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To Infinity and beyond

Question: What’s bigger than bazillion? Answer: Infinity!!! But Infinity is one of those words that’s really hard to define – What exactly does it mean to say that something is Infinite? Is Infinity even a number?

Photo by stichnin

What is Infinity?

Imagine that someone tries to get you to define infinity by asking you to name an infinitely big number – How should you answer?

999999999999999999?
9999999999999999999999999999?
9999999999999999999999999999999999999999999999999999?

No, clearly none of these are infinitely big numbers! In fact, as soon as you try and say a specific number, there’s no way you can really be talking about something that is infinite – You can always just add another digit to what you’ve said to make it bigger. The problem reminds me of that game you play as a kid where you dare someone to do something (actually, come to think of it, I still play it now):

‘I dare you to do it’
‘I double dare you’
‘Too bad because I triple dare you’
‘You lose sucker – I dare you a million times!’
‘No you lose! I dare you infinity times’
‘Well I dare you to do 50 press-ups infinity times plus one – Beat that Grandad!!!’

Photo by kamiraze

So, if infinity isn’t a number, what is it? Well, people usually think about it as a concept instead. Infinity is something so big that it has no end. This is a pretty vague definition but, as we’ve seen, trying to pin things down leads to problems!

Achilles and the Tortoise

One of the first Philosophers to look at the concept of infinity in detail was Zeno of Elea. Zeno lived in Ancient Greece from c. 490 BC until c. 430 BC. He told several paradoxical stories about infinity and the story of Achilles and the Tortoise is one of them:

Photo by megwwan

Imagine that Achilles is having a race against his slowest friend, Mr. Tortoise. Because Achilles is so confident that he’ll win the race, he gives the tortoise a head-start of 100 meters. So the race begins and both Achilles and Mr. Tortoise run as fast as they can. After a little time, Achilles has run 100 meters. But during this time, Mr. Tortoise moves forward a little too (we’ll say 10 meters). Now imagine that Achilles runs the next 10 meters to catch up with the tortoise – But by the time he has run this distance, Mr. Tortoise will have moved a little farther on again. And because this process can go on and on, it looks like however far Achilles runs, he will never catch up with the tortoise – It will always take Achilles some time to reach the point where Mr. Tortoise was before, and by the time he reaches that last point, the tortoise will have moved on a little bit! So the story states that while the gap between Achilles and the tortoise might get smaller and smaller, Achilles will never be able to catch up with his green-shelled friend!

Photo by rav_bunneh

Whilst it may not be clear straightaway, the problem Zeno expresses in his story relates to the concept of infinity – Namely, the problem involves Achilles trying to complete an infinite number of tasks in order to catch up with the tortoise. By the time Achilles reaches point 1 (the point at which the tortoise starts the race), the tortoise will have reached point 2. And by the time Achilles reaches point 2, the tortoise will be at point 3. And by the time Achilles reaches point 3, the tortoise will already be at point 4. And so on to infinity. So, in the story, it looks like Achilles must somehow complete an infinite number of steps (reaching points 1, 2, 3 etc.) in order to catch up with his friend.

But Zeno’s story is clearly absurd – We all know that Achilles would easily be able to catch up with the tortoise in real life! So what’s wrong with the story? Post your answers in the comments below, I need your help and infinite wisdom to answer this paradox!

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Photo by antibiotyx

written by tomas_bates

4 comments

  1. carmenism

    carmenism

    Interesting article. There is the obvious fact that the tortoise would have to be moving at a speed greater than or equal to Achille's fastest running speed (which is not possible unless Achilles is quite slow or the turtle is quite fast). And of course no person or animal could take an infinite number of steps except in a riddle; people get hungry or tired or die, so the number of steps one will take is finite.

    But anyway, I don't think the story describes infinity as much as it describes a limit. A limit is an infinite series, but it is quite possible for an infinite series to converge at a finite number. A function may have its limit at infinity, though not necessarily. Here I would definitely say this limit converges at a finite number. I think the paradox is trying to talk about how if you divide a space forever then it can tricky to determine what the sum of those divisions is. For example, (1/2)+(1/4)+(1/8)+(1/16)+... actually sums to 1!

    Say Achilles moves 10 times faster than the tortoise. So when Achilles has moved 10 meters from the starting point, the tortoise has moved only 1 meter, though he'll be 100+1=101 meters from the starting point, so the distance between them is 101-10=91 meters. When Achilles has moved 100 meters from the starting point, the tortoise will have moved only 10 meters, placing the tortoise 100+10 meters from these starting point. Then the distance between them is 110-100=10. As Achilles moves another 10 meters (placing him 110 meters from the starting point), the tortoise will move only 1 meter (placing him 111 meters from the starting point). Then the distance between them is 111-110=1 meter. As Achilles runs 1 meter ahead (now 111 meters from start), then the tortoise moves (1/10) meter (now 111.1 meters from the start), so the distance between is 111.1-111=0.1 meter. So it seems like he will never catch up. However, when Achilles runs 10 meters at that point in time, the tortoise moves only 1 meter, so the difference is 112.1-121=-8.9 meters. In other words, Achilles is ahead!

    So the distance Achilles must travel to catch up is 100 + 10 + 1 + 0.1 + 0.01 + ... + 10^(2-n) + ... where n is the number of seconds, which sums to 111.11m! He needs to travel a finite distance to catch up. So he is set as long as the track is at least 111.11 meters long.

    Here is another way of looking at it... It is reasonable to assume both the paces of Achilles and the tortoise will not increase as they get tired, so it is safe to assume that both are moving at a constant speed. Say Achilles is moves taking one steps per second where each step is 1 meter. Say the tortoise moves only one step per second and moves only (1/10) meter. So in n seconds, Achilles has moved n meters, while the tortoise has moved only (1/10)*n meters. Since the tortoise received a head start, the distance he has covered in n seconds is f(n) = 100 + (1/10)*n. The distance Achilles will cover in n seconds is g(n) = n. Set f(n) = g(n) to see when they converge and you'll find n = (1000/9) seconds. So, after 1000/9 seconds, Achilles will pass the tortoise. He will have moved (1000/9) = 111.11 meters in that time, so if the track is longer than that distance, he will win for certain!

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  2. basho

    basho

    @carmenism: OMG! :O

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  3. tomas_bates

    tomas_bates

    @carmenism - thanks a lot for your thoughts on this, very cool ideas :-)

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  4. stouf

    stouf

    @carmenism: Impressive ! And you should send your CV to the NASA ! : ))) And you're perfectly right... Hum I believe you... : )
    But maybe Zeno was trying to say that even looking infinitely closely at the moment Achilles beats the tortoise, he's still behind. Maybe ?

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This is the original article written in: English. It is also available in: 中文(繁體版), Deutsch & Spanish.