Xjyuk

My point is that there is no superposition of dead and alive. The cat will cause the probability function to collapse long before we open the box. What am I missing here? Isn't the cat capable of collapsing the probability function? If not, why?

According to the Many Worlds view, no cat, nor even a human scientist inside the box, will "collapse the wavefunction". The scientist, the cat, and the radioactive particle are all components of the universal wavefunction, which simply branches when the radioactive particle both decays and does not decay. To an observer outside who can't see inside the box or interact in any way with what is in the box, the scientist both dies with the cat and survives with the cat, until the observer peeks inside the box. At that point, the outside observer's wavefunction branches because it has become correlated with the wavefunction of all that's inside the box. The idea that "consciousness" causes wavefunction collapse has no meaning. The scientist who subjects himself to the cat's fate is conscious (aware) in one branch that he survived; in the other branch he might briefly be conscious/aware that he is dying.

A theorem of von Neumann says that it doesn't make a bit of difference whether you model the cat (or anything else along the causal chain between closing the box and opening it to observe the cat) as capable of collapsing the wave function. You'll make exactly the same testable predictions no matter where along the way you place the collapse.

So feel free to posit that the cat collapses the wave function. Or to posit that only a human has that power. And if you prefer one story while your neighbor prefers another, let a thousand flowers bloom.

This is why I generally prefer a "mostly subjective" viewpoint of quantum mechanics at it is really, despite looking at all the alternatives, the only one that fits the closest to the mathematics of the theory as given with no other adulterations (other ideas like MWI, Bohm, etc. really are "different theories" in that they play with the maths and seem to have a fixation on eliminating and explaining away the collapse concept, and thus really are "empirically equivalent theories" [except when they're not!]. My thinking has been that we need to take it blunt, at face value, and see where that leads.).

On subjective account, the wave function belongs to you, the one outside of the box. It models, your information or knowledge about the state of affairs in the box. The transition from "live cat" to "live or dead cat" to "dead cat" starting from the initial state is just showing how your best knowledge, without looking into the box, changes. The "superposition" just means you don't know.

That said, the theory does force us to admit that there really is something "odd" going on "in reality", otherwise it would just be doable with classical mechanics. But that "oddity" is more that the Universe seems to have an information limit that prevents the answers to all questions about a system from existing with perfect information at all times. Problems identified with subjectivity of wave function in literature seem to be hung up on the idea that if you take it as subjective you are taking it as subjective with some further assumptions on what the "real" reality should look like that often amount to sneaking classical mechanics in the back door, instead of letting the maths guide you as to what you can/can't say thereabout (which is that, if we make no such further assumptions, except perhaps relativistic causality, then you have to say that however it exists, physical parameters have "limited resolution" - limited, even fractions of a bit of, information.). Moreover, this frustrates us from being able to know - except perhaps at some moments - what information is really there with truly faithful-to-"reality" detail, but we should not confuse that with "no detail". We just need to qualify things, and not get caught up in black-and-white, all-or-nothing, either/XOR thinking.

In this case, though, as @WillO says, a lot of scenarios may be consistent with the reality of the situation. The question, moreover, bears witness to a misconception in that it is assuming a "magical" property of "observation" due to the collapse postulate, and that the wave function does objectively belong to the system. Yes, you can try that, and yes, then you get this idea (or you could say this evidences a problem with the idea), but you don't need to, and if you stick wholly to that the wave function belongs to each subject involved, then there is no need to posit this. As a subject, the cat may be assigned a wave function talking about the information it has regarding the contraption that is going to kill it. Of course, soon after that one "collapses" then there won't be any more wave function anymore because this subject, the information-bearer, has been terminated.

Hence, from that point of view, it makes no sense to ask this question because the wave function models your knowledge. The cat can't do anything to that. Well, maybe it can - it lets out one final scream as it dies, you hear that through the box and update your knowledge accordingly :)