# The naughty infinity

Infinity is not fully understood and the steps towards grasping it end in problems and paradoxes…

**Zeno’s paradox**

The runner can never reach the goal — claims Zeno of Elea. How does he justify such a statement?

According to him, the runner has to reach first half of the distance in order to arrive at his goal place. Next half of the half so 1/4. After that 1/8, 1/16 and so on..*ad infinitum*. It would therefore take him an infinite amount of time to reach his destination.

For us, this problem is solved today as we know that we may place an infinite number of elements in a finite section.

# Grand Hotel paradox

The Paradox of the Grand Hotel, often known as the ‘Infinite Hotel Paradox’ ’ was first described by German mathematician David Hilbert (1862–1943). It’s a thought experiment about the nature of infinite numbers that yields some unexpected outcomes.

This hotel has countless rooms and all of them are already occupied. Imagine you land at such hotel, asking for a free room and surprisingly, you hear that this won’t be a problem — all that has to be done is to ask the guests to move to the next room: guests from room 1 should move to room 2, guests from room 2 will go to room 3, etc. Every guest from room n moves to room n+1.

This way the first room will become empty.

This process can be repeated for any finite number of rooms. Let’s think about a family of 5 people arriving at our magical hotel.

Even though this hotel was fully occupied, it was still possible to find a room.

In this case every guest would move from n to n+5.

But what if there were an endless number of people looking for rooms? This is also not a problem. This time, each guest would be asked to relocate to a room that is twice their room number, so room 1 would go to room 2, room 2 to room 4, and so on, with each guest relocating from n to 2n. The odd-numbered rooms would be freed up as a result. Because there are an endless amount of odd numbers, we can accommodate an infinite number of new guests.

This is not the end of the possbilities. If one day the hotel is visited by countless cars, and in each countless guests… it will still be possible to place them according to the assignment 2^n * 2^c (c is the car number).

The person from room 2 will move to 2² * 3⁰ = 4, the person in seat 4 in the car 3 will move to 2⁴ * 3³ = 16 * 9 = 144.

# Thompson’s lamp

Imagine it is 2 o’clock am and you have a lamp that is currently turned on. You are being asked to play with the lamp up until 3 o’clock pm. At 2:30 you are turning it off and at 2:45 you are turning it on. At 2:25 and 30 sec you are turning it off and so on.. every time taking an action at the half of estimating time. What will be the status of the lamp at 3? Will the lamp be on or off?