Thanks in advance for any helpful comments! As this is all theory and abstract, please don't leave comments about the practicality of this (unless you feel that it is really relevant). Comments like "infinity is an impossibility anyway" don't really help here. Some people accept and some reject infinity. So for the sake of this post please let's all accept it for a moment.
Here I'll use Hilbert's Grand Hotel paradox to try to illustrate my question: Lets say there was a Hotel with infinitely many rooms. And let's say that every day, I decide to change to the room to the left of me (and there are infinitely many rooms to my left). Every day I get into a new room. Now, let's suppose I have been doing this for an INFINITE amount of time. For infinitely many days have I been changing the room into a new room of the hotel with infinitely many rooms. Does this mean I have been in ALL rooms? If I have been doing this for infinitely many days, would I still have infinitely many rooms to visit that I didn't visit yet? Because I'm thinking, sure, you can never reach the end of infinity. BUT, the infinity of how many rooms there are is the same size of infinity as how many days I have been staying at the hotel, right? And to my logic, in order for me to NOT have visited all rooms, I would have to not have stayed at the hotel for an infinite amount of days.
If I have visited the same amount of rooms as there are rooms, so I have visited infinitely many rooms in an hotel with infinitely many rooms, I would have to have visited all rooms, right? And second question: Is there any issue with "sizes of infinity" in this question? Like, am I maybe missing something that indicates that the sizes of the two infinities (infinitely many rooms and infinitely many days) do actually differ? Because I'm thinking, hotel Rooms are counted like 1, 2, 3, 4... and so are days.
Please note that wrote that I already HAVE BEEN staying at the hotel for an infinite amount of days. Not that I am staying at the hotel for an infinite amount of days. Not sure if it makes a difference, but I still wanna point it out.
Again, thanks in advance for any useful answers!