Hilbert's Hotel Paradox Calculator

Understanding Hilbert's Grand Hotel Paradox

Hilbert's Grand Hotel paradox is a famous thought experiment devised by German mathematician David Hilbert in 1924 to illustrate the counterintuitive properties of infinite sets. Unlike a finite hotel that has a maximum occupancy limit, a hypothetical hotel with a countably infinite number of rooms ($\aleph_0$, or aleph-null) can accommodate additional guests even when every single room is already occupied.

How the Infinite Hotel Accommodates New Guests

The key mathematical principle behind Hilbert's Hotel is that countably infinite sets can be put into a one-to-one correspondence (bijection) with proper subsets of themselves. Depending on how many new guests arrive, the hotel manager applies different relocation algorithms:

1. Accommodating a Single New Guest ($k = 1$)

When one new guest arrives at a fully booked infinite hotel, the manager asks the guest currently in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, and generally the guest in Room $n$ to move to Room $n + 1$:

$$f(n) = n + 1$$

This shifts all infinitely many current guests simultaneously, vacating Room 1 for the new arrival. In set theory terms, adding one element to a countably infinite set does not increase its cardinality: $\aleph_0 + 1 = \aleph_0$.

2. Accommodating $k$ Finite Guests

If a finite group of $k$ travelers arrives, each current resident in Room $n$ shifts to Room $n + k$:

$$f(n) = n + k$$

This simultaneously frees up the first $k$ rooms ($1, 2, \dots, k$) for all $k$ new arrivals.

3. Accommodating an Infinite Bus of Guests ($\aleph_0$)

Suppose an infinitely long bus carrying countably infinite passengers ($\aleph_0$) pulls up. The manager instructs every current resident in Room $n$ to move to Room $2n$:

$$f(n) = 2n$$

This moves current occupants into all the even-numbered rooms ($2, 4, 6, 8, \dots$). This leaves all infinitely many odd-numbered rooms ($1, 3, 5, 7, \dots$) vacant. The passenger at seat $m$ on the incoming bus is assigned to Room $2m - 1$:

$$g(m) = 2m - 1$$

Because there are infinitely many odd numbers, every bus passenger receives a private room: $\aleph_0 + \aleph_0 = \aleph_0$.

4. Accommodating Infinitely Many Infinite Buses ($\aleph_0 \times \aleph_0$)

When an infinite convoy of buses arrives, each carrying infinitely many passengers, the hotel manager has multiple mathematical strategies:

• Prime Factorization Method: Assign existing hotel guests in Room $n$ to Room $2^n$. Passengers on Bus $k$ at seat $m$ are placed into Room $p_{k+1}^m$, where $p_i$ is the $i$-th prime number (Bus 1 goes to $3^m$, Bus 2 goes to $5^m$, Bus 3 goes to $7^m$, etc.). By the Fundamental Theorem of Arithmetic, no two guests are ever assigned to the same room.
• Cantor's Diagonal Pairing Method: By treating the original hotel as Bus 0, every guest can be represented as a pair $(b, g)$ where $b$ is the bus index and $g$ is the seat number. Cantor's pairing function maps $(b, g)$ to a unique positive integer without leaving any unoccupied rooms.

Summary of Relocation Formulas

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