Yup. Fair comment. I was being rather loose with history, being an enthusiast for Canotr’s work, I was rather glossing over the rest tof it.
One could also easily imagine a number line with points on it which do not correspond to that particular system of quantities given the unfortunate name “real numbers”; e.g., one with distances greater than every integer, or with positive distances less than every positive rational. But whatever. That’s a distraction. We don’t even need the concept of “real numbers” to state or settle the problem. It still comes out to just:
No. Why would you think so? If you tell us what led you to think this, then we can show where such reasoning went wrong. Otherwise, there’s nothing more which can be said. This assumption is erroneous, and thus there is no real paradox.
Bah! Now I’m correcting my correction - in so far as What I was correcting wasn’t what was written. 
My bad.
Still true, the reals are the algebraic and the transendentals, but the original quote was correct - decimals are rationals. And if you restrict your thinking to these you don’t need to worry. I was going one further, and pointintg out that there are more numbers than these.
This is essentially the definition of a Limit in calculus. Well, maybe not the definition, but Limits are used to represent these types of things.
I did. In fact, I cited some of them in my last post.
It’s difficult to put one’s head around the idea that there can be infinite points within something finite. That’s pretty much it. You are to be commended if it doesn’t boggle your mind. It boggles mine.
Thanks to everybody who participated. It was very interesting to me. Continue as you please. Apologies to those who are somehow put off that a topic was discussed that’s been discussed before.
so much for that college edumacation.