Lesson · Paradoxes and Their Exits · The infinite

Achilles and the Tortoise

Achilles runs ten times as fast as a tortoise that has a head start. Each time he reaches where the tortoise was, it has moved on, so the gaps shrink and never reach zero. The gaps are a geometric series with ratio the speed ratio. The lab sums them and finds the limit, 1000/9, which is where Achilles passes the tortoise, and it finds no limit when the speeds are equal.

Open the interactive lesson Practice

gaps: 100, 10, 1, 1/10, ... ratio r = tortoise speed / Achilles speed total run = 100 / (1 − r) r = 1/10 gives 1000/9; r = 1 gives none gaps never reach 0; their sum is finite

A race with endless gaps and a finite finish

Core ideas

What this lesson establishes

The previous lesson's series returns with speeds attached. What is new is the ratio, and what it decides.

Each gap is the last gap times the speed ratio

While Achilles covers a gap, the tortoise covers that gap times the ratio of its speed to his. So the next gap is the last times r, and the gaps form a geometric series with that ratio.

The sum of the gaps is how far Achilles runs

He runs each gap in turn, and the total of the gaps is the distance he has run when he draws level. If the series has a limit, that limit is the meeting point.

The ratio decides whether there is a meeting

For a ratio below 1 the sum is finite, and Achilles passes the tortoise. For a ratio of 1 or more the gaps do not shrink, the series has no limit, and he never does. The paradox's form is the same in both.

The material

The race that never seems to end

The argument, the series of gaps, and the case in which Zeno is right.

Achilles runs ten times as fast as a tortoise, and gives it a head start of 100 metres. Zeno argues that Achilles can never catch it. To pass the tortoise he must first reach the place it started, and by then it has moved on. He must reach that place, and by then it has moved on again. Each stage leaves a gap, and a gap that is still positive means the tortoise is still ahead.

Before Achilles passes the tortoise he must reach the place the tortoise has just left.
Each time he does, the tortoise has moved on, so there is a gap that is still positive.
There are infinitely many such stages, and the gap never reaches zero.
So Achilles never passes the tortoise.

Premise three is true, and it is worth saying so: no stage ends with a gap of zero. The conclusion does not follow from it, and the series shows why. Achilles runs 100 metres while the tortoise runs 10; then he runs 10 while it runs 1; then 1 while it runs a tenth. The gaps are 100, 10, 1 and so on, each one tenth of the last.

100 + 10 + 1 + 1/10 + … limit = a / (1 − r) = 100 / (1 − 1/10) = 1000/9

The ratio r is the tortoise's speed divided by Achilles' speed, 1/10 here, and it is the same each stage because the speeds do not change. The first term is the head start. The lab sums the gaps exactly. After six stages Achilles has run 111111/1000 metres, with 1/9000 still to cover, and it reports the limit as 1000/9 metres, a little over 111. That is the total distance Achilles has run when he draws level, and each stage of Zeno's argument takes place before that distance. After the last gap of the list, which there is not, he would be past. After all of the gaps together, which is the limit, he is level, and from there he is past.

The same figure comes from ordinary algebra. Achilles runs at 10 and the tortoise at 1 from a start 100 ahead, so he draws level when 10·t = 100 + t, at t = 100/9 seconds, having run 10·t = 1000/9 metres. The series and the equation agree because they describe one race. What the series adds is the diagnosis: premises one to three describe stages that all occur before the meeting, and say nothing about whether the meeting comes.

The second preset shows how the ratio controls the limit. Achilles runs at 10 and the tortoise at 9, so r = 9/10. The gaps are 100, 90, 81, which sum to 271 after three stages, and the limit is 1000. A race that closes more slowly takes more stages and a longer run, and still ends.

The third preset is the case in which Zeno is right. If the speeds are equal the ratio is 1, every gap is 100, and the sum of any number of them is 100 times the number. The lab reports no limit. The tortoise stays 100 metres ahead, and Achilles does not catch it. The structure of the argument is unchanged. What differs is the ratio, and so the series, and so the third premise's conclusion.

Which exit is this? As the argument was set out, its three premises are true and the conclusion does not follow from them: the move from the third premise to the conclusion is a step that only looked valid, which is exit two. It looked valid because it borrowed a premise that was never written down, that a list of stages with no last member must go on for ever, in distance as in time. Write that premise in and the argument becomes valid, and the exit becomes the previous lesson's: deny the added premise, and the series is the refutation. The two descriptions are one repair. A step that fails only because it leans on an unstated premise is faulted by stating the premise and denying it, and which name the repair gets depends on how much of the argument was put into standard form, which is why that is the first move of the method. The equal-speed case is the reminder that the denial is conditional on the ratio: at r = 1 the unstated premise is true of this race, and the tortoise stays ahead.

Interactive

Exact partial sums

n terms added as fractions, against the limit when there is one

The gaps of a race, summed exactly

The first term is the head start and the ratio is the tortoise's speed over Achilles'. Read the limit: it is the distance Achilles runs when he draws level. Move the number of terms to see how much of that distance the first few stages cover. Then change the ratio to 1/2, so that the tortoise is half as fast, and see the limit become 200. In the third preset the ratio is 1 and the limit tile reads none.

Worked example

Series

First term a (geometric)

Ratio r (geometric)

Terms n6

Bankroll cap (St Petersburg; empty for none)

Term n—

Sum of n terms—

Limit—

Remainder—

Method

Setting up the race as a series

Four moves. The setting-up is the philosophy, and the arithmetic is only the check.

Write the first gap

It is the head start. If Achilles runs 100 metres to reach the starting place, then a = 100.

Find the ratio from the speeds

While Achilles runs a gap, the tortoise runs that gap times its speed over his. That fraction is r, and it does not change from stage to stage.

Sum the series, or find that it has no sum

If r is below 1, the limit is a / (1 − r). If r is 1 or more, the gaps do not shrink, and the series has no limit.

Interpret the limit

A limit is the distance Achilles has run when he draws level. A series with no limit means the gap never closes.

Name the exit and its price

The stated premises are true, and the step to the conclusion borrows one that is not: that a list with no last stage goes on for ever. Fault the step, or state that premise and deny it; the repair is the same, and its price is small.

Gaps and limits

Score 0 / 0

Choose the answer you can justify.

Head start 100, speeds 10 and 1

Achilles runs at 10 metres a second, the tortoise at 1, and the tortoise starts 100 metres ahead.

stage 1: A runs 100, T runs 10, gap 10 stage 2: A runs 10, T runs 1, gap 1 stage 3: A runs 1, T runs 1/10, gap 1/10 gaps: 100, 10, 1, 1/10, ... a = 100, r = 1/10 limit = 100 / (1 − 1/10) = 1000/9 check: 10t = 100 + t gives t = 100/9, run = 1000/9

The series and the equation give the same answer, as they must. Achilles passes the tortoise at 1000/9 metres from where he began, about 111.1, at 100/9 seconds. Every stage of Zeno's list falls before that point.

Common mistakes

Concluding that Achilles never catches the tortoise because the gaps never reach zeroThe gaps never reach zero, and that is true. What it does not give is that he never draws level. The gaps sum to 1000/9 metres, and Achilles reaches that point and goes on. The meeting is the limit of the stages, and it is not any stage. It is like the end of a road that has infinitely many milestones.

Trusting the paradox's form to decide the raceThe same form, the same infinite list of positive gaps, describes the equal-speed race in which the tortoise does stay ahead. The form cannot tell the two apart and the ratio can. Look at r, not at the number of stages.

Mixing up the two speed ratiosThe ratio of the series is the tortoise's speed over Achilles', which is below 1 when he is faster. Dividing the other way gives 10, a series of growing gaps, and a limit that does not exist for a race that clearly ends.

Check your understanding

✓

Finish when you can set up a race as a series and compute where it ends.

Given a head start and two speeds, you should be able to write the first gap and the ratio, state the limit of the gaps as an exact fraction or say that there is none, check it against the equation for the meeting time, and say which exit the result takes, naming the unstated premise it denies.

Note: The race is a paradox whose exit is settled by a computation, which is unusual. The lesson that follows keeps the series and removes the speeds, and finds a case in which the sum is finite and the question being asked still has no answer.