So if you're A[ristotle], then given a line segment between two points, you can keep cutting it and keep finding points, none of with (of course) touch. And in your mind, therefore, you have a series of line segments separated by points. What you can't do is consider all possible cuts, because that kind of realised infinity is foreign to his way of thinking.I think the idea of points 'appearing' when you divide the continuous is foreign to Aristotle's intention (at least at Physics 231a21). Rather, you divide the continuous and you get more continuous, period. You don't 'find' any points after a finite number of division, for the 'points' could only appear when the process of division is complete, which (for Aristotle) can never happen.
In which case, the final step is to go back and say, given that definition / idea, is his original proof valid? I think that, given that, his original result is valid, but vacuously so: he refuses to consider completed infinities, and a line, to be made of points, needs an infinite number of points, which he has ruled out, therefore a line isn't made of an infinite number of points. But only because of his artifical restriction on the meaning of infinity.
Remember that Aristotle doesn't talk about 'points'. He talks about the 'indivisible'. You start with the idea of a continuous thing as something which when divided gives two continuous things. It follows logically from this that the continuous is not indivisible, since it is part of of its definition that it can always be divided. It also follows that no finite process of division will yield anything that cannot be divided further.
If we then define 'composed of' as the relation between the continuous and any set of parts that result from any process of division, it follows that the continuous is composed solely of parts which are continuous when the process is finite. I.e. no points, no 'indivisibles' at all. Just many bits of continuous. Now add the assumption, which Aristotle thinks is impossible, that the process of division can be completed, and by definition (a) the process cannot be finite, from our original definition (b) what is left over will be indivisible, otherwise the process would not be complete and (c) the original continuous thing will be 'composed' of these indivisible thingies, from our definition of 'composed'.
That is, it’s not that the points start appearing as soon as you start splitting the marble. Rather, you only get more bits of marble. But if you keep bashing away hard enough so as to get millions of tiny grains of marble, a heap of fine sand, you can visualise where the process is heading – do this infinitely many times and those little grainy atoms as it were turn into real atom which cannot be further subdivided. Then, and only then, do the points appear. For points are indivisible.
On William's claim that Aristotle has an 'artifical restriction on the meaning of infinity' that's completely wrong. Aristotle understands the same as we do: an infinite process is one that cannot be completed in a finite number of steps. But he also holds that such a process cannot be completed at all, because it is infinite.