I have not been on this site very much, I must admit, but I stumbled on the question Can I have visited all rooms of Hilbert's infinite hotel? and before I put forward why I am posting about this question, I would like to make a few points.
Mathematics is all about possibilities
In a way, mathematics is scientific, because it is a study of what is possible, even in theory. You cannot write anything mathematical about something which cannot possibly occur.
There can only be one answer to the question I highlighted and that is No. It would be impossible, as I answered today
It just stands to reason that when you have an infinite number of rooms, you cannot possibly stay in every room by changing rooms each day.
The number of days you have stayed will always be finite. Even if you were able to stay for eternity, no matter when you assess the number of rooms there will always still be an infinite number of rooms still to stay in.
This question is a hypothetical question and it seems there can be no possible answer which would satisfy the OP.
The reason why I say this is because in the comments, the OP states that
For this thought experiment it is crucial to say that I have been staying at the hotel for an infinite amount of days. That seems weird to some people but I don't really wanna share my thoughts behind this question, it is simply important to assume that I have already been staying at this hotel for an infinite amount of days.
How is this question able to be on-topic?
Your length of stay to date will always be finite even if hypothetically you could stay for eternity. You cannot possibly stay in every room.
It would be pointless to say
Ah, but what if it were possible to have stayed an infinite number of days?
because the theory just isn't possible.
What have I missed here?
OK, to close a question for this reason you may need to first prove the impossibility, but...
Should we not be asking for mathematical foundation for the hypothetical question in the question when asking hypothetical questions?