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.
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
.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,
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
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.
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.