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  • Here's something I like to think about: say you want to try to list just all the real

  • numbers between 0 and 1, I mean, you can't actually list them because there's infinitely

  • many, and you can't even make a scheme for listing them because there's uncountably infinitely

  • many (see cantor's diagonal proof), but just 'cause something's a little bit impossible

  • doesn't mean you can't have fun trying it out and learning from all the ways it fails.

  • So, to list all the numbers, maybe you start by making a tree of possibilities: start with

  • 0., then here's all the combinations of the first digit, 0 through 9, and then after 0

  • can come any digit 0-9, and after 1 could come any digit 0-9 and so on, and we'll just

  • exponential tree this number thing and get all possible combinations of digits!

  • I call it: the digitree!

  • No wait, that's the name of a company.

  • Digitry?

  • As in, let us engage in some mathemagical digitry! nope, also a company.

  • Anyway every node on this digit-tree represents a number, here's .75, and .314, and the number

  • of nodes grows really fast but that doesn't change that theoretically we can list the

  • nodes, line them up layer by layer to count everything on the tree, here's the 1st layer,

  • and then there's 10 more in the layer below it and 100 more in the layer below that and

  • then 1000... the numbers get big, but as long as it's finite it can be counted.

  • so as we go infinitely down this tree there's eventually an infinite number of nodes but

  • it's a countable sort of infinity because each node has a counting number that can be

  • assigned to it, and thus we list an infinite number of numbers!

  • So... is every number between 0 and 1 included?

  • I mean, we have all combinations, right?

  • For any number, you just go down the digits until you get to the last one, and that's

  • the node for that number.

  • Which works fine, if there's a finite number of digits and you CAN stop at the last one.

  • Then there's tricky numbers like pi, or I guess pi/10 since we're working between 0

  • and 1, whatever, but you can still use the same scheme... all you have to do is follow

  • the pi path: branch 3, branch 1, branch 4, branch 1, and so on.

  • You'll never get to the last digit, but you can still go down the infinite pi path.

  • In fact, any real number can be thought of as an infinite path going down this infinite

  • tree.

  • .3 repeating, for example, means you take the third branch each time.

  • On your usual finite tree every node has a single path that goes to it, and every path

  • goes to a single node.

  • There's a correspondence between the number of nodes and the number of paths.

  • So at first think, you might assume that's still the case with the infinite tree: that

  • if the number of nodes is countable and every real number is included on this tree, then

  • this shows that the number of real numbers is actually countably infinite!

  • But mathematics is much weirder than that, and just because something's true for all

  • finite cases of things doesn't mean it's true once infinity gets involved.

  • Sure, if your number has a finite number of digits, it has a path along those digits that

  • ends on one of those countably infinite nodes.

  • But for numbers with infinite paths, there is no corresponding node.

  • The pi path can't end on a node before it reaches the last digit of pi, and there is

  • no last digit of pi.

  • The moment your tree becomes infinite, there's more paths than nodes.

  • Like, a lot more.

  • There's a countably infinite number of nodes, and a countably infinite number of branches,

  • but an uncountably infinite number of paths that don't correspond to nodes at all.

  • And this is true not just for this infinitree, oh, there's the word, infinitree, but also

  • other kinds of infinitrees like an infinite binary tree, ternary tree, n-tree, geometree,

  • poetry...

  • So, like, the number of circles in this appellonian gasket is the smallest kind of infinity, a

  • countable infinity, like nodes on a tree.

  • But the number of infinite spirally paths that go along circles down into the infinite

  • depths of this thing is uncountably infinite.

  • If you think that's weird, you should check out the Cantor set.

  • Or if even that's not weird enough for you, throw in a little axiom of choice and then

  • see how super weirdatronical things can get.

  • Anyway what I really like thinking about is: what if you made up some scheme where there

  • is a node corresponding with pi?

  • I don't know exactly how you'd do it yet but y'know, math is all about making stuff up

  • and seeing what happens so why not, if this node were the pi node it must have an infinite

  • path of infinite digits between it and the rest of the tree.

  • The old node listing method assumes every layer of nodes is finite, simple powers of

  • ten, but if you somehow let your number of layers go past any finite number, then your

  • node-counting is going to encounter layers of nodes with infinite nodes, 10 to the power

  • of infinity nodes to be precise, and your counting method will get stuck in the depths

  • of the infinitree.

  • Oh no, turns out infinitree is trademarked.

  • Ugh, see, this is why math is hard.

  • You make up all this cool stuff and then you want to give it cool names but the cool names

  • are taken and then you end up calling things generic and misrepresentative names like "imaginary"

  • numbers and "real" and "complex" and, just, why would you call a type of number that?

  • It makes no sense.

  • Only "infinitree" makes sense.

  • Anyway, the reason I like imagining there's a "pi" node is that then I get to try to imagine:

  • what does pi's neighborhood look like?

  • Like, what number is in the node next to the Pi node?

  • Up here, neighboring nodes have just the last digit changed, but there IS no last digit

  • of pi.

  • Maybe there is no neighboring node, just uncountably many nodes infinitely far apart?

  • Or maybe the neighboring node also equals Pi, maybe kinda like how .9repeating equals

  • 1, like, the neighboring pi node is pi plus .0repeating1, not that .0repeating1 is really

  • a thing, but if it were a thing it would equal zero and so this would be pi too, so maybe

  • there's infinite pi nodes before all those increments of .0repeating1 add up to something

  • measurable and then you get to the other numbers.

  • Maybe.

  • And usually if you go one node up, the number there has one fewer digit, but for pi infinity

  • digits minus one digit is still infinity digits and still exactly Pi, and I guess if you go

  • up an infinite number of branches you'll get ones that have finite numbers of digits, just,

  • there's no step-by-step path that gets you there.

  • Like, this is connected to here through an overly-infinite path, but you can't actually

  • get from one to the other.

  • I guess topologically it would be like, this pi path is an infinite ray going towards infinity

  • digits, but once you make the path include infinity and go beyond it, it's no longer

  • a regular ray, which even though it's infinitely long would still never reach down to here,

  • but a long ray, the "long" in "long ray" being another amusing yet confusing generic technical

  • math term that means, like, really long in the only sense in which "long" can have meaning

  • in topology 'cause like the whole point of topology is that length doesn't matter, but

  • still, couldn't they have called it "double infinilong" or something?

  • Because I feel like "infinilong ray" would better capture the way the ray is so long

  • that even though it's continuous, parts of it aren't actually connected.

  • And by "not connected" I mean "not connectoline-able," a word which would better capture the way

  • that a regular infinite line is too short to connect this node to this one.

  • And what about the layer after this infinitieth layer?

  • Is it still all Pi here in the infinite pi neighborhood, and an infinilong distance to

  • the right is the infinite e neighborhood, and it's all .3 repeating over here, because

  • .3 repeating is still .3 repeating even if you add in a 7 after infinity digits?

  • Or what if we go down to an uncountably infinite number of layers, what sort of numbers would

  • we find there?

  • It's like, it shouldn't make sense, but it kind of does, and probably if we used surreal

  • numbers, by which I mean the all-the-numbers numbers, we could make this infinilong infinitree

  • make both more and less sense, but, like, even regular real numbers, by which I mean

  • infinitable-digity-numbers (as opposed to rational numbers which is so close to being

  • a good name but couldn't they just have called them ratio-able numbers), yeah, real numbers

  • shouldn't make sense in the first place because allowing infinite digits causes such conceptual

  • weirdness, but unfortunately for sanity infinitable-digity numbers like pi just seem to come part and

  • parcel with the universe and all the universe's weird things that make it do universe stuff,

  • like, space and waves and time, none of which make any sense either but there you are.

  • Anyway, just something I like to think about sometimes.

Here's something I like to think about: say you want to try to list just all the real

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B2 中高級

無限樹是超級奇怪的 (Infinite Trees Are Super Weird)

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    林宜悉 發佈於 2021 年 01 月 14 日
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