Does Veritasium's/Hilbert's Infinite Hotel Really Run out of Room?

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[coweatsman]

I'm in bed in a room and for the infinite time that night I am being moved to another room. Guess there may be an infinite number of rooms but there is no sleep for any of the infinite number of guests.

[Sh-gsku]

can be assign A 1 and B 2 then we can fix AB infinity for example AAABBBABABAB =111222121212

[SouthernersSax]

The best way I can think to address your concern is as follows:

The manager has to start somewhere since there were no assigned seats on the bus and therefore no established order to them, so the first room may as well go to the first guy in line.

[theanonymousseeker]

I wonder if there are underground rooms like the negative numbers. Also, an infinite number of infinite buses could work for an infinite number of Hilbert hotels. These would be examples of Omega 1 since only one of these infinite objects would be infinitely smaller.

[platosbeard]

Do you ever have to put belt and braces on R being uncountable? From friends I tend to get something along the lines of "if we can add in an infinite set with no issue, can't we just keep doing that infinitely?" We can use a few rounds of induction to prove it, but it's not very satisfying and peeps don't believe it 😅😆

[josephparsons]

@Easy Theory

I am confused by your assertion that the diagonalization argument as presented by Veritasium is incomplete. This seems to be a standard proof by contradiction. If a premise derives a contradiction, then the premise must be false.

Let f be an arbitrary function that takes in the room number and outputs the name of a monster (assume the original hotel is empty, so it only has monsters with the characters A and B). Assume that there exists some function f that accommodates every single monster; in other words, every monster on the big bus has a room number (*). Then find a new monster name such that the n-th character of the monster's name is the opposite of the character of the monster in room n, for all positive integers n. Now the question remains: what room is this monster (call them Monster X) in? Clearly, the monster must be in a room, since we assumed that that was true at the * point. But no matter what room k you claim the monster is in, the monster cannot be in the room k, since we have already shown that Monster X's name is different from the name of the monster occupying room k at the k-th character, and two monsters with different names cannot have the same room. Hence, we arrive at a contradiction: f must have as its output every single monster, so Monster X must be in a particular room, but at the same time, no matter what natural number n you select, Monster X is not in Room n. This apparent contradiction shows us that an f that can accommodate every single monster could not exist, since the idea of it is inherently self-contradictory.

This assumes standard axioms of logic, such as the assumption that every statement must be either true or false, but not both, etc. In some logic systems, this could break down.

[apoxalypsewhen]

im having my math finals next month... dude its not possible to be interested in math with my teacher ... Her classes are boring ! You can not tell yourself yeah im gonna study math because she will makes you hate math ! Math is so interesting, how can you make it boring ??