Simply Beautiful Art's realm of calculus and analysis

While messing around with the idea of ordinal collapsing functions, I stumbled upon an interesting simple function: $$C(0)=\{0,1\}\\C(n+1)=C(n)\cup\{\gamma+\delta:\gamma,\delta\in C(n)\}\\\psi(n)=\min\{k\notin C(n),k>0\}$$ The explanation is simple. We start with $\{0,1\}$ and repeatedly add it...

Leaky Nun

what is sportsmanship? can I go like

Simply Beautiful Art

I wanted to use this function

But I guess its a little bit late for that

Leaky Nun

@SimplyBeautifulArt well that function isn't that fast

Nilknarf

12:18 AM

Did mine pass that?

Leaky Nun

BB(googol), I win

Simply Beautiful Art

No uncomputable functions

Nilknarf

Nope, I've tried that before XD

Leaky Nun

:c

Nilknarf

simply got mad at me

XD

Simply Beautiful Art

12:19 AM

Cuz the game ends really fast that way

Leaky Nun

alright

Simply Beautiful Art

@Nilknarf Play the antimatter dimensions

Leaky Nun

seriously, I need to analyze that function :P

Simply Beautiful Art

Uh

Nilknarf

What is it?

It's taking longer than a second...

Simply Beautiful Art

12:20 AM

His number is $\approx f_{\omega+1}\rm(googol)$

Leaky Nun

well, Phi(G,G) = Phi(Phi(G),G-1) = Phi(Phi(Phi(G)),G-2) = ... = Phi^G(G)

ninja'd

Nilknarf

Correct

Simply Beautiful Art

Well, the point is that if someone makes a new function

You don't go copying it

Nilknarf

Is someone else gonna go?

Leaky Nun

your function is omega+1 in two arguments

you shouldn't have wasted it by providing only two arguments

Nilknarf

12:21 AM

:(

I'm bad at ordinals :P

Simply Beautiful Art

I'll go

Nilknarf

ok

Simply Beautiful Art

$$f_{(a,b)}(n)=\begin{cases}n+1,&a=b=0\\ f_{(a,b-1)}^n(n), &b>0\\ f_{(a-1,n)}(n), &\rm else\end{cases}$$

Leaky Nun

holy god

Nilknarf

Oo

I assume the superscript is iteration, and not exponentiation

Simply Beautiful Art

12:24 AM

My number shall be$$f_{(1,2)}\rm(googol)$$

And yes, iteration

Number is $=f_{\omega+2}\rm(googol)$

Nilknarf

Oof, I don't even know how big that is

Leaky Nun

oh god

Simply Beautiful Art

Hm

Leaky Nun

I'm so uncreative

you guys create functions

@SimplyBeautifulArt even created the fastest function

how do you even create those

Simply Beautiful Art

It is approximately $$\newcommand{glog}{\rm googol}\Phi(\glog,\glog,\glog)$$

Leaky Nun

12:25 AM

I'm just going to go with Goodstein(googol)

Simply Beautiful Art

@LeakyNun Magic and training

Leaky Nun

@Nilknarf did you just create a new benchmark function

Simply Beautiful Art

Hm

googol = 10^100

Leaky Nun

oh, so Phi(G,G,G,...,G) is f_(omega+n-1)(G) where n is the number of arguments

Simply Beautiful Art

≈ 1000^33

≈ 2^330

Nilknarf

12:27 AM

hahaha

Leaky Nun

go on @SimplyBeautifulArt

Simply Beautiful Art

≈ 2^2^8

= 2^2^2^(2+1)

Goodstein(googol) ≈ $f_{ω^{ω^{ω+1}}}(3)$

3 tower's of ω

Leaky Nun

how does that work?

Simply Beautiful Art

You replace the 2's with ω's

Leaky Nun

fair enough

Simply Beautiful Art

12:29 AM

Taking the leading ω^x power

It is approximately f_x(3)

I gotta go eat

Leaky Nun

@Nilknarf your turn :P

Simply Beautiful Art

Go get addicted to antimatter dimensions

Nilknarf

Ugh, I think $\Phi$ is dead

It won't load! D:<

Simply Beautiful Art

:'(

Nilknarf

Okay, I gotta go now

Leaky Nun

12:34 AM

@Nilknarf :'(

@Zacharý welcome!

Zacharý

What is the purpose of this room? Just to converse about stuff for calculation?

Leaky Nun

@Zacharý well I mostly come here to learn about ordinals

Zacharý

Or would you say you ordinal-ily come here to learn about ordinals?

Leaky Nun

stares

@Riker hi

Zacharý

@LeakyNun (& @Riker) And you have a lot to learn, ~~Commander~~ Leaky Nun.

Leaky Nun

12:43 AM

@Zacharý sure

Zacharý

(I have no clue, I had to do a star trek quote)

Simply Beautiful Art

I'm back @LeakyNun

Leaky Nun

@SimplyBeautifulArt welcome back

your turn

Simply Beautiful Art

Gah, welp

Hm

Leaky Nun

@yaa09d hi

Simply Beautiful Art

12:56 AM

I don't have any spectacular notations I can make off the top of my head that surpass ε_0

Oh wait, I do

Oh wait, nvm

Have you been playing antimatter dimensions? @LeakyNun

Leaky Nun

@SimplyBeautifulArt shamefully, yes

Simply Beautiful Art

Hehe

How far did you get?

Leaky Nun

I just reset the game for the second time

so I'm aiming for 6D

Simply Beautiful Art

LOL

I have no life

Leaky Nun

you?

> How much antiwood could an antiwoodchuck chuck if an antiwoodchuck could chuck antiwood?

Simply Beautiful Art

12:58 AM

I've already done 1 antimatter galaxy + 8 resets

Leaky Nun

> The fatter catter satter on the antimatter.

@SimplyBeautifulArt indeed no life you have

Simply Beautiful Art

x'D

Leaky Nun

@SimplyBeautifulArt where did you learn all this ordinal ~~sith~~ rabbit hole

Simply Beautiful Art

idk

Mostly on Discord I think

Zacharý

Well I'm out ...

Simply Beautiful Art

1:00 AM

Cya @Zacharý

Leaky Nun

@SimplyBeautifulArt from whom?

Simply Beautiful Art

Mostly me

Leaky Nun

what

Simply Beautiful Art

Some assistance from others

what what?

Leaky Nun

that's amazing

Simply Beautiful Art

1:01 AM

:P

Should I do a dimension sacrifice?

Leaky Nun

is there an uncomputable countable ordinal?

Simply Beautiful Art

I'm doing it

Leaky Nun

or non-recursive countable ordinal?

Simply Beautiful Art

Yeah

Leaky Nun

gamma?

Simply Beautiful Art

1:02 AM

No, any computable ordinal is countable

Leaky Nun

that isn't incompatible with what I said

Simply Beautiful Art

Gamma ($φ(1,0,x)$) is computable

@LeakyNun But the Church-Kleene ordinal, for example

In fact, any ordinal between the Church-Kleene and $\omega_1$

Leaky Nun

hmm

you're constantly blowing my mind, you know

i study maths in imperial

and you still blow my mind

Simply Beautiful Art

imperial?

:P Well, blowing minds is what I try to do

Leaky Nun

@SimplyBeautifulArt imperial college london

Simply Beautiful Art

1:06 AM

Btw

There's this achievement on antimatter dimensions to reach infinity (a.k.a. 10^308) using only the first dimension and timespeed

Leaky Nun

but why

Simply Beautiful Art

I'm scared

x'D

90 byes of the 8th dimension!

Woot woot!

@LeakyNun and that's cool

Leaky Nun

@SimplyBeautifulArt I mean, groups, topology, cohomology, algebra, they're all cool

but nothing compares to ordinals

it's the most beautiful thing I've ever encountered

Simply Beautiful Art

lol

Leaky Nun

seriously

Simply Beautiful Art

1:09 AM

say

How do you feel about ordinal collapsing functions?

Leaky Nun

what is your statistics?

You have made a total of 39.88 Qt antimatter.
You have done 2 soft resets.
You have 0 Antimatter Galaxies.
You have played for 7 hours, 53 minutes and 50 seconds.
If every antimatter were 65.84 M zeptometers cubed, you would have enough to make a proton.

@SimplyBeautifulArt I haven't read much about them yet

Simply Beautiful Art

lol

You have made a total of 50.48 OQa antimatter.
You have done 9 soft resets.
You have 1 Antimatter Galaxies.
You have played for 19 hours, 44 minutes and 52 seconds.
If every antimatter were a planck volume, you would have enough to make 16.02 M hypergiant stars.

^ 16 M hypergiant stars

Zacharý

... what the heck ... are these actual terms?

Simply Beautiful Art

x'D

@Zacharý ivark.github.io

Get addicted

Leaky Nun

@SimplyBeautifulArt I'm just making protons

you're making stars

Zacharý

1:15 AM

So ... it's cookie clicker without clicking on a cookie.

Leaky Nun

@Zacharý enough with the puns stares

Zacharý

@LeakyNun That is not a pun, it's a comparison to a popular (or at least it WAS) popular, phenomenon.

Leaky Nun

> What do you mean, more than two dimensions??? We're on a screen, clearly there are only 2 dimensions.

@Zacharý what do you mean?

Zacharý

Look up Cookie Clicker.

Leaky Nun

I see

> If you open an anti-lootbox, are you selling random possessions for in-game currency?

> If an alien lands on your front lawn and extends an appendage as a gesture of greeting, before you get friendly, toss it an eightball. If the appendage explodes, then the alien was probably made of antimatter. If not, then you can proceed to take it to your leader. -Neil deGrasse Tyson

Zacharý

1:27 AM

Wouldn't it explode on contact with the air?!

Leaky Nun

> A new chat server has been created for Antimatter people to spy on Matter people, and the world has fallen into chaos and discord

Zacharý

*and it was on Discord, of course

Leaky Nun

> But to an antimatter person, wouldn't they be matter and we antimatter?

Zacharý

WTF is this from?

Simply Beautiful Art

@Zacharý Join the game

Zacharý

1:33 AM

I am on it now

Simply Beautiful Art

> If every antimatter were a planck volume, you would have enough to make 3.25 K Oort clouds.

wtf?

I'm half-way to infinity

Probably means I broke past 10^150 antimatter

> Antimatter is honest, matter makes up everything

Leaky Nun

@SimplyBeautifulArt I reset the third time: now waiting for 7D

Simply Beautiful Art

x'D

Zacharý

How long have you guys been playing for??

Simply Beautiful Art

> You have played for 20 hours, 11 minutes and 40 seconds.

Zacharý

1:38 AM

Where does it say this?????

Simply Beautiful Art

I reset

Under statistics

3 minutes later...

@LeakyNun So much boost lol

Leaky Nun

@SimplyBeautifulArt I'm creating a function that is heavily inspired by binary-Veblen

Simply Beautiful Art

:P

Leaky Nun

one might ask what the point is, to which I reply "I have no idea"

Simply Beautiful Art

same

I reset and I've already got some walls of chinas

Zacharý

1:45 AM

Question: does this use e^t for its algorithm or something?

Simply Beautiful Art

Now making asteroids

@Zacharý probably

Zacharý

#dank-maymays

Leaky Nun

Pattern-matching notation (discard-upon-first-match)
$\delta(0,0,0) = 0$
$\delta(0,y,0) = \sup\{\delta(0,b,0)+1 \mid b < y\}$
$\delta(x,y,0) = \sup\{\delta(a,y,\omega) \mid a < x\}$
$\delta(x,y,z) = \sup\{\delta(x,\delta(x,y,c),0) \mid c < z\}$

Simply Beautiful Art

:P

Leaky Nun

a more rigorous/elegant approach to Veblen function

explore :P @SimplyBeautifulArt

Simply Beautiful Art

1:46 AM

Wait what do you want me to do?

Leaky Nun

wait, there might be some error

let me fix it first

Simply Beautiful Art

I'm making dwarf planets

Leaky Nun

Pattern-matching notation (discard-upon-first-match)
$\delta(0,0,0) = 0$
$\delta(0,y,0) = \sup\{\delta(0,b,0)+1 \mid b < y\}$
$\delta(x,0,0) = \sup\{\delta(a,0,\omega) \mid a < x\}$
$\delta(x,y,0) = \sup\{\delta(a,\delta(a,b,\omega)+1,\omega) \mid a < x, b < y\}$
$\delta(x,y,z) = \sup\{\delta(x,\delta(x,y,c),0) \mid c < z\}$

@SimplyBeautifulArt I want you to explore this function lol

Simply Beautiful Art

:|

Leaky Nun

domain is $(\omega+1) \times \operatorname{On} \times (\omega+1)$ for now

Simply Beautiful Art

1:49 AM

$\delta(0,y,0)=y$

Leaky Nun

@SimplyBeautifulArt what

Simply Beautiful Art

$\delta(x,y,z)=z$ nestings of $\delta(x,\dots,0)$ with initial value of $y$.

$\delta(x,y,0)=\omega2$ nestings of $\delta(a,\dots,0)$ with an initial value of $b<y$ and $a<x$.

> If every antimatter were a planck volume, you would have enough to make 298.74 K Suns.

Leaky Nun

how is sun after star

star is obviously bigger than sun

Simply Beautiful Art

I reset earlier mate

I'm at red giants right now

$\delta(x,0,0)=\omega$ nestings of $\delta(a,\dots,0)$ with $a<x$ and initial value of $0$.

Hey @Secret

:P Still playing antimatter dimensions?

> If every antimatter were a planck volume, you would have enough to make 1.3 hypergiant stars.

$\delta(1,0,0)=\delta(0,0,\omega) → \delta(0,\delta(0,0,n),0) = \delta(0,0,n) → ... =0$

Leaky Nun

oh gg

lemme fix

Simply Beautiful Art

2:02 AM

> None of this matters

@Zacharý How you holding up over there? Lol

> If every antimatter were a planck volume, you would have enough to make 2.2 nebulas.

Enabled cancer notation

You get an achievement for unlocking galaxies with cancer notation enabled

> If every antimatter were a planck volume, you would have enough to make 2.8 Oort clouds.

@AkivaWeinberger Play antimatter dimensions with us

> How does NASA organise a party? They planet.

> If every antimatter were a planck volume, you would have enough to make 3.9 Local Bubbles.

What do you guys think a local bubble is?

oh

> According to no known laws of aviation, there are multiple ways a bee should be able to be swallowed up by antimatter

I haven't seen that one before

Simply Beautiful Art

2:35 AM

> If every antimatter were a planck volume, you would have enough to make 1.01e3 galaxies.

Simply Beautiful Art

2:53 AM

Reset!

Leaky Nun

10:38 AM

$\delta(x,0,0) = 0$
$\delta(0,y,0) = y$
$\delta(x,y,0) = \sup\limits_{a<x\\b<y} \delta(a,\delta(x,b,0)+1,\omega)$
$\delta(x,y,z) = \sup\limits_{c<z} \delta(x,\delta(x,y,c)+1,c)$

@SimplyBeautifulArt $\delta(\omega,2,0)=?$

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$Mathematics$ Simply Beautiful Art's realm of calcu…