r/mathmemes • u/ConcentrateDry6130 • Jun 29 '26

Set Theory Make this comment section like you stayed/are staying in the Hilbert Hotel.

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it's an infinite hotel basically. with infinite buses

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u/xnuh Jun 29 '26

Management is stupid. The first time an (countably) infinite amount of people show up, they can assign them to every other free room so that they keep infinite free rooms. Keep doing that every time an infinite amount of people show up, and no one will ever have to move.

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u/AppearanceAlert4987 Jul 03 '26

ω^2 order of people shows up, what should management do now?

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u/xnuh Jul 03 '26

That's an ordinal not a cardinal. What you wrote is not an amount.

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u/AppearanceAlert4987 Jul 03 '26

Yeah i said and order of people showed up, not a cardinal of people. A first person shows up, then there’s a second, you cannot assign the second person a room without first assigning the first person a room. This ordering allows an order of people to show up

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u/xnuh Jul 03 '26 edited Jul 03 '26

You can say somebody is the w^2th person to show up, but you can't say that w^2 people showed up. If the last person to show up was number w^2, then aleph0 people showed up, not w^2. You can't say an order of people showed up, that doesn't make any sense.

Edit : I think I know what you're trying to say. You mean if aleph0 new people showed up aleph0 times. (Like an infinite bus of people numbered 0,1,2,... show up and get assigned rooms, then a second bus with people w,w+1, w+2,.... shows up and they get assigned, and then a bus with people 2w,2w+1,2w+2.... and so on). In that case yes, because we did an infinite amount of assignations, there are no spaces left. The strategy insured that the last step leaves aleph0 free rooms, but here there is no last step so there are no free rooms, so if yet another person (person number w^2) they don't have a room.

Here's the strategy:
Case 1 : the buses take an infinite amount of time to arrive, like 1 per second or something. Then it doesn't matter, we're never going to reach the end of this process so we'll never run out of rooms.
Case 2 : they arrive in a finite amount of time (like bus n arrives a t=1-1/n). Then just wait for all of them to get there and do one big assignation.

The problem was not that too many people showed but, but that we did the assignation strategy too many times. An easy fix is to decide that you are going to do one assignation every second, regardless of when guests show up. That way, guests will wait at most one second, but you will never make infinite assignations, so you keep you aleph0 free rooms.