well-defined state'.

Yes, that's why I pointed out above that what you were calling "cat" is actually "atom + cat" in an entangled state. And the atom, cat, and you have all interacted with other things in the past, so yes, from the MWI point of view the most likely thing is that nothing in the universe is in a definite state by itself; everything is entangled. Only the whole universe will be in a definite state on this view.

If your response to that is "that doesn't seem like it would work", welcome to the club of people who don't think the MWI works.

If all entangled states are needed to define the actual state, then you have a system that is indistinguishable from no state at all.

No, you don't. A system with $10^{123}$ (to throw out some kind of number for the whole universe) degrees of freedom that are all entangled is very different from nothing at all, which has zero degrees of freedom.