O Paradoxo do Hotel Infinito

Introduction

This tutorial explores the concept of the Infinite Hotel Paradox, introduced by mathematician David Hilbert. The paradox challenges our understanding of infinity, illustrating how some infinities can be larger than others. By the end of this guide, you will grasp the fundamental ideas behind this intriguing mathematical concept.

Step 1: Understanding Infinity

• Infinity is not a number but a concept representing an unbounded quantity.
• There are different types of infinity, and some infinities can be considered larger than others.
• The Infinite Hotel Paradox serves as a thought experiment to illustrate these differences.

Step 2: The Setup of Hilbert's Hotel

• Imagine a hotel with an infinite number of rooms, all occupied.
• Each room is numbered: Room 1, Room 2, Room 3, and so forth.
• Even when fully booked, the hotel can still accommodate new guests.

Step 3: Accommodating New Guests

• To welcome a new guest, move the current guest in Room 1 to Room 2, Room 2 to Room 3, and so on.
• This action frees up Room 1 for the new guest, demonstrating that an infinite number of guests can always be accommodated.
• This highlights the counterintuitive nature of infinity: you can add more elements without reaching a limit.

Step 4: Accommodating Infinite New Guests

• Now imagine a bus arrives with an infinite number of new guests.
• Move the guest in Room 1 to Room 2, Room 2 to Room 4, Room 3 to Room 6, and so on, shifting each guest to the room number that is double their current room number.
• This leaves all odd-numbered rooms (1, 3, 5, etc.) available for the infinite new guests.

Step 5: The Concept of Different Sizes of Infinity

• The paradox illustrates that while we might think of all infinities as equal, there are indeed different "sizes" of infinity.
• For example, the set of natural numbers (1, 2, 3, ...) has a different cardinality than the set of real numbers (which includes decimals and fractions).

Conclusion

The Infinite Hotel Paradox elegantly demonstrates that infinity is not uniform. By exploring how an infinite hotel can accommodate more guests, we learn about the fascinating nature of different infinities. This thought experiment encourages deeper thinking about mathematical concepts and their implications in real-world scenarios. Consider further exploring topics in set theory and the philosophy of mathematics to expand your understanding.