How to have mind-boggling fun with infinity

[rocketpj]

Reading this thread has been somewhat like the time I got lost in the forest as a kid.

At first I was looking at interesting things, then more interesting things, then suddenly I found myself completely lost and wondering what the hell is going on.

[Jimichan]

That makes too much sense, though. Calling it the average of the two is the joke.

[OtherMichael]

You cannot subtract infinity from a number, because infinity is not a number – see the original post.

Infinity is defined, sometimes, as { 1…n, n+1 } (or zero, or -1, or pick any number). Go up to any particular number. Then one more. Then one more. Then one more. Ad infinitum.

It’s not a quantity.

Infinity is not intuitive unless you’ve played around with it a lot (don’t count me in with that lot). Rudy Rucker has written some good non-fiction on infinity, and some good fiction with infinity. I like the higher concepts of math, without, I suppose, really understanding them. Math is hard. But it ain’t boring!

[davide405]

Cardinality is a property of a set, it is not the set itself.

One of the flaws of using numbered balls for the initial examples in this article is that people get fixated on the balls and the process of putting them in and taking them out.

The interesting assertion in this article, the thing that makes it “mind-boggling”, as I have said above is [quote]the cardinality of the set of all natural numbers is the same as the cardinality of the set of natural numbers that are perfect squares of an equal or lesser natural number[/quote]

The two sets in question are

• {1, 2, 3, … n} which is the natural numbers
• {1², 2², 3², … n²} which is the set of squares of natural numbers

In that layout, it may be a bit easier to see that for every element in the first set, there is a corresponding element in the second. That is to say, they have identical cardinality.

Yes, it’s counter-intuitive, but the everyday intuition of dealing with small, finite, countable sets breaks down when exposed to infinite sets. Specifically [quote] In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. One example of this is Hilbert’s paradox of the Grand Hotel. Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset …
{emphasis added}[/quote]

I strongly recommend the article on the Grand Hotel for a much more thorough explanation than I can give here.

Edit to fix the broken link:

Hilbert's paradox of the Grand Hotel

[Eksrae]

It could be a matter of who was going to St. Ives.

[OtherMichael]

@Eksrae

It could be a matter of who was going to St. Ives.

You know, that bugs me. Couldn’t they all be going to St. Ives? It stands to reason that an unencumbered traveler is likely to travel faster than one carrying seven sacks, and thus the solo body meets the group body as he overtakes them, nu?

[OtherMichael]

Meanwhile, in Philosophical Gotham:

(source)

[redesigned]

they have a lot of balls asking such a question…

[daneel]

ℵ₀?

[davide405]

@kupfernigk

The problem with the question as posed is that people keep trying to perform invalid operations on sets of size \aleph_00, like subtracting one subset from another, or proceeding to the “last” element. When these errors are removed it becomes obvious that the “answer” that no elements are left cannot be right, because no matter how we try to do it, an invalid operation is needed.

No.

The problem with the question as posed is that people keep getting stuck in the algorithmic presentation of the problem.

When we look at the problem in an algorithmic way it seems “obvious” that the number of balls in the drawer increases without bound, and that the number of balls in the box also increases without bound, but at a slower rate.

But the balls and drawer and box are NOT the point of the article. The point of the article is the counter-intuitive (hence “mind-boggling”) notion that two countably infinite sets, one of which is a proper subset of the other, actually have identical cardinality.

The balls and drawer and box are NOT the “answer”. They are a sign that points toward the answer. A sign that says “Los Angeles — 462 miles” is NOT the same as the city.

No invalid operations are performed. You certainly can remove a subset, even a countably infinite subset from a set with cardinality of Aleph-null. It’s just that the results of the operation don’t line up with our everyday experience with finite sets.

Apparently it’s so mind-boggling that even though the answer is categorically correct, many of us are incapable of getting past the imperfect analogy that was used to introduce the problem,

Conspiracy Theories

[Raita]

All this arguing when the answer is so simple:

Infinity is 200mg of ketamine.

[chenille]

@kupfernigk

Hence my point; you cannot subtract the cardinality of one set with ℵ0 elements from another set with ℵ0 elements and get 0 elements, which is what the article clearly states to be the case.

You can’t subtract the cardinality because it’s not a well-defined operation on infinite values, but you can still subtract the sets themselves, and in this particular case the empty set is the result. Like davide405 said, sets and their corresponding cardinal numbers are different things.

@kupfernigk

Furthermore, two sets with ℵ0 elements DO NOT have identical cardinality, because you cannot establish such an identity (they are infinite sets).

That’s also not true. A cardinal number isn’t formally an equivalence class, because trying to apply them to sets causes problem, but it works very much like one. Two sets have the same cardinality if there exists any one-to-one mapping between them; sets are usually defined to be infinite when they have the same cardinality as a proper subset.

[davide405]

@kupfernigk

Suggesting that I am a delusional conspiracy nut really does not add anything to your post, either.

Note the words of the character in the first panel: “Please, stop, because seeing this happens to you breaks my heart.” The intended suggestion was that I’m arguing helplessly with a smart person who is blind to the glitch in their thinking on this topic.

What is so hard about this proposition

• Every natural number can be squared.
• The square of a natural number is itself a natural number.

Therefore

• The set of all natural numbers EXCEPT those that don’t have their square in the set of natural numbers is the empty set.

[andy_hilmer]

@davide405

The intended suggestion was that I’m arguing helplessly with a smart person who is blind to the glitch in their thinking on this topic.

There’s that “aha!” moment where the size of the two sets, one which seems to be a subset of the other, suddenly make sense as being essentially the same, regardless of the operations which distinguish them. I think the best way to present this was when you just listed them:

@davide405

{1, 2, 3, … n} which is the natural numbers
{1², 2², 3², … n²} which is the set of squares of natural numbers

That’s the only way I’ve ever been able to make sense of this seeming paradox. Otherwise I end up going down the same rathole, focusing on the operations rather than the meaning.

[wrecksdart]

It’s what killed the radio star, maaaan.

[kupfernigk]

@davide405

Note the words of the character in the first panel: “Please, stop, because seeing this happens to you breaks my heart.”

You do realise that you come over as extremely arrogant?
You almost have me persuaded, but I would hate to be a student in your class. The way people behave in xkcd is not the way things happen in real life.

@davide405

The problem with the question as posed is that people keep getting stuck in the algorithmic presentation of the problem.

This is why I am only almost persuaded. Perhaps it is because the maths I know is applied rather than pure, but to me the original problem in its algorithmic form is different from the problem you are solving. Of course, the original problem is impossible in reality because the number of particles in the universe is finite, but regarded purely as a theoretical exercise, I do not see why it can simply be ignored. You could argue that the algorithmic statement includes an arrow of time, while the all at once statement does not; in other words, the dimensions of the two problems are different.
For me to agree with you, you would need to show without hand waving why the algorithmic aspect of the problem can simply be ignored.

[davide405]

@kupfernigk

The way people behave in xkcd is not the way things happen in real life.

I dunno, I think he pretty much nails my own behavior right here…

Duty Calls