The 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.
x^2 is not exponential; it's quadratic. 2^x is an example of an exponential function.
The parent comment was alluding to the idea of set cardinality (https://en.wikipedia.org/wiki/Cardinality). Two sets have the same cardinality if you can establish a bijection (a one-to-one mapping) between elements of one set and elements of the other. The set of all natural numbers is said to have a "countably infinite" cardinality.
It turns out that for any countably infinite set S, the set S x S is also countably infinite (see Hilbert's hotel: https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...). For example, the set of 2-tuples of natural numbers (1, 1), (1, 2), (1, 3), ..., (2, 1), (2, 2), ... is the same size as the set of natural numbers. So in this sense, "endless*endless = endless". Whereas the set of infinitely-long tuples of natural numbers is "uncountably infinite;" it has a cardinality greater than that of the set of natural numbers. Thus, "you have to go exponential"; i.e. "endless^endless".
Huh. Most of those make sense to me, but infinity == infinity being true definitely feels like risky business. Algebraic limits is full of even some pretty trivial scenarios where infinity divided by a lesser-infinity turns out to be a real number— those cases where the two infinities are definitely not equal to each other.
>>> 1/0
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ZeroDivisionError: division by zero
Still you do have infinity and nan in python - because these are part of the floating point spec.
>>> float('inf') - float('inf')
nan
>>> float('inf') == float('inf')
True
>>> NAN=float('inf') - float('inf')
>>> NAN == NAN
False
However that's not mathematics, it's computers (these are even stranger...)
I have my own little programming language - PYX [1] - and i don't allow this madness (even if it is a violation of the floating point spec ;-)
pyx
> mathconst.Infinity - mathconst.Infinity
Error: results in 'not a number' - that's not allowed here
#(1) mathconst.Infinity - mathconst.Infinity
|....................^
> 1/0
Error: Can't divide by zero
#(1) 1/0
|..^
[1] PYX - https://github.com/MoserMichael/jscriptparse - it's supposed to be an educational programming language, where I am trying to have detailed error messages, my side project.
Dividing by 0 or -0 is a valid floating-point operation because there's an infinity in the number system, and JS uses double precision floating point for all numbers. Python has an integer type and a double type, and division by 0 is disallowed for integers, but okay for doubles.
> those cases where the two infinities are definitely not equal to each other
It's been a while since I was doing this in a classroom, but I feel like those things you're thinking are nonuniform infinities could just as well be thought of as entities comprised of infinity and a (perhaps implied) coefficient. Divide out infinity to reveal the coefficient. (And the same for powers/logs, etc.)
In this model, infinities are indeed uniform (quite similar to a constant), though they are often augmented in any of infinitely many ways.
I can just imagine someone coming up with the infinity symbol, and arguing (as we are here) about whether a circle represents the idea. No, not quite; it needs something more... another circle should suffice, and connect them seamlessly. Yes, yes. This looks much more infinite than a mere circle.
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 language is math.
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.
The first one is basically "Fear and Loathing on the Campaign Trail '72" for anthro majors, and then he gets less focused somehow. He'd be my personal Kilgore Trout if we didn't have contemporary science fiction.
> 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.
If you try hard enough, you can find similar questions, that do type-check. You can talk with 6yo children about them if you want. Still, I stand with my answer. I would say this (also I think this is the best thing to say/I am capable of).
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)
Terminology and strict definitions aside, it is quite intuitive. A commenter below also pointers this out:
> If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).
The only thing that is a little off here to me is that I don't think there is mathematical notation for "many" or "few". And yet infinity does have mathematical notation and is used in some equations, no?
It's an analogy, meant to show the similarities between two things in a limited way, to illustrate an idea. They do not have to be exactly the same in every way.
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.