Achilles races to catch a slower runner—for example, a tortoise that is crawling in a line away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least as far as the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run at least to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones. So much the worse for the claim that any kind of motion really occurs, Zeno says in defense of his mentor Parmenides who had argued that motion is an illusion.3

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There's more than seven numbers, but you get the idea. . .

1. 0.00003430454657612443......

2. 0.10000000000000000000......

3. 0.32683268326832683268......

4. 0.35109845576243954373......

5. 0.45272885996767235436......

6. 0.33333333333333333333......

7. 0.31265585317839327284......

1. 0. 0 0003430454657612443......

2. 0.1 0 000000000000000000......

3. 0.32 6 83268326832683268......

4. 0.351 0 9845576243954373......

5. 0.4527 2 885996767235436......

6. 0.33333 3 33333333333333......

7. 0.312655 8 5317839327284......

0.1171349...

Since ancient Greek person, Anaximander, first came up with the idea and called it Apeiron, ∞ has always been something of an ambiguous concept, one that escapes easy definition. It's definitelyand there's definitely more to it than that, but it's not always clear what that is. Almost everyone has a sense of ∞, but you could argue that it still lacks a strict definition. Throughout history, people have continually had different ideas of what ∞ is, and we can see this by looking at two famous arguments, both of which imply completely different interpretations of the concept.One way we can look at ∞ as as a sort of process. As if ∞ was a sort of way of saying 'just keep adding one continuously and don't stop'. This is exactly the view that Aristotle takes back in the third century BCE when he argues against Zeno's Paradox. Bradley Dowden summarises the most famous form of Zeno's paradox as follows:Effectively, Zeno is arguing that any distance Achilles moves, the tortoise will have also moved, and that Zeno has to move an infinite number of small distances. Therefore, the tortoise will move some distance infinite times. But, Aristotle responds, "while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility". Effectively, to say a distance can be divided up into infinite parts, does not say anything particular about the distance itself, but simply goes to show that we can continuously divide it. It is not the distance therefore that is infinite, but our act of dividing which does not end.In this way, Aristotle sees ∞ as a process. Saying something is infinitely divisible, it effectively saying something about what we can do to it, not about the thing itself. ∞ is a sort of process, one of endless division/addition.The German mathematician Georg Cantor, viewed ∞ in a completely different way, more like an actual number than a process of endless division/addition. We can see this in his assertion that there are multiple different infinite numbers of different sizes. Despite sounding extremely complicated, the proof itself is actually beautifully simple. Cantor just shows that there are more numbers between 1 and 0 than there are whole numbers in totalTo do this, first we need to imagine that we have all the whole numbers written down in order:Then, next to them, we write a decimal number for each one:So now, for every whole number we have a number between one and zero. All Cantor needs to do next is show that we can find another number between 1 and 0 that we don't already have. To do this, we select the first digit from the first number, the second digit from the second number, etc:Now to make a new number, let's just add one to each digit (if it's nine, we'll take it down to zero) and put them in order. So here we'd end up withWe know this is a new number, because it's been changed from all the other numbers. For instance, it can't be the first number, because its first digit is different. Similarly, it can't be the second number, because the second digit is different. Therefore, we have a completely new number. But we can't put this number next to a whole number because for every whole number there is already a number between 1 and 0.Cantor concluded that therefore there are more numbers between 1 and 0 than there are whole numbers, despite there being an infinite amount of both.It's not completely clear what the implications are of saying there's more than one ∞, but it's definitely something that would be impossible if we treated ∞ as a process, like Aristotle, rather than as a number, like Cantor.There's no great reason to think that either Cantor or Aristotle is more justified than the other in the way they consider ∞, nor is there any reason to think that they can't both be justified. ∞ doesn't seem to be strictly defined enough a concept for one of them to be right, and the other wrong. Perhaps the concept ∞ simply can't be strictly defined, and perhaps it is simply a practice, or a symbol, that can be used differently in multiple contexts, just as a toothpick and a cocktail stick can turn out to be the same piece of wood...............................1. Wikipedia, Anaximander (2017) 2. Technically, Zeno's paradox is actually a series of multiple paradoxes, but since they're all more or less the same and only involve subtle variations on the same theme we can think of them as making up one paradox. Zeno effectively sat back on the success of one idea to generate multiple writing and in this way was the originator of the franchise sequel.3. Bradley Dowden, Zeno's Paradoxes (2017) 4. Aristotle, Physics, Book 6 (C 350 BCE) 5. Georg Cantor,(1891)6. Vihart has a great video on the topic ( Vihart, Proof some infinities are bigger than other infinites, (2014)