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Start for freeWelcome to the Infinite World of the Hilbert Hotel
Imagine being the manager of a hotel with an endless number of rooms, each numbered sequentially from one to infinity. This isn't just any hotel; it's the famed Hilbert Hotel, a thought experiment that stretches our understanding of infinity and mathematics.
The Basics of Infinite Accommodation
At first glance, managing such a hotel seems straightforward. With infinite rooms at your disposal, turning away guests should never be an issue. Initially, when all rooms are occupied by an infinite number of guests and a new guest arrives, you might think you're out of options. But here's where knowledge of infinity comes in handy. By asking each guest to move down one room - guest in room one moves to room two, and so on - room one becomes available for the newcomer.
This strategy also applies when a bus carrying a hundred people arrives. Simply shift everyone down a hundred rooms to accommodate them all. But what happens when an infinitely long bus with infinitely many passengers pulls up?
Dealing with Larger Infinities
When faced with this challenge, you instruct every guest to move to a room number that is double their current room number (room one to room two, two to four, etc.). This maneuver frees up all odd-numbered rooms for new guests from this infinite bus.
The Challenge of Unbounded Infinities
The real complexity begins when not just one but an infinite number of buses carrying infinite passengers each arrive. To manage this scenario, envision an infinite spreadsheet listing all current hotel guests and incoming buses. Assigning rooms now involves creating a unique identifier for each person based on their vehicle and seat position.
By zigzagging across this spreadsheet and straightening it out like pulling on a string, you transform this complex array back to a single line - aligning each individual with their own unique room in your infinitely expansive hotel.
A Limit Within Infinity?
However, things take an unexpected turn with the arrival of another type of infinite bus - an 'infinite party bus' where every passenger has a uniquely infinite name made up only from two letters 'A' and 'B'. These names are not just long; they are endlessly repeating sequences like 'ABBA', 'AB AB AB...', making them uncountably infinite.
When attempting to assign rooms using your usual method by matching names with room numbers directly - you encounter something peculiar. By manipulating sequences diagonally across your list (flipping 'A' to 'B' and vice versa), you can always create a new name that doesn't match any existing sequence on your list.
This revelation shows that while both sets (rooms in Hilbert Hotel and names on the party bus) are infinite, they are not compatible in size or type; some infinities indeed surpass others! The countable infinity (rooms) cannot cover the uncountable infinity (names), revealing limits within what seemed limitlessly accommodating.
Implications Beyond Mathematics
The discovery that infinities come in different sizes isn't just mathematical navel-gazing; it has profound implications across sciences and even influenced technological advancements leading up to modern computing devices.
The paradoxes and solutions within the Hilbert Hotel not only challenge our comprehension but also inspire awe at the complexities hidden within concepts as familiar as 'infinity'. It serves as both brain teaser and gateway to higher mathematical thinking.
Article created from: https://youtu.be/OxGsU8oIWjY?si=QAdBykCVAvKxM3p7
