[antasvara]

Take something like the integers (1,2,3,etc.). They are infinite; given an integer, you can always add 1 and get a new integer.

However, there are "gaps" in that number line. Between 1 and 2, there are values that aren't integers. So the integers make a number line that is infinite, but that has gaps.

Then we have something like the rational numbers. That's any number that can be expressed as a ratio of 2 integers (so 1/2, 123/620, etc.). Those ar3 different, because if you take any two rational numbers (say 1/2 and 1/3), we can always find a number in between them (in this case 5/12). So that's an improvement over the integers.

However, this still has "gaps." There is no fraction that can express the square root of 2; that number is not included in the set of rational numbers. So the rational numbers by definition have some gaps.

The problem for mathematicians was that for every infinite set of numbers they were defining, they could always find "gaps." So mathematicians, even though they had plenty of examples of infinite sets, kind of assumed that every set had these sorts of gaps. They couldn't define a set without them.

Cantor (and it seems Dedekind) were the first to be able to formally prove that there are sets without gaps.

[dkarl]

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?

[layer8]

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, since given any pair of points, there would always be a gap between them. “Clearly” they can’t be forming a contiguous line.

[lupire]

Right, but that's the opposite of what the Quanta article says. The article says that Cantor and Dedekind discovered infinity in bounded intervals. What they discovered (really, what they concocted) was uncountable infinity.

[littlestymaar]

Quanta messing things up isn't a particularly rare occurrence, unfortunately.