Wow, I got more than I hoped for. Thanks a lot for your long, thoughtful reply. After reading it a few times, I realized there's little I can disagree with. I suppose this sentence perfectly expresses what I was trying to say:
each and every number ever represented in the decimal system, whether on paper, in a mind, or in a computer, has had only a finite number of digits in its representation
The human mind is funny. There seems to be a dissociation between logic and ideas; our ability to discover truths about things that don't exist often plays tricks on us. For instance, the idea that the number 3.333... is a correct representation of 10/3 is often used as a trick to "prove" that 9.999.... = 10. I'm not sure what that means, it seems to me the "paradox" can easily be solved by arguing that you can't start multiplying 3 by 3.333.... before you write down all the 3's.
The axiom of choice, in my amateurish interpretation, seems to say that not only you can think of a number so small you can't even think about it, you can also think of a number that's even smaller than that. Which is funny when you think about it.
You said, "Once you allow for fractions, it becomes impossible to state there's a logical limit to how much you can divide a quantity."
Watch me... There is a logical limit to how much you can divide a quantity. There,...according to you, I have just done the impossible.... Just kidding. I was just having fun with language, exploiting the ambiguity that Wittgenstein tried to reveal.
That was funny! I had to read it a few times before I got it.
Seriously, the key question here is, What do you mean by "quantity"? There are, IMHO, three separate possibilities: 1) the amount of something real 2) some measure of a mathematical concept in a system that includes the AC, and 3) some measure of a mathematical concept in a system that excludes the AC.
Do you think the concept "real" means anything in mathematics? Is there any difference between, say, "three apples" and "three real apples"? What does the "real" in front of "apples" mean?
It was not the continuous math that revealed the graininess of nature, but the direct examination of nature herself. In fact, the observations of nature are inconsistent with a continuous math.
But isn't that equivalent to saying that a continuous math is not self-consistent? It seems to me any self-consistent set of ideas must agree with observations, since all self-consistent systems are tautologies. Tautologies can't be invalidated, not even by experiments.
Now this really excites me. I agree completely with your answer, but I think I read more into it than you intended. The reason it excites me so is your inclusion of the personal pronoun. As I have asked on many other occasions, "What do you mean by 'we', white man?"
Well, that's a tough question. I suppose I really mean "I" since I don't know how other people think, I just assume they must think as I do.
I think it is the "one" you mentioned when you said "a finite segment of line can have as many points as one wishes". Yes, literally "the One".)
So that we don't get into metaphysics, all I meant was that "all finite segments of line have an undefined number of points". That gets rid of "the one".
I would suggest the following version: "What am I saying here? Simply, that while the AC is  a guiding principle for abstract mathematical thinking[, leading to contradictions if you accept it and discrete systems if you don't], it's not true [or useful] as a guiding principle for real problem-solving.
Hmm... do you think a discrete system for math would be free of contradictions?
Thanks for your post...ML