Solving Zeno's paradox
Adam: Ok, we have seen that different sizes of infinite sets can exist by thinking about size as pairing things up. We've seen that adding one to an infinite set makes it the same size and today I'd like to try and deal today with some of the problems raised by Zeno. Would you like to explain the paradox of the arrow so I can see how best to respond?
Plato: Ok, suppose an arrow is fired at a target. Before it can hit the target it has to get half way there, before it can get half way there it has to get a quarter of the way there and so on. So it needs to do an infinite number of things before it can hit the target. Doing an infinite number of things takes an infinite amount of time. So the arrow will never hit the target.
Adam: It's easiest to answer this problem by thinking of something else first. I'd like to try and work out what it means for an infinite sequence to end up somewhere “at infinity”.
Plato: Can you give an example of what you mean by “at infinity”?
Adam: Sure, think of the sequence 1/1, ½, 1/3, ¼, 1/5, 1/6, … and so on forever. I want to say that if you carry on down this sequence that “at infinity” it will end up being 0.
Plato: Nonsense. Everything in that sequence is bigger than 0, it cant end up at infinity.
Adam: If you think that then I'm going to ask you a question. You think this sequence always stays away from zero?
Adam: By how much? I mean how close does it get to zero?
Plato: I dont know, something small, say it gets down to 0.0000000000001 but no closer.
Adam: I'm sorry to have to contradict you, but 0.0000000000001 is the same as 1/1000000000000, so every term in the sequence after that fraction will be closer to zero than that.
Plato: Ok, but what about 0.0000000000000000000000000000000000000001 or something really small like that?
Adam: Sorry, every number you think of I can always find a point in the sequence where everything is less than it. It might help to think of a graph.
I say that 1/x ends up being zero “at infinity” because no matter how close to zero your red line is, I can always find a point in green so that 1/x is always closer to 0 than that as we get closer to infinity.
Plato: Hang on though. I've only given you numbers with finite decimal expansions. We found out last time you can have numbers with infinite decimal expansions. So what about the number 0.00...0001 with an infinite number of zeros? You cant find any terms where 1/x is less than this, I say this is the place 1/x gets to “at infinity” not 0.
Adam: The problem is that that's not a number. You say the set of zeros in this number is infinite right?
Adam: And we said that an infinite set is the same size as one with an extra element in it?
Adam: So then the number 0.000...01 is the same as the number 0.000...001 with an extra zero.
Plato: I guess so.
Adam: But this number is ten times smaller. How can that make sense? You cant have a number being 10 times smaller than itself.
Plato: No, that doesn't make sense. I guess I'll have to agree that 1/x goes to 0 “at infinity”. But I'm confused. We've never talked about what things are like at infinity here, not really, only what they're like on the way to infinity.
Adam: Well observed. It's important to view infinity not as some definite place a long way away, but rather as a direction of travel. It's not about the end, because there is no end, we can only talk about what happens on the way.
Plato: Ok, but how does this answer Zeno?
Adam: The objection raised by Zeno is that you cant add up an infinite number of actions to produce something finite right?
Plato: That's right. You cant add ½, ¼, 1/8, 1/16, 1/32 … and so on forever (the distances the arrow has to go before getting to the target) to make something finite. The answer has be be infinitely big.
Adam: Ok, let's try and use the same argument as last time. I say that the sum you get from adding all these numbers is 1, but you say that eventually it must get larger than this. Now I challenge you to name the point in the sequence of sums where it goes over 1.
Plato: By the sequence of sums you mean the number you get by adding a small number of fractions together? So it would be ½, ¾, 7/8, 15/16, 31/32, … I dont know where this goes over 1, but it must happen eventually.
Adam: Not so, you can work out what each term is, by writing the sequence as 1-1/2, 1-1/4, 1-1/8, 1-1/16, 1-1/32 etc. This can never go above one. We can see this in a diagram.
No matter how many smaller rectangles we add we can always fit them in the same picture.
Plato: So you can add these motions up to make 1. … I see, so the arrow can do an infinite number of things, because those infinite number of things add up to make one. But what about Zeno's other paradox. That at any instant the arrow isn't moving, so how can it be moving when you add all those instants together?
Adam: That I'll have to deal with next time, but the approach is the same as last time. We dont consider what things are like “at infinity” as if infinity is a definite point on the horizon. We think about what things are like “at infinity” by thinking about what they are like along the way.