Achilles and the Supremum

[Rasalhague]

KG Binmore talks about Zeno's paradox of Achilles and the tortoise to motivate the idea of suprema for sets of real numbers:

Since Achilles runs faster than the tortoise, the tortoise is given a head start of x₀ feet. When Achilles reaches the point where the tortoise started, the tortoise will have advanced a bit, say x₁ feet. [...]

The simplest way to resolve this paradox is to say that Achilles catches the tortoise after he has run x feet, where x is the 'smallest real number larger than all of the numbers x₀, x₀ + x₁, x₀ + x₁ + x₂, ...' [...]

This solution, of course, depends very strongly on the existence of the real number x.

i.e. on what he calls the continuum property.

But can't Achilles can catch the tortoise even without the continuum property, e.g. on a race track of rational numbers? Let x be Achilles' position, and y that of the tortoise. Let u be Achilles' speed, and v that of the tortoise.

$$x = 0+ut;$$

$$y = \frac{1}{2}+vt;$$

$$x,y,u,v,t \in \mathbb{Q}.$$

Let the race begin at t = 0. If u = 1, and v = 1/2, Achilles will catch the tortoise at x = y = 1.

Is the problem with $\mathbb{Q}$ that there exist combinations of parameters for which Achilles can pass the tortoise without at any time being at the same point as the tortoise? Is that the paradox which the real numbers, with their guarantee of a least upper bound, resolves?

 

[HallsofIvy]

Yes, that's a very good insight into the problem.

 

[AC130Nav]

Yes, that's a very good insight into the problem.

But has nothing to do with Zeno's paradox.

 

[Rasalhague]

But has nothing to do with Zeno's paradox.

Feel free to elaborate!

My question was about Binmore's use of the story of the race to motivate the idea of a least upper bound. I was trying to pin down what it is about a race along, say, the rational numbers (which lack the continuum property) that makes such a race paradoxical (to Binmore) in a way that a race along the real numbers is not (because they have the continuum property).

I can appreciate that Zeno may have used the story in a different way, and that for him perhaps the paradox lay simply in the idea of an infinity of intermediate points. Is this what you wanted to draw our attention to?

 

[CellCoree]

I find Binmore's use of Zeno's paradox to introduce the concept of suprema for sets of real numbers to be a clever and effective way to illustrate the importance of the continuum property. The paradox itself highlights the potential issues that can arise when dealing with infinite sets and the need for a mathematical framework that can handle them.

Binmore's solution relies on the existence of the real number x, which is the smallest real number larger than all of the numbers x0, x0 + x1, x0 + x1 + x2, and so on. This concept of a "least upper bound" is crucial in understanding the behavior of infinite sets and is a fundamental property of the real numbers. Without it, the paradox cannot be resolved.

The example provided by Binmore of a race track of rational numbers also highlights the limitations of using just rational numbers. While it is possible for Achilles to eventually catch the tortoise on this track, there are combinations of parameters where he can pass the tortoise without ever being at the same point as the tortoise. This is the paradox that the real numbers, with their guarantee of a least upper bound, can resolve.

In conclusion, the use of Zeno's paradox of Achilles and the tortoise to introduce the concept of suprema for sets of real numbers is a clever and effective way to illustrate the importance of the continuum property. It highlights the limitations of using just rational numbers and the need for a mathematical framework that can handle infinite sets.