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Origins -

David Hilbert conceived the Infinite Hotel as a thought experiment to challenge our understanding of infinity. By creating the idea of a paradoxical hotel with an infinite number of rooms, he demonstrated how infinity behaves in ways that defy logic.

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Endless rooms -

In order to understand the thought experiment, Hilbert asks every person to imagine a hotel with an infinite number of floors and an endless supply of rooms. Despite this, the hotel has a diligent night manager who must constantly reorganize guests to make space for new arrivals.

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A fully booked hotel -

One night, Hilbert’s Infinite Hotel is entirely full with an infinite number of guests. But when a new traveler arrives at the fully booked hotel, the night manager does not turn him away, even though every room is taken.

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Welcoming -

In order to provide the new guest with a room in the fully booked hotel, the night manager moves each existing guest from their current room to the next. The guest in room 1 moves to room 2, while the guest in room 2 moves to room 3, and so on. As such, each guest is moved from their current room of "n" to room "n+1," which ultimately frees up room 1 for the newcomer.

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Making space -

One night, a tour bus arrives with 40 guests, and the night manager applies the same strategy on a larger scale. Every current guest moves from room "n" to room "n+40," vacating the first 40 rooms. Mathematically, this shows that infinity can always absorb any finite number.

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An infinitely large bus -

The night manager faces a greater problem when an infinitely large bus arrives carrying an infinite number of passengers. Simply shifting guests by a finite amount is no longer effective. He must develop a more sophisticated method to ensure every new guest receives a room.

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Even-numbered rooms -

To solve the problem, the night manager moves every existing guest from room "n" to room "2n." Every guest is ultimately placed in an even-numbered room, and all the odd-numbered rooms are left vacant for the infinite number of passengers on the bus.

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Booming business -

Despite the constant arrival of new guests, the Infinite Hotel’s revenue remains paradoxically the same. Since the number of guests is always infinite, the hotel’s nightly earnings are always infinite as well.

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An infinite arrival of infinite buses -

One night, the night manager faces an unprecedented challenge when an infinite number of infinitely large buses arrive, each carrying an infinite number of passengers. This situation presents a new level of complexity that requires an even more advanced mathematical approach.

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Prime numbers -

To accommodate this overwhelming influx of guests, the night manager turns to prime numbers for the answer. Essentially, prime numbers can only be divided by the number 1 and by themselves. As such, they would include digits like 5, 13, 47, and 89.

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Euclidean math -

After considering prime numbers, the night manager recalls a mathematical fact from ancient Greece in which the mathematician Euclid proved that prime numbers are infinite, and so he decides to use them as a method for assigning rooms in a way that prevents overlap.

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The first prime number -

The existing guests in the Infinite Hotel are reassigned using the first prime number, which is 2. Each guest in room "n" moves to room "2^n" (2 raised to the power of their room number). This move exponentially spreads out the existing guests and creates vast amounts of available space.

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Placing the first bus’ passengers -

The new guests who have offloaded from the first infinite bus are assigned rooms based on the second prime number, 3. A passenger in seat number "n" moves to room "3^n" (3 raised to the power of their seat number). This ensures they do not overlap with the guests already in rooms.

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Repeated pattern -

Each subsequent bus follows the same rule using the next prime number on the list. The second bus uses prime 5, the third bus uses prime 7, the fourth uses prime 11, and so on. This ensures that all guests from every bus receive unique rooms without ever encountering conflicts, infinitely.

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No room overlaps -

The strategy works because prime numbers have unique properties. Since each room assignment is based on the exponents of these prime numbers, no two guests will ever end up in the same room, no matter how many new arrivals appear. This can go on infinitely.

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Empty rooms -

Surprisingly, not every room gets filled under this system. Some numbers, like room 6, remain empty because they are not prime powers. This highlights another strange property of infinity, which is that expanding numbers infinitely still leaves gaps.

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Countable infinity -

The Infinite Hotel operates within the realm of countable infinity, meaning that each room and each guest can be numbered in a sequence (1, 2, 3, etc.). This type of infinity, known as aleph-zero (ℵ₀), is the smallest level of infinity and is manageable using structured techniques.

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The classification of infinities -

Georg Cantor, a German mathematician who revolutionized our understanding of infinity, introduced the idea that not all infinities are equal. He showed that countable infinity (like the hotel’s rooms) is different from uncountable infinity, such as the infinity of real numbers.

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Higher orders -

The structured methods used in Hilbert’s Hotel only work because the hotel’s infinity is countable. If the hotel dealt with real numbers (which can include negatives and even decimals), the same strategies would fail, as real numbers cannot be listed in a sequential way like whole numbers.

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Real numbers -

If the hotel accepted real numbers as room assignments, things would become far more chaotic. There would be rooms assigned to fractions, irrational numbers like pi, and even negative numbers, which creates an impossibly complex booking system.

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What happens if guests check out? -

If an infinite number of guests suddenly check out, the hotel remains infinite but now with an infinite number of vacant rooms. The night manager could reassign the remaining guests by shifting them back, yet (as strange as it may sound) the number of rooms would still be endless.

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Why it matters -

Beyond being a fun paradox, the Infinite Hotel plays a crucial role in understanding the mathematical properties of infinity. It helps students, philosophers, and mathematicians grasp fundamental ideas about infinite sets and how they can work in peculiar ways.

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Real-world applications -

Though it’s a thought experiment, the principles behind Hilbert’s Hotel have real-world applications in areas like computer science, physics, and cosmology. Concepts such as infinite memory allocation and the nature of an expanding universe relate directly to how infinity is handled in Hilbert’s Hotel.

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Philosophical implications -

Hilbert's Hotel also influences fields like cosmology. It challenges our understanding of the universe's structure and the concept of actual infinities, prompting discussions about the nature of reality and the infinite.

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The Big Bang -

Scientists have asserted and proven that the universe was created via a Big Bang approximately 13.8 billion years ago. Using Hilbert’s Hotel, they also consider the possibility that (much like infinite guests coming and going from the hotel) the universe perpetually crashes in on itself and then expands, all in a rinse-and-repeat cycle lasting trillions of years.

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A finite universe -

Some philosophers and theologians argue that the paradoxes arising from Hilbert's Hotel suggest that an actual infinite cannot exist in reality, thereby implying that the universe must have a finite beginning. This line of reasoning has also been employed in theological debates about the existence of a creator.

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Quantum mechanics -

Researchers have explored parallels between Hilbert's Hotel and quantum mechanics, particularly in trying to establish infinite dimensions in the universe. This analogy helps in understanding complex quantum systems and the mathematical structures that describe them.

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Resource allocation -

The paradox can also possibly serve as a metaphor for resource management, illustrating how infinite resources can always be reallocated to meet endless demands. Although entirely theoretical, it does prompt discussions about efficiency and optimization.

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Struggling with infinity -

The Infinite Hotel serves as a reminder that every person’s intuition is shaped by finite experiences, which makes true infinity difficult to comprehend. While we can manipulate infinity mathematically, it is an intellectual challenge to grasp its true nature, which is why Hilbert’s Hotel continues to captivate thinkers even a century after its conception.

Sources: (TED-Ed) (Britannica) (ScienceABC)

See also: The most significant schools of philosophy

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