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Comment by irrational | Hacker News Reader

>To explain the idea to a child, I would focus on the principal idea: whether finite or infinite, a number is even when it can be divided into pairs. For finite sets, this is the same as the ability to divide the set into two sets of equal size, since one may consider the first element of each pair and the second element of each pair.

The answer this quote came from is amazingly obtuse, but it does make me think that infinity must be even since infinity can be divided into 2 pairs, each of which is of equal size since both are infinity.

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In mathematics, you can define things in different ways to get different answers. Ways of defining things tend to be highlighted as true (in at least some context) if they are interesting and useful, and ignored if not. I don't think the definition based on "dividing into pairs" is particularly interesting or useful in the context of the child's understanding of numbers, because it's too vague to be useful, and it doesn't lead to any insights.

The definition based on transfinite ordinals explained in the same answer does seem interesting, and I wouldn't be surprised if it were useful. I think this is a case of simplification gone wrong, where everything interesting was lost in the translation to more accessible terminology.

A more honest thing to say to a child would be that the way even and odd are defined only make sense for finite numbers. It's true for the definition they know, and it introduces them to the important insight that logical rules that are created for one kind of thing might not work when applied to something else. I think this would be more accessible and stimulating for a six-year-old than giving them a half-baked verbal imitation of a result from transfinite mathematics.

They'll be thrilled later if they study math and discover that there are definitions of "infinity" and "even" that yield an answer to their childhood question.

An even more honest thing to say is that infinity when used as a number is a hack introduced by mathematicians to make notation and reasoning simple in some cases, but that it can be dangerous in other cases, like any other hack. If you want to use infinity in a safe way, then use limits around your expressions.

(And this quickly resolves the case of this article, since lim x->inf x-2*floor(x/2) does not exist).

That's true in calculus, and probably a lot of other applied mathematics contexts where rigor tends to get swept under the rug, but it's not completely fair since there are versions of "infinity" that are defined and used rigorously. (The transfinite ordinals and cardinals mentioned in the Stack Overflow article are the example I'm familiar with.)

It's not a hack to create a new set and work out rules for how to use it which are both internally consistent and support easy morphisms with more familiar sets.

It may not be easy, but it's hardly a hack. It's one of the big ways math works, really. Are negative numbers a hack? Rational numbers? Algebraic numbers? Well then neither is the two-point compactification of the reals or extending the natural numbers into the ordinal numbers.

These are things with very precise models and interpretations. No hacks at all.

But nobody said that hacks cannot have precise interpretations. It's the unreasonable cognitive load that is the problem.

> it does make me think that infinity must be even since infinity can be divided into 2 pairs, each of which is of equal size since both are infinity.

This is true,

but the same is true of (infinity - 1)

Therefor infinity must also be odd.

The concept of "infinity - 1" doesn't exist. Subtraction isn't defined for ordinals. Furthermore even if you try to define it, it doesn't work for limit ordinals.

If you are thinking about the difference between

    [0,1,2,3,…]

and

    0, [1,2,3,4,…]

Then I regret to inform you the former is omega and the latter is 1+omega which is the same as omega. In other words attempting to subtract one from infinity by removing from the front results in infinity.

> In other words attempting to subtract one from infinity by removing from the front results in infinity.

And I regret to inform you that if you read more carefully, you will find that my comment above makes use of that very same property of infinity. Not only do I already know it; that's the joke.

Specifically, that statements about omega are also statements about 1 + omega. The parent post saying "I think that infinity must be even" is such a statement. Regardless of if it's true or not, well-defined or not, coherent or not, it's equally all that about (infinity - 1).

Should I also spell out that an argument that "n - 1 is even" is also an argument that "n is odd" ?

Infinities aren’t comparable for equality… are they?

Not if you aim to pass your exam.

Sure they are. You can define a one-to-one mapping, they're equal.

You can define a one-to-one mapping between the sets {1 2} and {3 4}, but I don't think anyone would say they are equal.

You’re thinking of isomorphic, not equal.

They meant "their cardinalities are equal". It's honestly an easy mistake to make, especially if typing on a small screen. Or especially if having a discussion where sizes of infinity are already being discussed.

equal in cardinality, yes, thanks.

In the general case, the comparability of cardinals relies on the axiom of choice. In other words, they are comparable, but they require a slightly unintuitive foundation to establish that they are always comparable.

But zero is in the middle, so it must be odd

Couldn't you trivially say zero is in "the middle" of any even split? 2 + 0 + 2 = 4?

Edit: perhaps you meant one is in the middle?

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