How an Infinite Hotel ran out of Rooms: the Hilbert Hotel

Welcome to a fascinating journey through the paradoxical world of Hilbert's Hotel, where infinity takes on a whole new meaning! Inspired by the ingenious thought experiment introduced by mathematician David Hilbert, this video explores how an infinite hotel can accommodate an endless number of guests—until it finally meets its match.

We'll start with a finite bus arriving at the hotel and see how the clever manager finds room for everyone. Then, we ramp up the challenge with an endless bus, and even more mind-bogglingly, a countably infinite number of infinite buses. Thanks to insightful explanations from Veritasium we'll discover how the manager uses mathematical tricks to fit everyone in.

But the real twist comes when an uncountable infinite bus with uncountable infinite seats arrives. Here, we delve into the difference between countable and uncountable infinities and why this scenario finally stumps the hotel's manager.

Join us as we explore the depths of infinity, inspired by the Veritasium video on Hilbert's Hotel and the thought-provoking movie 'A Trip to Infinity.' This video is a must-watch for anyone curious about the boundaries of mathematics, philosophy, and the infinite universe.

Don't forget to like, share, and subscribe for more intriguing content. Let's dive into the infinite possibilities of Hilbert's Hotel!
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[BlueZulfishBoiii]

bro earraped me with that intro 💀
"infinitely long" is wild my boy

[jamestagge]

The problem exists in trying to manage a proposition in which infinity is being paired with finite concepts because they cannot be joined. There is no such thing as infinity within materiality. The concept cannot be realized as complete, i.e., a complete infinity within the context of an instant in time of its consideration. It can only be considered a progression to continue on forever, but quantifiable at any point along that infinite line/progression of time.

All the business about bijection is irrelevant when we are considering infinity in its application. Hilbert proposed an hotel with infinite rooms and infinite guests BUT with all the rooms full. This means that there can be no shifting of guests to make room from others for as the unending nature of the rooms is considered that new guests might be accommodated, the same consideration must be exactly made for the infinite guests for “all the rooms are full”. This qualification couples each room in the infinite progression to a guest which precludes the proposed shifting.

As for all the bugle oil about bijection, consider, if these two infinites are supposed to be the same size (conceptually a ridiculous notion) as per the proposition (if not, all the rooms could not be full) then by definition if the infinity of rooms is employed to make room for new guests then the infinity of existing guests would have to correspond to the manipulation of the infinite rooms that it might remain that “all the rooms are full”. One cannot have it both ways, including Hilbert.
His paradox is sophomoric and a contradiction of his own making. Its piffle.


Any thoughts?