Can Achilles Catch the Tortoise Without the Continuum Property?

[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?