I think you are getting away from my point, which pertains to what the article said, which is that mathematicians thought there were "gaps". What mathematician? Can I see the original quote?

The linguistic sleight-of-hand is what I challenge. What is this "gap" in which there are no numbers?

- A reader would naturally assume the word refers to a range. But if that is the meaning, then mathematicians never believed there were gaps between numbers.

- Or could "gap" refer to a single number, like sqrt(2)? If so, it obviously is not a gap without a number.

- Or does it refer to gaps between rational numbers? In other words, not all numbers are rational? Mathematicians did in fact believe this, from antiquity even ... but that remains true!

Regarding this naive construction you are referring to: did it precede set theory? What definition of "gap" would explain the article's treatment of it?