Zeno's paradox revisited

[Quantum Ninja]

Here's another variation of Zeno's paradox that occurred to me today.

Consider the number line of all real numbers. At the origin, a particle is at rest. Now suppose the particle moves to the right (in the positive direction). What is the first point to which the particle moves after leaving the origin?

If you say something like 0.1, then the particle had to first move through 0.01 to get there. And before getting to 0.01, the particle had to first move through 0.001. In fact, there is no point after leaving the origin that could be described as the "first" point the particle occupies. How, then, is motion possible in continuous space/time?

The problem does not arise if we assume time and space are discrete. Any ideas how to resolve this issue if we assume the opposite (that both time and space are continuous)?

[tronvillain]

If the particle is moving at one unit per second then at one second it will be at 1, at 0.1 seconds it will be at 0.1, at 0.01 seconds it will be a 0.01 and so on all the way back to zero. That there is no such thing as "the first point to which the particle moves after leaving the origin" does not appear to be a problem. Any problem with such a lack is simply an artifact of wanting a descrete answer when the subject is continuous.

[pmurray]

The thing that resolves many of these paradoxes is understanding the idea of orders of infinity. There is a higer order of infinity of points on a line than there are natural numbers. This means that you can never enumerate the points on a line, and that's why you can't identify a boint "next to" another.

You even get the same problem with fractions, which have the same cardinality as natural numbers but which are densely packed on the real number line.

Do a web search on "Cantor's diagonal slash", "aleph-null", and "densely packed".

[Kat_Somm_Faen]

That is why Diogenes is kewl!

I had read somewhere that upon hearing this "paradox" of Zeno's his reply was to silently stand up and walk :)

[Quantum Ninja]

Originally posted by Kat_Somm_Faen
That is why Diogenes is kewl!

I had read somewhere that upon hearing this "paradox" of Zeno's his reply was to silently stand up and walk :)

Yeah, we were talking about Zeno's paradox on the first day of my logic course and my professor did basically the same thing. He presented Zeno's paradox, then pointed out two arbitrary points in the air, calling them A and B, and then moved a water bottle from one to the other. It was pretty humorous the way he did so.