> Before their papers, mathematicians had assumed that even though the number line might look like a continuous object, if you zoomed in far enough, you’d eventually find gaps.

I'll try to interpret this sentence.

We all have some mental imagery that comes to mind when we think about the number line. Before Cantor and Dedekind, this image was usually a series of infinitely many dots, arranged along a horizontal line. Each dot corresponds to some quantity like sqrt(2), pi, that arises from mathematical manipulation of equations or geometric figures. If we ever find a gap between two dots, we can think of a new dot to place between them (an easy way is to take their average). However, we will also be adding two new gaps. So this mental image also has infinitely many gaps.

Dedekind and Cantor figured out a way to fill all the gaps simultaneously instead of dot by dot. This method created a new sort of infinity that mathematicians were unfamiliar with, and it was vastly larger than the gappy sort of infinity they were used to picturing.

You can construct sequences of rational numbers where the limit is not rational (eg it's sqrt 2)

Trivially, the sequence of numbers who are the truncated decimal expansion of root 2 (eg 1.4, 1.41. 1.414, ...) although I find this somewhat unsatisfying.

With the real numbers there are no gaps. There are no sequences of reals where the limit of that sequence is not a real number

Complete just means the limit of every sequence is part of the set. So there’s no way to “escape” merely by going to infinity. Rational numbers do not have this property.

How to construct the real numbers as a set with that property (and the other usual properties) formally and rigorously took quite a long time to figure out.

> Give x < y, it's easy to construct x + (y-x)(sqrt(2))/2.

That's only obviously irrational if x and y are rational.

The density does not dictate cardinality which is what this article is about.

The continuum. Connectedness.

I have an opinion about the editorial style of Quanta that I don't think it's popular here (judging by how often they get upvoted), but I think it's a symptom of that.

They cover science, but the template they follow pretty consistently is a vague title that oversells the premise and then an article filled with human-interest details and appeals to implications. This makes it easy for everyone to follow along and have an opinion, but I feel like science is a distant backdrop and never the actual subject.

In this article, what's the one tidbit of scientific knowledge that we gain? Dedekind's and Cantor's work is described only in poetic abstractions ("a wedge he could use to pry open the forbidden gates of infinity"). When the focus is writing a gossip column for eloquent people, precision doesn't matter all that much.

Thank you! After Benj Edwards and Kyle Orland's Ars Technica article they published using AI (while saying they didn't), and all the while their article was about an AI agent publishing a hit piece on Scott Shambaugh (matplotlib maintainer), I feel like I now assume journalists are using AI and things need to be fact-checked just as we do for our AI interactions.

I appreciate hearing about details like this and getting the source directly. I hope Kristina Armitage and Michael Kanyongolo from Quanta Magazine respond and you can update us!

Scott's Blog on Hit Piece: https://theshamblog.com/an-ai-agent-published-a-hit-piece-on... Ars Editor Note: https://arstechnica.com/staff/2026/02/editors-note-retractio... Ars Retraction: https://arstechnica.com/ai/2026/02/after-a-routine-code-reje...

Are you citing your own book?

I'm here for the 19th century drama. Imagine the head lines!

    Cantor's Continuity Credentials Cancelled: Clear Cut Copy Cat Case!

This comment made me think of this xkcd 2501

https://xkcd.com/2501

There really is an xkcd for everything

> since most HN readers should know who Cantor and Dedekind are.

Show up with your hands here if you didn’t know either Cantor or Dedekind.

Hard disagree

I’ll go out on a limb and say the majority of HN users at this point do not know the context and implications of the impact of Cantor - would probably have only heard the name in the context of mathematics but no deeper

I’d go further and say the majority have not ever heard of the name Dedekind

I am not a mathematician; I barely knew who Cantor was and had never heard of Dedekind. I would have likely not read the article without the title being so sensational. Your assumption sits upon the tip of your nose.