Once upon a time in a mathematical world far far away, there lived a wise mathematician named David Hilbert. He was known for his love of numbers and his ability to think outside the box. Each day, he would explore the mysteries of mathematical concepts and find creative ways to make them understandable for everyone.
One day, as he was deep in thought about the idea of infinity, he had a brilliant realization. He realized that infinity was not just another number, but a never-ending story waiting to be discovered.
And so, he created a story, a tale so magical and so profound that it would captivate mathematicians and laypeople alike for centuries. He called this tale, “Hilbert’s Hotel”, a story about a hotel with ∞ rooms!
But the story of infinity seldom starts here.
Father: You’re causing a bit too much fun. Can you come here for a minute?
Child: (stomps over) What do you want, Dad?
Father: I have a story for you. It’s about a runner named Achilles.
Child: Who’s Achilles?
Father: Achilles was a great runner who lived a long, long time ago. He wanted to race a tortoise, but the tortoise had a head start. Do you think Achilles could catch up to the tortoise?
Child: Yes he’s super fast, right?
Father: That’s what most people thought, but there’s a twist to the story. The philosopher Zeno came up with a paradox that says that Achilles could never catch up to the tortoise no matter how fast he ran.
Child: Why not?
Father: Well, Zeno said that every time Achilles reaches the place where the tortoise was, in that period of time, the tortoise has already moved ahead a little bit.
Father: And every time Achilles reaches that new place, the tortoise has moved ahead a little bit more. So, there’s always a smaller distance to cover but the race would never end.
Every mathematics enthusiast secretly knows where infinity resides. Depending on the type of logic we use, it is possible to conclude two or more possibilities about the existence of infinity’s precise address.
Imagine checking into an infinite hotel with an infinite number of rooms, all occupied. You might think finding a room for you is a hopeless task. But not for the clever receptionist! They have a trick up their sleeve.
With a flick of their wrist, the receptionist tells the guest in room 1 to move to room 2, the one in room 2 to move to room 3, and so on. Ad infintum!
In a flash of math-a-magic, all the guests have shuffled down the hall and made room for you in room 1.
What’s the catch? The hotel has ∞ rooms, and every room is always occupied. Yet, whenever a new guest arrives, the hotel is able to accommodate them, even though it was already full.
This mind-boggling scenario shows that infinity is not your typical number and behaves in strange and unexpected ways. Most numbers change when added to one another. But Infinity added to One is also Infinity.
The thought of Infinity has always made me anxious.
How was this magic possible, you might ask?
Well, David Hilbert had devised a clever solution. Each guest in the infinity hotel, represented by the symbol n, would simply move to the next room, represented by the equation n + 1, making room for the new arrival in the first room. And so, the hotel was always full, represented by the equation ∞, but always had room for at least one more.
This equation shows us that even though infinity is unbounded, it can still be increased by one. This raises more questions about the nature of infinity and the limits of mathematical concepts than it answers.
Infinity is probably the only other symbol that reaches itself when added to anything or even itself. Probably a close cousin of Zero given the double resemblance.
However, it is important to note that infinity is not a natural or real or complex number, so adding or subtracting numbers from infinity does not make mathematical sense unless we have defined the number space we are working in.
For example, we may be working in the extended real line or the extended complex plane, where we are then allowed to add and subtract numbers from infinity.
Consider this equation ∞ + 1 = ∞.
Are both infinities in this equation the same kind of infinity? Or are they of different kinds? With surface symbolic resemblance that could be deeply deceptive?
In mathematics, one is allowed to choose. The easier choice is to assume that all infinities are of the same type.
Some choices in mathematics lead to hidden truths. Other choices are not encouraged as they lead to contradictions or dead ends.
Child: That’s confusing.
Father: It can be, but it’s also a fascinating idea that has been studied and discussed for thousands of years. It makes you think about the nature of reality and what is truly possible. These days, we call this race Zeno’s paradox.
Child: Does this mean that Achilles could never catch the Tortoise?
Father: Well, that’s up for interpretation. Some people believe that Achilles could catch the tortoise, while others believe that he could never reach it. There are a few different solutions, some people argue that Zeno’s paradox is based on a misunderstanding of infinity. Others say that it’s a problem with our concept of time.
Child: What is Infinity?
Father: Infinity is a concept that represents an unbounded quantity. It’s greater than any finite number. The idea of infinity is important because it helps us understand things like Zeno’s paradox.
In mathematical terms, infinity is represented as an unbounded quantity that goes on forever, represented by the symbol ∞. This concept of infinity is a key component of many mathematical theories, including those related to limits, sequences, and series.
For example, consider the sequence of natural numbers, represented by the symbols 1, 2, 3, 4, …, ∞. This sequence goes on forever, with no end in sight. If we started counting today, one can only hope that it ends before time itself.
Child: Don’t we see Achilles win the race in real life?
Father: There are many ways to resolve the paradox at the heart of this, but one of the most popular is to say that motion is continuous, not composed of an infinite number of finite steps. This means that Achilles could in fact catch the tortoise if he moves continuously, without stopping.
Child: That makes sense. But why did Zeno then think that motion was made up of infinite steps?
Father: Zeno believed that if motion was continuous, then it would be impossible to define a starting and ending point, and therefore it would be impossible to measure. He used the paradox of Achilles and the Tortoise as a way to demonstrate this point.
Child: That’s interesting. But it is possible to define a starting and ending point.
Father: Yes, that’s one way to look at it. Another way to resolve the paradox is to say that the infinite number of steps that the paradox assumes are not actually infinite, but rather very small. This means that they can be treated as a continuous motion, just as we see in real life.
Some of the challenges with understanding infinite sets come from their contra-intuitive nature. For example, the statements “there is a guest in every room” and “no more guests can be accommodated” are not equivalent when there are infinitely many rooms. When k many people arrive at the infinite hotel and need k rooms, we simply ask them to move to rooms k + 1, k + 2, etc. This frees up the first k rooms for the new guests.
What if an infinite number of busses each containing an infinite number of persons arrive at the fully booked hotel and they all want a room? If an infinite number of guests arrive, we can’t just move the current guests because that would take infinitely more movements. Instead, we move the guest in room 1 to room 2, the guest in room 2 to room 4, and so on. This makes all the odd-numbered rooms available, and we can place the infinitely many new guests in those rooms.
This way, all the odd-numbered rooms become availableand empty, and we can place the new guests in them. This means that even though the set of even numbers is a subset of the set of natural numbers, they still have the exact same number of elements or cardinality.
Child: So, then what was Zeno trying to prove with this paradox?
Father: Zeno was trying to prove that motion is an illusion and that reality is unchanging. He believed that if motion was real, then it would be possible to define a starting and ending point, and therefore it would be possible to measure. But he also believed that if reality was unchanging, then motion would be impossible.
Child: So, he used the paradox of Achilles and the tortoise as a way to demonstrate this point.
Father: He believed that in order to move from one place to another, you first have to cover half the distance, and then half of what’s left, and so on. And since there’s always a smaller distance to cover, you can never actually reach your destination.
Child: That’s a pretty bold statement.
Father: Yes, it is. Zeno was a very influential figure in the development of ancient Greek philosophy, and his ideas have been studied and discussed for centuries. Even today, people are still trying to understand and resolve the paradox that he introduced.
Child: So, what else did Zeno believe?
Father: Zeno was a philosopher who was associated with the school of thought known as Eleaticism. He believed that reality is unchanging and that change is an illusion. He also believed in the idea of the unity of opposites, where opposite qualities are really just two aspects of the same thing.
If we compare two piles of stones and want to determine which pile has the greatest number or “count” of stones, we can pair them up by repeatedly taking one stone from each pile until there is only one or zero piles left.
At that point, the process terminates.
If there is a single pile left, that was the one with the greatest number of stones originally, and if they match up completely, yielding no piles when the process terminates, then that means that all their stones could be paired up two by two, and they would have had the same number of stones in them to start with.
When we generalize this concept to infinite sets, we get the same scenario as with the stones, but the pairing is done with bijective functions instead of small stones.
Child: Wow, I never knew that an old story about a race could be thought-provoking.
Father: That’s the beauty of stories, they can teach us so much about life and the world around us.
Child: I still believe that Achilles would surely win the race.
Father: I understand why you might think that. A lot of people were skeptical of Zeno’s ideas when he first introduced them. But over time, his paradoxes have become some of the most studied and debated problems in philosophy.
Child: Really? Why are they so important?
Hilbert introduced the Hilbert Hotel paradox in a 1924 lecture. The tale of Hilbert’s Hotel was a profoundly creative way of exploring the concept of infinity in that era. It challenged the minds of mathematicians and laymen alike, and showed them that infinity was not just a number, but a never-ending story waiting to be told.
David Hilbert came up with this analogy to explain the contra-intuitive nature of infinite sets and transfinite arithmetic. Hilbert wanted to show how to think of infinity and how to work with it in a structured way. This was a mathematical taboo before him.
Years went by, and the tale of Hilbert’s Hotel became famous throughout the mathematical world. It was used as a teaching tool to help students understand the abstract concept of infinity, and inspired mathematicians and enthusiasts alike to explore new ways of thinking about this fascinating and time less concept.
Father: Well, Zeno’s paradoxes are interesting because they challenge our understanding of time and space. They show us that our minds can sometimes play tricks on us, and make us believe things that aren’t necessarily true.
Child: Hmm, that’s kind of like when I thought I could fly off the roof and into the pool, but then I got scared and realized it wasn’t a good idea.
Father: Exactly! And just like that, Zeno’s paradoxes encourage us to think critically and question our assumptions. They force us to examine our beliefs and consider alternative perspectives.
Child: That’s pretty cool. So, how do people solve these paradoxes?
Father: Sometimes the best way to solve a paradox is to keep an open mind and keep thinking about it. This is just one way to think about it.
There are surely infinite other ways.
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