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?