I just don't understand why this was disturbing. Prior to the construction of the reals, the existence of irrational and transcendental numbers was disturbing, because they showed that previous constructions (rational numbers and algebraic numbers) were incomplete. If those gaps were disturbing, a construction without gaps should have been satisfying, reassuring, a resolution of tension. Was there some philosophical or theological theory that required the existence of gaps, that claimed that a complete construction of the number line was mathematically impossible, because of some attribute of God or the cosmos?

I think the issue was that most irrational/transcendental numbers aren’t finitely representable. This means that they are mathematical objects which, each of them individually, somehow consist of an infinity (e.g. an infinite decimal expansion). They are the result or end point of infinitely many steps (e.g. a converging sequence) that you can’t actually reach the end of in practice, and for most of them can’t even write down a finite description on what steps to perform, and which therefore arguably doesn’t “really” exist.

Another point of contention was the notion that the continuous number line would be formed out of dimensionless points. Numbers were thought of as residing on the line, but it was hard to grasp how a line could consist solely of a collection of points, as given any pair of points, there would always be a gap between them. “Clearly” they can’t be forming a contiguous line.