I don't think that aleph0 + 1 is odd, because I can make a 1-1 correspondence between two of its subsets. They're ALL even, by that definition!
Is infinity an odd or even number? (2011)
281–290 of 398 posts
Re: Is infinity an odd or even number? (2011)
#282npm i is-even const isEven = require("is-even"); console.log(isEven(Infinity)); TypeError: is-odd expects a number.
I read the title and I sarcastically thought "ask JavaScript!" Your comment didn't disappoint. Importing the is-even package is the cherry on top.
Even with plain Javascript, `Infinity % 2` evaluates to `NaN`, as it should.
And if you implement, e.g.:
function IsEven(x) { return x % 2 == 0; }
function IsOdd(x) { return x % 2 == 1; }
Re: Is infinity an odd or even number? (2011)
#283Earlier quoted context omitted.
The way I would explain it to a 6 year old would be like this: There are natural numbers, like 0,1,2 and so on. Natural numbers can be odd or even. There is no such natural number as infinity. Therefore the question if 'infinity' is odd or even is meaningless. It does not even type-check. In math people like well-formed questions, and generally don't like ill-formed questions.
The fallback metaphor I use in these situations or similar ones, "What's outside of the universe" for example, is the old, "What's North of the North Pole?" Then you explain that we can create questions and statements in our languages which don't have logical, mathematical or physical validity. Although we can often describe scientific and technical concepts in common languages, that's just a translation, the real la…
Carlos Castaneda is at his most interesting when he wrestles with "what's outside of the universe" paradoxes since his informants seem like they're able to not only hold mutually exclusive concepts but exist in a relationship between them. They'd have an internally consistent idea about what's North of the North Pole and could explain it to you in terms you might understand.
He's given me quite a bit to think about in regard to NULL and the assumptions I make around the concept, which is fascinating in itself because his books are hot garbage.
Re: Is infinity an odd or even number? (2011)
#284Earlier quoted context omitted.
Yes never, not in school, not in analysis, and certainly not in numerical analysis. You've proved my point. It's either π or 3.14. Except that the latter is a rational number :)
> Yes never, not in school, not in analysis, and certainly not in numerical analysis. Weird, I just assumed that was normal in most education systems. I don't know how you'd get a sense of the rough scale of various common irrationals, without having some idea what they look like when represented in decimal notation. Such representations are normal starting not later than when we start seriously working with circles,…
Can you write here an irrational number, not in the form of a letter or a formula?
For me a "decimal number" is a number represented in base 10. Which is why I was asking. I even googled and there is no real definition.
Re: Is infinity an odd or even number? (2011)
#285Earlier quoted context omitted.
The fallback metaphor I use in these situations or similar ones, "What's outside of the universe" for example, is the old, "What's North of the North Pole?" Then you explain that we can create questions and statements in our languages which don't have logical, mathematical or physical validity. Although we can often describe scientific and technical concepts in common languages, that's just a translation, the real la…
Carlos Castaneda is at his most interesting when he wrestles with "what's outside of the universe" paradoxes since his informants seem like they're able to not only hold mutually exclusive concepts but exist in a relationship between them. They'd have an internally consistent idea about what's North of the North Pole and could explain it to you in terms you might understand. He's given me quite a bit to think about i…
I was so confused at your glowing review until the redemption of the last sentence.
Re: Is infinity an odd or even number? (2011)
#286Re: Is infinity an odd or even number? (2011)
#287Earlier quoted context omitted.
"Countably infinite" makes zero sense to me. Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers. I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem. And pairing one infinite list with another infinite list doesn't make either one any mo…
> Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers. Exactly correct! This holds true of everything you can generate stepwise, even infinite sets. Cantor proved that you cannot "generate" (stepwise) all Reals between 0 and 1. Any infinite set you can generate stepwise is Countably Infinite. > I'll put Cantor and his proof in a box…
> And then infinitely later
There can be no 'and then' after infinitely later.
I don't see why stepwise is important but that must be the key to Cantor's proof.
If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned.
The ordering could be entirely random, I don't see how it makes a difference. There will always be enough integers to match.
Is it that my black box metaphor is cheating by coercing a truly 'parallel' generation of decimals into a linear operation? But even then, if I'm getting exponentially bigger chunks of new decimals, I can provide equally large chunks of integers... so it still doesn't make sense to me. Infinity is infinity and you cannot count it.
Re: Is infinity an odd or even number? (2011)
#288Earlier quoted context omitted.
But that seems redundant with countable/uncountable sets, because then every countable infinite set would be even (e.g. rational numbers), and every uncountable infinite set would be odd (e.g. real numbers). It's also not clear to me what justification there would be for a "preference" for the "even" category that way -- it seems arbitrary. Why not be odd if there exists a way to pair things off such that one is left…
I think the reals are also even: If x is rational pair it as you would in the rational case (which we assume is even - I haven't proven this). Otherwise pair it to -x, and thus the reals are even. Being "even" seems like a much more interesting (and simpler) property of a set. I don't see what use there could be to know that you could pair things off, with one element left over. When you extend the notion you do have…
If you want to pair positives with negatives, then reals would still be odd, as long as zero is unsigned. Zero is the unpaired element, hence odd.
But it all just seems silly. We can say the set of positive integers is even because we can come up with a pairing of elements, while the set of positive reals is odd because we can't come up with a pairing? Where's the mathematical utility in that?
Re: Is infinity an odd or even number? (2011)
#289The way I would explain it to a 6 year old would be like this: Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.
The way I would explain it to a 6 year old would be like this: There are natural numbers, like 0,1,2 and so on. Natural numbers can be odd or even. There is no such natural number as infinity. Therefore the question if 'infinity' is odd or even is meaningless. It does not even type-check. In math people like well-formed questions, and generally don't like ill-formed questions.
> In math people like well-formed questions, and generally don't like ill-formed questions.
This is not so simple, though. Ill formed questions can be interesting as a motivation to formalise them (ie make them well-formed) in generalising/abstracting concepts into new concepts. Eg how even/odd has been generalised to transfinite numbers.
Re: Is infinity an odd or even number? (2011)
#290Earlier quoted context omitted.
This is true until you introduce transfinite numbers.
That might be a bit too advanced for a 6 year old perhaps.
On the contrary, it's entirely natural. The technical definition is quite intuitive.
"There are many infinities! The smallest one is bigger than all the counting numbers, so you can't count up to it, but it's out there! We call it omega. You can make bigger infinities too, like omega + 1!"
Kids LOVE that, and it's good math too! (But gets tricky quickly, because addition of transfinite ordinals is not commutative, and standard transfinite ordinals don't allow subtraction)
It's easy to draw as a "number tree" too:
root
/ \
/ \
1,2,3,4... omega, omega+1,...
