Are you saying we can experience “true infinity”?
The Paradox of the Grand Hotel
Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.
[edit]Finitely many new guests
Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.
This is silly, because the underlined “and so on” can never be completed.
To start from the beginning, praxeologically an “infinite” hotel is simply a hotel with so many rooms that I’m sure I’ll never see the end of them, and I’m sure that everything I ever do with the expectation that I will “run out of rooms” or “do something to every last room” will result in disappointment. (If “infinite hotel” meant anything else, I would have to ask how I can possibly find any meaning in it.)
But for the very same reasons, I can be equally sure that the act of moving each guest will never be completed. So the final sentence in the Wikipedia article pays no rent (no pun intended), since I can most definitely not expect to make room for any new guests. I could do it by some sort of broadcast to all guests at once, so that they all move in unison, but by hypothesis (in bold above) I cannot ever be sure my message reached all the rooms, as I would have to experience the impossible to do so: I would have to experience a flow of information that I could never expect to end.
When “infinity” and “infinite set” are rationally interpreted in a way that actually means something in terms of anticipated future experience, there are no eery paradoxes, just normality.