someguy1 said:

You know for several pages you've been throwing out random math phrases you've read on Wikipedia and asking me to put them in context for you.

The quotes aren't from Wikipedia and I haven't had the least expectation that you will put them in context for me. Nor have I asked you or anyone else to do that. Your arrogance is a given, it seems. You can't help yourself.

To the rest:

The ideas I've been trying to explore go like this: the real line is continuous, so is physical motion. With calculus you can define a set of instantaneous positions and velocities for a classical object with motion. Calculus usually defines the derivative in terms of limits, the epsilon-delta method uses real numbers, not infinitesimals (mainly because there's only one infinitesimal real). But the derivative can be defined in other ways; if you want to invoke infinitesimals you "need" a map from them to the standard reals.

Also, when you see a textbook saying dy/dx means an infinitesimal change in y and in x in the reals, they don't mean actual infinitesimals but very small real numbers (one supposes).

What calculus seems to say is, it doesn't matter if the real line is after all a subset of the hyperreal line, or if we call one or the other an abstraction of reality. Zeno's problems with continuity of motion seem to have more than one solution. And Zeno was just saying there's a problem if we assume real (actual) distances are infinitely divisible. (that is, into 1/2 the distance, then 1/2 the remaining distance,

*ad infinitum*), so he thought, an object in motion has to pass through an infinite number of "checkpoints" in a finite time. He got that wrong, but why?

Is it because the real line isn't, after all, infinitely divisible? Or is it because an object's motion really isn't "from point to point"?