I am learning the fundamentals of mathematics.
A bit background: This article says that "The mathematical paradox about infinite sets" envisages Hilbert's Grand Hotel:
"...a hotel with a countable infinity of rooms, that is, rooms that can be placed in a one-to-one correspondence with the natural numbers. All rooms in the hotel are occupied. Now suppose that a new guest arrives – will it be possible to find a free room for him or her? Surprisingly, the answer is yes."
My issue is around this two statements:
- "All rooms in the hotel are occupied"
- "a new guest arrives"
From an excellent answer here, I gather that 1. is taken to mean that the hotel is hosting an infinite set of guests and that 2. means things have changed, we now have to reassign every room again to accommodate a new infinite set of guests (eg: the ones before + 1).
I saw other threads and answers. But the "new" set is just the same old set. Why not simply tell them all to momentarily leave and then to come back in to their new room? that is, remap the new set every time as the first set, and have no need to come up with fancy algorithms to reassign every new addition to the set guests to host tonight.
Finally, How is it useful to say things like:
- The "hotel is completely full"?
- Each and every room is occupied (or all rooms are)?
*[I have changed this question many times. Trying to keep it in one single question.]
To me, it sounds like you are considering two possible statments about a hotel
and feeling that the two statements can never be simultaneously true. Indeed, for real-world finite hotels, the two statements are opposites of each other. But for an infinite hotel, it is possible for both to be true at the same time (indeed the second statement is always true for an infinite hotel).
Put another way: for finite hotels, we use "full" to mean "there's a guest in every room", and we also use "full" to mean "they can't fit in another guest". But these two senses of the word "full" are not equivalent for infinite hotels. Part of the value of the Hilbert hotel thought experiment is to help us notice when our intuitions about finite sets are incorrect intuitions for infinite sets.
This clash with intuition is a perfectly reasonable cognitive dissonance to have while learning about infinite sets! It seems like that discomfort is making you lean in the direction of not accepting infinite sets as well-defined mathematical objects (and you wouldn't be the first to feel that way). But infinite sets are crucial to mathematics—without them we couldn't even talk about intervals of real numbers like $[0,1]$. I would encourage you instead to lean into the discomfort and use it to help refine your intuition in ways that will pay off for future mathematical studies.