Comment by helen___keller | Hacker News Reader

Comment by helen___keller | Hacker News Reader

I'm used to blogs, magazines, and the internet butchering this topic, so it was refreshing to see a reasonable introduction.

Only complaint is the attempt to build intuition came very close to making an incorrect statement:

> But there’s something unsatisfying about declaring the size of the set of real numbers to be the same “infinity” used to describe the size of the natural numbers. To see why, pick any two numbers, like 3 and 7. Between those two numbers there will always be finitely many natural numbers: Here it’s the numbers 4, 5 and 6. But there will always be infinitely many real numbers between them, numbers like 3.001, 3.01, π, 4.01023, 5.666… and so on.

> Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation.

The alluded property (the set of real numbers is dense) is not related to cardinality, in the sense that the rational numbers are dense but also countable. But I appreciate that the author is trying to motivate the remaining explanation about diagonalization, which can be a tricky topic for beginners.

Diagonalization might be a tricky topic for beginners, but that is also who we teach it to. It is usually in your introduction to mathematical proofs class, and even there it is taught in the first couple of weeks.

There is an additional problem with the argument as written, as there are in fact "relations" on the "decimal expansions" of real numbers. for instance .099... = .1000....

Notice that none of the places in this expansion agree with each other so this might indeed be the element you construct from the following list:

`.1000... .0100... .0010... ...`

You might say, ok well pick a number different from a_i and also different from 0 or 9, which does indeed get you a number not in your list, but it really begs the question of "why doesn't this work in binary?", and is also far less intuitive than factoring the problem into two steps of |N|~=|P(N)|=|R|. This approach is more general anyway.

> for instance .099... = .1000

wait, so how does diagonalization account for this? The argument, as i recall, was that every real number in the "fake list" differs from the generated real number by at least 1 position in the decimal expansion.

But doesn't this argument depend on the assumption that every real number has a unique decimal expansion?

Well, the short answer is that there aren't that many related decimal expansions at most two for any expansion, and a countable number of pairs, but it's a problem with the argument.

I was equally annoyed by this part. You are very generous with your very close to making an incorrect statement. I would straight away say it’s both misleading and wrong.

The author specifically states, and I quote:

>no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between ... this doesn’t mean that the sets of real numbers and natural numbers have different sizes

What is misleading or wrong about that? What specific statement is incorrect?

No matter how close any two distinct rational numbers are to each other, there will always be infinitely many rational numbers in between. But the cardinality of rational numbers and natural numbers is the same.

That's correct and it's precisely what the author states, and I quote:

>this doesn’t mean that the sets of real numbers and natural numbers have different sizes

They state that only as a disclaimer, after a bunch of other statements that hint at the opposite. Just adding a disclaimer to misleading writing merely makes it confusing and unclear for those who are paying attention, and still misleading for those who don't notice the disclaimer.

I think the writer might have been meaning to imply that there's a spectrum of properties from countably infinite to dense to uncountable, where each property is stronger than the previous one. But this is actually not the case. The Cantor set is uncountable, but nowhere dense. So density is not just something that's in between countability and uncountability---it's an orthogonal thing.

I would say your criticism is akin to the curse of knowledge [1], wherein you know all of the relevant background information and so you have difficulty putting yourself in the position of someone who does not.

The author is writing the article for someone who could very well be learning this for the first time and may think to themselves "Hmm... this is something unusual and counterintuitive.". Since you are not that reader the author's writing could come across as annoying, but it's not wrong or misleading, it's a way for the author to hint to the reader that they are empathizing with them and will address this unintuitive notion further (which the author does).

[1] https://en.wikipedia.org/wiki/Curse_of_knowledge

That’s an interesting quote you got there. Let’s look at the surrounding sentences, shall we? Emphasis mine.

> But there’s something unsatisfying about declaring the size of the set of real numbers to be the same “infinity” used to describe the size of the natural numbers. To see why, pick any two numbers, like 3 and 7. Between those two numbers there will always be finitely many natural numbers.

> Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation.

So no, as rightfully pointed by other commenters, the article implies that density somewhat indicates that two cardinalities which is straight wrong as the rational are dense in the natural yet of the same cardinality. They are even dense in R which entirely defeat the point of the analogy. Density and cardinality are not related at all.

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