If you have the memories of every single human up to that point, then you don't know which of them you are.
This depends on the mechanism of attaining all these memories. In that world, it COULD be that you still know which memories are privileged, or at least which ones include meeting God and being in position to be asked the question.
I mean, I'm with you fundamentally: it's not obvious that ANYTHING is truly objective - other people can report experiences, but that's mediated by your perceptions as well. In most cases, one can avoid the confusion by specifying predicting WHAT experiences will happen to WHICH observer.
1An idea I've been playing with recently:
Suppose you have some "objective world" space . Then in order to talk about subjective questions, you need a reference frame, which we could think of as the members of a fiber of some function , for some "interpretation space" .
The interpretations themselves might abstract to some "latent space" according to a function . Functions of would then be "subjective" (depending on the interpretation they arise from), yet still potentially meaningfully constrained, based on . In particular if some structure in lifts homomorphically up through and down through , you get exactly the same structure in . (And these obviously compose nicely since they're just spans, so far.)
The key question is what kind of space/algebra to preserve. I can find lots of structures that work well for particular abstractions, but it seems like the theory would have to be developed separately for each type of structure, as I don't see any overarching one.
I am having trouble following you. If little-omega is a reference frame I would expect it to be a function that takes in the "objective world" (Omega) and spits out a subjective one. But you seem to have it the other way around? Or am I misunderstanding?
isn't a reference frame; rather, if is a world then aka are the reference frames for .
Essentially when dealing with generalized reference frames that contain answers to questions such as "who are you?", the possible reference frames are going to depend on the world (because you can only be a real person, and which real people there are depends on what the world is). As such, "reference frames" don't make sense in isolation, rather one needs a (world, reference frame) pair, which is what I call an "interpretation".
This summarizes a (possibly trivial) observation that I found interesting.
Story
An all-powerful god decides to play a game. They stop time, grab a random human, and ask them "What will you see next?". The human answers, then time is switched back on and the god looks at how well they performed. Most of the time the humans get it right, but occasionally they are caught by surprise and get it wrong.
To be more generous the god decides to give them access (for the game) to the entirety of all objective facts. The position and momentum of every elementary particle, every thought and memory anyone has ever had (before the time freeze) etc. However, suddenly performance in the game drops from 99% to 0%. How can this be? They have more information, they know everything!
If you have the memories of every single human up to that point, then you don't know which of them you are. Who is "you" in the "What will you seen next?"? Before the extra information was added you knew which human you were, it was made very obvious by the memories and information at your disposal. But, given the memories of everyone, and all that other information, you suddenly require an additional piece of information to answer the question.
God: "What will you see next?"
Participant: "I know what every human will see next, but I don't know which one I am".
G: "You have been given all the information there is. How can you not know?"
Idea
"What will you see next?" is a subjective question. All objective facts put together is not enough to answer it, because an addition piece of information "Which person am I"? is needed. This final piece of information is subjective, and arguably in some materialistic sense doesn't really exist. But, the question is also subjective, so it should not be surprising that subjective information is needed to answer it.
You can make the situation more extreme in various ways. Instead of providing a snapshot of the universe at a particular time the god could provide the participant the entire history of the universe from beginning to end (a block universe). Then they need to know not only which person they are, but at what time. If you take Many Worlds seriously then a participant given the wavefunction of the entire universe would also need to ask "Which branch am I in?"
In everyday life this doesn't really matter, because we know which human we are, and through continuity we trace that label out into the future and past. It only comes up when we expect our subjective experience to split.
If you are like me you have wondered about a replication machine, or of some future where you can upload your mind and have a copy of you in the machine. These situations often feel paradoxical, because we intuitively want to ask questions like "What will I see next? The simulation, or more fancy MRI machine?" We find that we don't seem to be able to give an exact answer to this question, (is it 50/50? Are we about to see "both"? if the digital copy has .zip file backups does that change the distribution?), and this seems confusing because we cannot identify any objective facts that are missing. The answer is that the "I" (and the "next") are anchored subjectively and that if they are under-specified the question is under-constrained.
Splitting in this way occurs in several theories and thought experiments with some examples below:
Applications
Sleeping Beauty: In the Sleeping Beauty problem a "halfer" thinks that every objectively distinct universe is equally likely, while a "thirder" thinks that every subjectively distinct universe is equally likely. (In the timeline where Beauty is awoken twice there is only one objective universe evolving over time, but Beauty has two separate subjective experiences of waking up in that universe).
Replication Machines: Do you weight the probability of futures where you will be replicated more highly than others? If I am going to be replicated until there are three of me, then there will be three subjectively distinct universes I could associate with that single objective one. How I assign likelihoods to different outcomes in these cases will come down to whether I count over the objectively distinct outcomes, or the subjectively distinct ones.
Born Rule in Many Worlds: If you take the Many Worlds Interpretation of quantum mechanics seriously then the Born rule doesn't in any way govern the dynamics of the objective universe. Through this lens the objective universe (the wavefunction) has a large number of different subjective universes associated with it (different branches of me) and the Born rule tells us something about how strongly each of those subjectively distinct universes is supported. It is a recipe for turning one objective thing into a probability distribution over subjective ones.
Quantum Immortality: The Quantum immortality argument rests on the assumption that we count over subjectively distinct universes. With the additional caveat that you should assign zero weight to objective universes in which you immediately die (and thus have no subjective experience). [For the record I think the Quantum Immortality argument is very bad, mostly because the additional stipulation that you should discard the timelines with zero subjective experience seems like nonsense].
Boltzman brains are another example of reasoning based on the multiplicity of subjectively distinct worlds, rather than objectively distinct ones. These arguments say either (1) The one true objective reality contains many copies of your subjective experiences, most of them thermal fluctuations, or (2) There are many objective realities consistent with your subjective experience, and in most of them you are a thermal fluctuation. I don't know if (1) or (2) is the canonical BB scenario, but they are very different arguments.
Turning the Tables
So far we have largely considered the case where one objective universe ([OU]), supports many subjective ones ([SU]). The opposite situation, where many distinct [OU]s all support the same [SU] is commonplace. For example in thermodynamics, a huge family of distinct [OU]s (where the air particles have different positions and momenta) are all consistent with a single [SU] where I feel the temperature and pressure of my room. In these cases it is uncontroversial that when we don't know which state things are actually in we should do probability theory by assuming all of the possible [OU]s are equally probable, not all of the [SU]s.
(The entropy of a system is the related to the number of distinct [OU]s possible given the [SU] we observe. If we wanted to be really weird, we could generalize this to the examples above with replication machines etc., but now with the [SU]s outnumbering the [OU]s, which would do something strange, maybe a negative entropy? Seems like nonsense?)
Conclusion
A single objective world can contain a large number of different, subjectively distinct perspectives. Many "weird feeling" arguments come down to deciding if situations that are subjectively distinct, but objectively identical, should be counted according to multiplicity of the objective or of the subjective. When looking at a strange thought experiment it can be useful to remember that the totality of all objective facts may be insufficient when trying to answer a question that is subjective.
I think this distinction between counts over subjective and objective states of the universe gets at something important.