- Info box: 30 Deepcold 5382
- Scarlet: Raising a number to the power of another number is like multiplying it by itself the same number of times.
- Julius: Yes, seen it before.
- Equation #1: 34 = 3 x 3 x 3 x 3
- Equation #2: 100100100 = 100100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
- Scarlet: But you know what’s really cool? If you do that just two times for 100, you’ll get a result that‘s so large that it’s impossible to write it down.
- Julius: Wow!
- Scarlet: I wonder if there’s a way to construct even larger numbers. None of the math books I’ve read mentioned anything.
- Julius: Couldn’t you simply add more layers?
- Scarlet: No, I mean a number so big that you couldn’t even theoretically write it down!
- Julius: I don’t think that’s possible.
- Scarlet: I think it is. With some kind of machine.
- Julius: A machine? Something like a mechanical slide rule?
- Scarlet: Not a machine made of steel. Rather some kind of logical apparatus.
- Julius: That sounds totally crazy!
A Sky Full of Stars 048
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23 thoughts on “A Sky Full of Stars 048”
Frith Ra
And she starts programming the computers that she hasn’t invented yet.
HKMaly
So did Ada Lovelace.
Manabi
Maybe she can get enough followers to create a computer out of people, like the Trisolarians did in Lixin Ciu’s Three Body Problem series. That was really cool to read about.
Hegel-Marx
Come on! Is that Scarlet on her way to developing axiomatic set theory, theory of infinite sets and the notions of ordinal and cardinal numbers!? 😮 Generalization of the concept of number to infinite sets, then the realization that there is more than “infinity”, that is, that there is an infinite layering of infinity? A theory that may not be known or forgotten at Lavarel?
Will she need it to understand how the Machine works!?
I would say it’s impossible for a nine-year-old to develop such a theory. But, last year, I would have said it’s impossible for an eighteen-year-old to become world chess champion, and that’s exactly what happened! 😯
OK, maybe it’s toward some theorems from programming theory that I don’t understand (P versus NP problem).
Shankar
In context, “number” here means finite: any kind of infinity (uncountable or otherwise) is trivially too large to write down.
If she’s just inventing the field of googolology, I’d guess something with Knuth’s up-arrow notation, or maybe the Ackermann function, but if she means uncomputability, it might be the “busy beaver sequence.”
Novil
She’s thinking of uncomputable numbers.
Flowers
A lovely article along these lines is “Who can name the bigger number?” by Scott Aaronson. More esoteric but also potentially of interest is “On Random and Hard-to-Describe Numbers” by Charles Bennett.
Hegel-Marx
“Googology”, OK! Thanks guys, I think I heard that term before, but I forgot. I just read a little about it. There you go, comic comics can be educational, too! 😉 And I learned something about axiomatic set theory on my own when I was in high school – half a century ago! 😫
BookWizard
She’s going to both save AND revolutionize the world!
Vicious Sand
Coloring books? Comic books? Nooo, it’s hard math just the way she likes it. It’s a good thing that Julius loves her. Does this count as tutoring?
At least this won’t lead to some sort of Golden Ratio panel-key math equation to unlock the amount of whateranium crystals she’ll need, right?… Right? Math in Sandra and Woo/Gaia comics does tend to end in nuclear explosions. Will we get a callback for a three for three?
MelkiorWiseman
Scarlett would squee with joy if she found out about Graham’s Number and the new notation which had to be implemented in order to be able to represent it.
twink potus
I love how Novil’s magic-tech systems end up so mathematical, this seems to be where the worldbuilding is going again.
In Gaia we had an equivalence between theorem discovering and spellcasting
Kaworu
They don’t have… Computers? 0_o
I’m learning more and more about the setting… 🙂
aaaaaa123456789
The funniest part is that we can show that irrepresentable numbers exist, but, for rather self-evident reasons, we cannot speak of them.
Suppose we use some set of symbols S to talk about math (including everything we use to write, letters, numbers, symbols, any markup needed, whatever). That set is finite. Therefore, for any natural number n, the set of texts of length n or less is finite. Some subset of those is the subset of texts that each describe a single number (such as “6”, “fourteen” or “the largest root of x² – 5x + 3”); that subset must also be finite, and, since the number described is a function of the text, the set of described numbers is also finite. Call that number D(n), for each upper length n.
Now consider some large n, such as 10²⁴. D(10²⁴) is a finite set of numbers. Some of those numbers are integers, and thus, since integers are discrete, there is a maximal integer in D(10²⁴). That value plus one is also an integer, and by definition, not in D(10²⁴).
But we have no way to speak of that value, whatever it is. This text doesn’t do it: if it did, considering this text is definitely shorter than 10²⁴ symbols, the number would be in D(10²⁴)!
Tauben
One consideration I always find fun is that for similar reasons most real numbers, although existent, can’t be proven to exist.
As proves are texts using some finite alphabet of mathematical symbols which are lined up finitely (a proof needs to have a start and an end) there are at most countably infinite valid proves.
Simultanously, we know that there are more than countably infinite real numbers.
Therefore, we know that for most real numbers r the theorem “r exists” can’t be proven, as otherwise we would have more than countably infinite proves, contradicting the former.
That is of course related to your point as the reason these are not provable is simply that we have no means of talking about r to begin with.
HKMaly
You are assuming that single proof only proves existence of single number. That is trivially false: plenty of proofs proves existence of infinite number of numbers at once.
HedAurabesh
There is a difference between (unique) representation and computability. The thought experiment you’ve given has the gap of not proving irrepresentability, as it disregards computation without unique representation.
It is trivially easy to let a finite set of symbols generate an arbitrarily large set of numbers — as long as you allow for computation over time and do not require the chosen number to be selectable (i.e. chosen).
A sentence describing an algorithm of “Let x(n+1) = x(n)+1 and x(0) = 0. Emit each result in whatever form you desire.” will generate (represent) arbitrarily large numbers if you let it run long enough. Any finite integer will eventually be printed by that function — that is eventually it will emit a number > D(n) for any number n.
Now, you are unable to specify WHEN that value will appear. In other words, the number can be represented, but not uniquely so.
Pevel
Hey Oliver hey Elli I would like to ask you few things about Lavarel world
1 from what point they count years, did they count ftom year 0 or 1
2 when foxes and wolves hit their equivment of 18?
Did I correctly understand that foxes live on avarage 60-68 and Wolves live 64-70 and cant be older than 76?
Aui
Ohh is she getting busy with a beaver? 🙂
Tony
I have been led to believe that two beavers are better than one. They´re twice the fun.
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