## Quick ordinal notation

Well, I somehow came up with an ordinal notation based off my previous unsimplifcation of Rathjen's big Psi while eating breakfast. I'm going to type it up quickly so I don't forget. Might contain typos or misrememberings.

Symbols: Binary Ξ, ternary Ψ, multi-ary +, constants 0 and K

And from here we enter There Is Only One Rule. This will be in future "OCF"s based on one and only one unchangeable rule.

G(α,β) is an abbreviation of "largest ordinal α uses except below β"

(ordinal more like Ordinal() which is the structures we use in the ordinal notation)

(If more than one case is matched, the one listed first has higher priority.)

if α

## Additive inaccessible OCF but unsus

Simultaneously define \(I_\alpha: \mathrm{Ord} \to \mathrm{Ord}\) and \(C_\alpha \subset \mathrm{Ord}\) for \(\alpha \in \mathrm{Ord}\) via the following transfinite recursion:

- \(C_\alpha = \{\kappa: \mathrm{cof}(\kappa) = \kappa \land \forall \beta \in \alpha (I_\beta(\kappa) = \kappa)\}\).
- \(I_\alpha(\beta) = \min\{\kappa: (\kappa \in C_\alpha \lor \kappa = \sup(C_\alpha \cap \kappa)) \land \forall \gamma \in \beta (I_\alpha(\gamma) \in \kappa)\}\).

Define \(\mathrm{IS}: \mathrm{Ord}^2 \times \mathbb{N} \to \mathrm{Ord}\) via the following transfinite recursion:

- If \(\gamma = 0\), \(\mathrm{IS}(\alpha, \beta, \gamma) = \alpha\).
- If \(\gamma = \delta + 1\) for some \(\delta \in \omega\), \(\mathrm{IS}(\alpha, \beta, \gamma) = \min\{\kappa \in …

## Should ThaAwesome's notations and numbers be removed from mainspace?

My reasoning is that it's very clearly ill-defined from a source which is neither reputable nor notable, and which was created as a response to deletion from sourcing from a blog post.

I'll also propose something more radical; that we should possibly change our guidelines for deletion or sources, so stuff like this doesn't end up on the wiki.

## Star Notation

Star Notation is a notation made by User:I AmNow.

\(a\star = (10 \uparrow\uparrow (a+1))^{100}\)

\(a\star_b = (\cdots((a\underbrace{\star)\star)\cdots)\star}_{b+1}\)

\(a\star_{b, c} = (\cdots((a\underbrace{\star_b)\star_b)\cdots)\star_b}_{c+1}\)

\(a\star_{\#b} = a\star_{\overbrace{a\star_{a\star_{\ddots}}}^{b+1}}\)

\(a\star_{\#b,c} = a\star_{\overbrace{b,a\star_{b,a\star_{\ddots}}}^{c+1}}\)

## A new paradigm of ordinal notations

A "determinacy-style ordinal notation" is a new paradigm of ordinal notations. They're called that since I constructed one which I guess reaches \(\mathrm{PTO}(\textsf{ZF} + \textsf{AD})\), believe it or not. However, with power comes complexity, so there's a lot of properties which need verifying for it to be a "proper" determinacy-style ON. In the following, I walk through the layout of a determinacy ordinal notation. If P進大好きbot sees this, I'll be very excited to see what they make!

- 1 Step 1
- 2 Step 2
- 3 Step 3
- 4 Step 4
- 5 Step 5

Construct a set \(\mathcal{T}\) of terms in the ordinal notation. They can be whatever you want! For example:

- Formal strings
- Finite sequences of natural numbers
- Natural numbers

Construct a binary relation \(\rightarrow^{\#}\) on…

## Bad OCFs I came up with

**[redacted]**

All ψs in this post are Multivariable Buchholz unless otherwise specified.

(ok, this one might not be that bad)

ψ_{α}(β) is the smallest ordinal not in
*C*_{α}(β)

\(\Omega_\beta \subseteq C_\alpha(\beta)\)

\(\forall\xi,\upsilon\in C_\alpha(\beta):\xi+\upsilon,\xi^\upsilon\in C_\alpha(\beta)\)

\(\forall\xi,\upsilon\in C_\alpha(\beta)\cap\beta:\psi_\xi(\upsilon)\in C_\alpha(\beta)\)

\(\forall S\subset C_\alpha(\beta):S\text{ does not satisfy the conditions above}\)

Ω_{0}=1

χ(α) is the smallest ordinal not in *C*(α)

0∈*C*(α)

∀ξ,υ∈*C*(α): ξ + υ, φ_{ξ}(υ),

Ω_{ξ}⊂*C*(α)

∀*S*∈*C*(α): *S* does not satisfy the conditions
above

…

## A definition of my array notation

Here are the rules of TPAN (Trial Purple Array Notation)

All \(\psi\)s are Extended Buchholz in my blog posts, unless otherwise specified.

__Warning: The following is informal and probably
ill-defined.__

An array is
{*a*_{0}*s*_{0}*a*_{1}*s*_{1}...*s _{k}*

_{-1}

*a*}, where the

_{k}*a*are either natural numbers, arrays, or sums of arrays. The

_{i}*s*are separators, which are explained below.

_{i}A separator is [*q*;^{k}], where *q* is
a countable or uncountable ordinal, expressed in the array
notation. If *k* is bigger than ω, write
[*q*/*k*`;].

## Addition and Sequences: Rayo string for n(3)

In this blog post, I will write a Rayo string for n(3). I chose this because it seems that it would take the least symbols out of all the combinatorial functions.

I must have screwed up somewhere. Can someone please check my work?

- 1 Idea
- 2 Letters
- 3 Addition
- 4 Sequence
- 5 Subsequence
- 6 \(n(k)\)
- 7 Conclusion

Ordered pairs are hard to work with, and ordered n-tuples are just hell. Therefore, to represent the Friedman strings, we must not use ordered n-tuples. Instead, we do something like this:

\[F=\{\{x_1,1\},\{x_2,2\},\{x_3,3\},\dots\}\]

To ensure there is no confusion, we must make the letters of the alphabet correspond to non-ordinals. The simplest choice is just to let \(A=\{1\}\), \(B=\{A\}\), and \(C=\{B\}\).

Once we define the constants, we can just ref…

## Attempt 2 at an additive inaccessible OCF

Due to the glaring flaws in my original attempt, I have made a new post. It seems to work similarly to multivariable Buchholz psi, diagonalizing on the subscript of Buchholz psi, with \(\psi_I\) not enumerating fixed points of \(\Omega_\alpha\) but rather additive principals but I'm not entirely sure.

Define \(\Omega_\alpha\) as follows:

- \(\Omega_0=1\)
- \(\Omega_{\alpha+1}=\text{next uncountable regular after }\Omega_\alpha\)
- \(\Omega_{\alpha}=\lim\{\Omega_\beta|\beta

## Extensible Illion System and my old -illions have been replaced

https://integralview.wordpress.com/2022/05/15/zillion-notation

## Ubersketch attempts to make an additive inaccessible OCF

Since there aren't many additive inaccessible OCFs, I will attempt to make one. I don't know much about inaccessible OCFs so this might be horribly broken.

Define \(\Omega_\nu\) as follows:

- \(\Omega_0=1\)
- \(\Omega_{\nu+1}=\text{next regular after }\Omega_\nu\)
- \(\Omega_{\nu}=\lim\{\Omega_\mu\}\) where \(\mu>\nu\) if \(\nu\) is a limit ordinal

Define \(I_\alpha\) as the \(\alpha\)th weakly inaccessible.

Define \(\mho_\nu\) as follows:

- \(O_\nu^0=1\)
- \(O_\nu^{n+1}=\{\alpha|\alpha

## Small improvements to bounds on values of Rayo's function

Almost two years ago, i posted a bound for values of Rayo's function: Rayo(266+20n) > 2↑↑n. Since then, not much has happened. I searched for "Rayo" in blog posts and couldn't find any better bounds. Also, if i remember correctly, the smallest n for which it's known that Rayo(n) > 2 was 59 until now. Although the improvements i found this week are small and kinda trivial, i think they are worth posting after such inactivity in this area.

Of course, Rayo's function is model-dependent, so i should say that in this blog post, i'm working in some "ideal" model, which is undefined but we mostly agree on these basic properties of it (modeling ZF, or even KP, should be enough for these improvements to work).

The bound i posted can be reduced to Ray…

## Is there a recursive analogue for cofinality?

Since there are recursive analogues of inaccessibility, regularity, and so on, one would expect for there to be a recursive analogue of cofinality but so far I've only found weird definition which have strange consequences like \(\text{cof}(\omega_1^\text{CK}+\omega_1^\text{CK})=\omega\).

## Low-level Rayo bounds

Last year, there was a lot of research on how to represent 65536 in the Rayo function, and Psi also calculated with proof some lower values of the function. However, I have noticed that there is no research on how to represent the smaller numbers. So here I will try to provide Rayo strings for arbitrary \(n\). Feedback is encouraged!

By von Neumann's definition, \(S(n)=n\cup\{n\}\). Which means that the predicate \(\text{IsSn}(x)\) should say something like \[n\in x\wedge n\subset x\wedge\neg\exists y(y\in x\wedge y\ne n\wedge y\notin n)\] Therefore, the corresponding Rayo string is \[\exists n((\text{Is}n(n)\wedge((n\in x\wedge (\neg\exists y((y\in n\wedge(\neg y\in x)))))\wedge\neg\exists y((y\in x\wedge((\neg y=n)\wedge(\neg y\in n))))))…

## Explanation of Buchholz's psi for beginners

Ordinal collapsing functions and their associated ordinal notations are important and elementary concepts in googology. Despite this, most explanations of OCFs are confusing and not so illuminating. To rectify this, I have written this blog post on Buchholz's psi, which is the most commonly used OCF in googology. This post will assume prior basic knowledge on ordinals in googology up to \(\varepsilon_0\).

Let us introduce a function \(\psi_0\). \(\psi_0(\alpha)\), then, constructs a set \(C_0(\alpha)\) of all ordinals constructible from finite additions, and finite applications of \(\psi_0\) on 0, as long as the input of \(\psi_0\) remains lesser than \(alpha\) itself, and outputs the smallest ordinal larger than anything in \(C_0(\alpha)\)…

## Redirects

Hello, BlankEntity here.

I believe we should actively put redirects in categories just like non-redirects (for example, with this edit).

I don't really have anything more to add to this.

## On club sets and normal functions

In googology, there are two notions, club sets, and normal functions which seem disparate but are actually deeply related in a way that I will go over in a moment.

A club set \(S\) of an ordinal \(\alpha\) is a subset of \(\alpha\) (using the von Neumann definition of an ordinal) such that...

- For every ordinal \(\beta\) lesser than \(\alpha\), there exists a \(\gamma\) in \(S\) such that \(\gamma\) is greater than or equal to \(\beta\).
- If the least ordinal \(\beta\) greater than anything in some subset \(T\) of \(S\) is lesser than \(\alpha\), then \(\beta\) is in \(S\).

You might be familiar with club sets from the definition of a weakly Mahlo cardinal.

A function normal in \(\alpha\), \(f\) is a function \(f:\alpha\rightarrow\alpha\) such t…

## Mirrors

One thing that bugs me (as evidenced by my only contribution here so far being the Weak Entropic Function and Strong Entropic Function) is people not understanding the whole "it doesn't count if you say 'bignumberillion plus one'" thing. But I don't have a word to explain it succinctly.

I propose using a term like either **Gentleman's Mirror**
(referring to the "gentleman's agreement" in the Big Number Duel to
not use any previously-established methods of increase), **Berry
Mirror** (referring to the Berry Paradox, which is the type of
fallacy these maneuvers are), or **Subthetoid** (referring to θ
as a replacement-inaccessible cardinal, as these maneuvers are just
the Replacement Axiom) to refer to numbers that are functionally
identical even if their valu…

## A fast-growing function diagonalizing proofs in a strong theory

We define a highly expressive theory \(\textsf{EST}\), and give a very powerful fast-growing function.

- 1 Language
- 1.1 Alphabet
- 1.2 Predicates
- 1.3 Free Variables & well-formed formulas
- 1.4 Abbreviations

- 2 Axioms
- 3 Fast-growing function
- 4 Questions

- Finitely many predicate bound variables \(\phi_1, \phi_2, \phi_3, \cdots\)
- Finitely many free variables \(x_1, x_2, x_3, \cdots\)
- Constant symbol \(\bot\)
- A function symbol from predicates to predicates \(\omega_\bullet\).
- Binary relation between predicates \(=\)
- Truth of a predicate \(\phi_n(\bullet)\)
- Unary function symbol from predicates to predicates \(S(\bullet)\)
- Binary function symbol from two predicates to one predicate \(\bullet_1 + \bullet_2\)
- Logical connectives \(\neg, \land, \lor, \implies, \iff\)
- Quantifiers …

## Starting admin noticeboard

I am announced to create an admin noticeboard using Wikipedia-based styles as I seen here, since I will keep track of incidents and for posting some information and issues of interest to administrators.

## 800th tier 3 illion is yoot not yot

I saw that Jonathan Bowers coin the name yootillion instead of yotillion for the 800th tier 3 -illion, as I see that there is at hatnote that distinguishes the name for that article. Why?

## The Ill III Tale

Ill III was the only son of a noble family. He invented a large number... But, his fiancée says, "Your number is ill-defined!" The son sets out on a journey to deny it.

## Strong Subscript Notation

**Strong Subscript Notation** is a googological notation.

x_{y} = x*10^{y}

**Rule 1.**If the Subscript value is 0, x is the value of result.

**Rule 2.**If the x value is 0, the result is 0.**Rule 3.**If the x value is x*10^{n}, x/10^{n}is the value of x/10^{n}+y.

1_{100} is googol.

1_1_100 is googolplex.

Note: you can make this but you gotta do not in blog post and include the original definition.

## Strong 3-var ψ function

Japanese version: https://googology.fandom.com/ja/wiki/%E3%83%A6%E3%83%BC%E3%82%B6%E3%83%BC%E3%83%96%E3%83%AD%E3%82%B0:Kanrokoti/%E5%BC%B73%E5%A4%89%E6%95%B0%CF%88%E9%96%A2%E6%95%B0

- 1 Overview
- 2 Strong 3-var ψ
function
- 2.1 Notation
- 2.2 Ordering
- 2.3 Cofinality
- 2.4 Fundamental sequence
- 2.5 FGH
- 2.6 Limit function
- 2.7 Standard form
- 2.8 Naming

- 3 Stronger 3-var ψ
function
- 3.1 Notation
- 3.2 Ordering
- 3.3 Cofinality
- 3.4 Fundamental sequence
- 3.5 FGH
- 3.6 Limit function
- 3.7 Standard form
- 3.8 Naming

- 4 Even Stronger 3-var ψ
function
- 4.1 Notation
- 4.2 Ordering
- 4.3 Cofinality
- 4.4 Fundamental sequence
- 4.5 FGH
- 4.6 Limit function
- 4.7 Standard form
- 4.8 Naming

We define Strong 3-var ψ function. Strong 3-var ψ function has improved fundamental sequence compared to Iψ function, which has improved cofinality and fundamen…

## Nerfing block duration of the 3-out rule

Update: 5/7/2022: Fixed some mistakes

I found out that under the section "Googology Wiki:Policy#How to
warn users", In that case, the admin will check the correctness of
the sources in the format. If the sources of three or more of them
actually show distinct violations, then the admin will block the
user at least three months for the first time, and at least
__ 99 years for the second time__. At one part in that
paragraph in the policy shows that the term "99 years" is
nonstandard block duration and can confuse with "indefinite
blocks", hence I will nerf that rule further under his blog post
minorly. In addition, the 3-out rule occurs very infrequently.

After that, I am going to nerf it due after this blog post, to at least three years for the se…

## A neat little recursion on FGH

In the following, a "function" is a map of type \(\mathbb{N} \to \mathbb{N}\), and we use \(\mathfrak{F}\) to denote the set of functions.

We define a map \(\begin{array}{ccc} f: \mathbb{N}^3 \times \mathfrak{F} & \to & \mathbb{N} \\ (s,m,n,x) & \mapsto & f_s^m(x,n) \end{array}\) like so:

- If \(m = 0\), \(f_s^m(x,n) = n\).
- If \(m = 1\):
- If \(s = 0\), \(f_s^m(x,n) = x(n)\).
- Else, \(f_s^m(x,n) = f_{s-1}^n(x,n)\).

- Else, \(f_s^m(x,n) = f_s^1(x,f_s^{m-1}(x,n))\).

We define a map \(\begin{array}{ccc} g: \mathbb{N}^2 & \to & \mathfrak{F} \\ (n,l) & \mapsto & h_l(n) \end{array}\) like so:

- If \(l = 1\), \(h_l(n)\) is the map \(k \mapsto k+1\).
- If \(l > 1\), \(h_l(n) = \{\begin{array}{ll} h_{l-1}(n) & \textrm{ if } n = 0 \\ p \mapsto f_p^1(h_{l-1}(n),p) & \te…

## Gamma one

I have found that the ordinal \(\Gamma_1\) satisfies the fundamental sequences of \(\{\Gamma_0+1,\varphi(\Gamma_0+1,0),\varphi(\varphi(\Gamma_0+1,0),0),\varphi(\varphi(\varphi(\Gamma_0+1,0),0),0),...\}\)? And I have found that iterating \(\varphi(\varphi(...,0),\Gamma_0+1)\) do not reach \(\Gamma_1\), but \(\varphi(\Gamma_0,1)\). Why the heck it was be that there!?!

## DLMAN linear series

Sneak peak: DeepLineMadom's Array Notation linear array series numbers (sources are yet to be added to avoid spamming manner)

- 1 Megaoogol
- 2 Megaoogol-un-boogol
- 3 Megaoogol-un-troogol
- 4 Megaoogola-kilaoogol
- 5 Megaoogola-kilaboogol
- 6 Megaoogola-trookilaoogol
- 7 Megaoogola-quadookilaoogol
- 8 Troomegaoogol
- 9 Troomegaoogol-un-boogol
- 10 Troomegaoogola-kilaoogol
- 11 Quadoomegaoogol
- 12 Quidoomegaoogol
- 13 Sidoomegaoogol
- 14 Gigaoogol
- 15 Gigaoogola-un-boogol
- 16 Gigaoogola-kilaoogol
- 17 Gigaoogola-megaoogol
- 18 Troogigaoogol
- 19 Quadoogigaoogol
- 20 Teraoogol
- 21 Petaoogol
- 22 Extaoogol
- 23 Eptaoogol
- 24 Ocaoogol
- 25 Enaoogol
- 26 Dakoogol
- 27 Hendoogol
- 28 Dokoogol
- 29 Tradakoogol
- 30 Tedakoogol
- 31 Pedakoogol
- 32 Exdakoogol
- 33 Epdakoogol
- 34 Ocdakoogol
- 35 Endakoogol
- 36 Ikoogol
- 37 Trakoogol
- 38 Tekoogol
- 39 Pekoogol
- 40 Extakoogol
- 41 Eptakoogol
- 42 Octakoogo…

## DLMAN 2-entry series

Sneak peak: DeepLineMadom's Array Notation 2-entry array series numbers (sources are yet to be added to avoid spamming manner)

- 1 Kilaoogol
- 2 Kilaboogol
- 3 Kilatroogol
- 4 Kilaquadoogol
- 5 Kilaquidoogol
- 6 Kilasidoogol
- 7 Kilaseptidoogol
- 8 Kilaoctidoogol
- 9 Kilanonidoogol
- 10 Kiladecidoogol
- 11 Trookilaoogol
- 12 Trookilaboogol
- 13 Trookilatroogol
- 14 Trookilaquadoogol
- 15 Quadookilaoogol
- 16 Quadookilaboogol
- 17 Quidookilaoogol
- 18 Sidookilaoogol
- 19 Septidookilaoogol
- 20 Octidookilaoogol
- 21 Nonidookilaoogol
- 22 Decidookilaoogol
- 23 Vigintidookilaoogol

**Kilaoogol** is equal to 10[1,2]100 = 10[100]10 in
DeepLineMadom's Array Notation. The number is slightly smaller than
boogol and gugold.

**Kilaboogol** is equal to 10[2,2]100 =
10[1,2]10[1,2]...[1,2]10[1,2]10 (with 100 10's) in DeepLineMadom's
Array Notation. The nu…

## Ultra-Googolplexian-Ultimate-Infinity

Ultra-Googolplexin-Ultimate-Infinity is type of math errors or
something but really large ones typically like this *f*
(*x*)=3*x*,*g*(*x*)=*x*-1 ⇒(*f* ∘
*g*)(*x*)=3(*x*-1)

you see there is not wrong with it but adding a divison will make to this really (REALLY) big number since it can be formed into like this ∰ (x2 + y2 + z2) dx dy dz or 100^^75 googolplexianity

## The Worst OCF ever made?

I don't actually understand OCFs, which will make this attempt at one that is intentionally weak funnier.

a and b are assumed to be natural numbers.

2 the omega fixed point rule: if there is an omega fixed point, replace it with Ω_Ω_Ω... with a y long chain of Ω if the omega fixed point is in x, an x long chain of Ω if the omega fixed point is in y, and make them both 1000 long if they are both.

this should go to 2 entry veblen if I'm not wrong, which... is really weak for an OCF. This was just made as a joke, so yeah, I probably did everything wrong, but it was fun to make

updated so omega fixed points would work

## My list of illions 1

Source: Pointless Googolplex Stuffs - My list of -illions

- 1 Tier 4 - Hypercelestial
- 2 Tier 5 - Greek quantitive prefixes
- 3 Tier 6 - Colorize
- 4 Tier 7 - Elemental
- 5 Tier 8 - Western Offshores & Western Europe
- 6 Tier 9 - Central and Eastern Europe
- 7 Tier 10 - Greater Middle East

Partially inspired by CompactStar's -illions (formerly known as Nirvana Supermind, with some modifications)

- Metillion = 10^(3*10^(3*10^(3*10^45))+3)
- Xevillion = 10^(3*10^(3*10^(3*10^48))+3)
- Hypillion = 10^(3*10^(3*10^(3*10^51))+3)
- Omnillion = 10^(3*10^(3*10^(3*10^54))+3)
- Outillion = 10^(3*10^(3*10^(3*10^57))+3)
- Barrillion = 10^(3*10^(3*10^(3*10^60))+3)
- Barrekalillion = 10^(3*10^(3*10^(3*10^63))+3)
- Barremejillion = 10^(3*10^(3*10^(3*10^66))+3)
- Barregijillion = 10^(3*10^(3*10^(3*10^69))+3)
- Bar…

## my -illion system

- 1 million to nonillion
- 2 decillion to nonadecillon
- 3 mamillion to nonanonillion
- 4 that is the end of my -illion system

i just use normal short scale numbers

add the m,b,quadr,quin,sex,sept,oct,non prefixes and add an "a" at the end and then the "decillion"

add the m,b,quadr,quin,sex,sept,oct,non prefixes and add an "a" at the end and then another the m,b,quadr,quin,sex,sept,oct,non prefixes

## Lightning cardinals: fixed

This is a very slight fix of my blog post regarding the lightning cardinal axioms. Note that they used to be called continuum cardinal axioms, but that caused confusion with \(\mathfrak{c}\) etc.

- 1 Cardinals
- 2 Axioms
- 2.1 Main axioms
- 2.2 Weakenings

- 3 Implication diagram
- 4 Results

For a cardinal \(\kappa\) and an ordinal \(\alpha\), let \(\kappa^{+^0} = \kappa\), \(\kappa^{+^{\alpha+1}} = (\kappa^{+^\alpha})^+\) and \(\alpha \in \text{Lim} \Rightarrow \kappa^{+^\alpha} = \sup_{\beta \in \alpha} \kappa^{+^\beta}\). It's easy to see that \(\aleph_\alpha = \aleph_0^{+^{\alpha}}\).

A cardinal \(\kappa\) is lightning if \(\beth_{\kappa+1} = \aleph_{\kappa^+}\).

A cardinal \(\kappa\) is weakly lightning if \(\beth_{\kappa+1} \geq \aleph_{\kappa^+}\).

For an ordin…

## Iψ function

Japanese version: https://googology.fandom.com/ja/wiki/%E3%83%A6%E3%83%BC%E3%82%B6%E3%83%BC%E3%83%96%E3%83%AD%E3%82%B0:Kanrokoti/I%CF%88%E9%96%A2%E6%95%B0

- 1 Overview
- 2 Iψ function
- 2.1 Notation
- 2.2 Ordering
- 2.3 Cofinality
- 2.4 Fundamental sequence
- 2.5 FGH
- 2.6 Limit function
- 2.7 Standard form
- 2.8 Naming

We define Iψ function. Iψ function has improved cofinality and fundamental sequence compared to 3-var ψ, which extended Ordinal Notation Associated to Extended Buchholz's OCF to 3 variables. I referred to 三関数 (三 function) defined by p進大好きbot in order to write the definition.

Iψ function calculator by Naruyoko

I put two examples of expansion which are different from 3-var ψ. The above expansion is by 3-var ψ, while the below one is by Iψ function.

- \(\psi_0(0,\psi_1(1,…

## ThaAwesome's Array Notation

a[b]c = a ↑ᵇ c

Example: 3[3,0]3 = 3↑↑↑3

a[b,1]c = a followed by a[b]c up arrows to c

a[b,2]c = a followed by a[b,1]c up arrows to c

a[b,3]c = a followed by a[b,2]c up arrows to c

a[b,n]c = a followed by a[b,n - 1]c up arrows to c

Graham's Number = 3[4,63]3

a[b,n,1]c = a[b,x]c, where x is a[b,n]c

a[b,n,2]c = a[b,x]c, where x is a[b,n,1]c

a[b,n,y]c = a[b,x]c, where x is a[b,n,y - 1]c

a[b,n,y,d]c = a[b,n,x]c, where x is a[b,n,y,d - 1]c

a[b,n,y,d,e] = a[b,n,d,x]c, where x is a[b,n,y,d,e - 1]c

abc = a[b,b,b,b,b,...b,b]c, with b numbers of b's

a[[[b]]]c = ab,b,b,b,b,..b,bc with b numbers of b's

a[b/0]c = a[[[[...b]...]]c with b numbers of brackets

a[b/1]c = a[[[...[b,b,b,b,b,...b]]...]]c with b number of brackets & b's

a[b/2]c = a[[[...b,b,b,b,b,...b]...]]c w…

## Large cardinals in ordinal analysis

A cardinal \(\kappa\) is called regular iff, for every function \(f: \kappa \to \kappa\), there is some ordinal \(\lambda < \kappa\) such that, for every \(\mu < \lambda\), \(f(\mu) < \lambda\).

A class of ordinals \(X\) is called unbounded in an ordinal \(\alpha\) iff, for every \(\beta < \alpha\), there is some \(\gamma < \alpha\) such that \(\beta < \gamma\) and \(\gamma \in X\).

A class of ordinals \(X\) is called closed in an ordinal \(\alpha\) iff, for every \(\beta < \alpha\) such that \(X\) is unbounded in \(\beta\), \(\beta \in X\).

A formula \(\varphi\) is called \(\Pi_n\) for a natural number \(n\) iff it is of the form \(\forall x_1 \exists x_2 \ldots Q x_n \varphi(x_1, x_2, \ldots, x_n)\), where \(\varphi\) contains no unbounded …

## Analysis of 1st system Taranovsky's Ordinal Notation

(Main Ordinal Notation System)

1st row - Cantor's notation or Veblen's notation or Buchholz's
notation

2nd row - TON

3rd row - TON in prefix form

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238…

## Naming every -illion between 1,000 and 10^1,000!

please argue with me

this is short scale, and basically all names are moved back one, with some edits mostly to make pronunciation easier. wip so don't comment until finished.

10^3: Unillion

10^6: Billion

10^9: Trillion

10^12: Quadrillion

10^15: Quintillion

10^18: Sextillion

10^21: Septillion

10^24: Octillion

10^27: Nonillion

10^30: Decillion

10^33: Undecillion

10^36: Bidecillion

10^39: Tridecillion

10^42: Quadecillion

10^45: Quindecillion

10^48: Sexdecillion

10^51: Septemdecillion

10^54: Octodecillion

10^57: Novemdecillion

10^60: Vigintillion

10^63: Unvigintillion

10^66: Bivigintillion

10^69: Trivigintillion

10^72: Quavigintillion

10^75: Quivigintillion

10^78: Sesvigintillion

10^81: Septevigintillion

10^84: Octovigintillion

10^87: Novenvigintillion

10^90: Trigintillion …

## The jumping ball function

- 1 Introduction
- 2 How does the JBF
works?
- 2.1 Why the 1st frame doesnt generates new balls?
- 2.2 How i could get the amount of balls per second?

The jumping ball function, aka TJBF for short, is a function made by Idontknow350. Its concept is very simple, yet it can create bestial numbers. Heres the explanation:

See the infobox GIF. It shows a ball jumping (Thats why the name exists lol). Imagine that in every frame excluding the 1st one the same GIF duplicates and adds another video that does the same: Add more videos. Everytime, more and more jumping balls are coming, and if you ask, the GIF is looped, so the frames never ends.

Because it could generate infinite balls in less than a plank time, and we dont want a overloading that fast.

The formula is…

## Announcement

I'm going to learn how to do ordinal analysis :)

After that, I'm probably going to completely overestimate my intelligence and try to analyse something stronger than \(\textbf{KP} + \Pi_1 \textrm{-coll}\), give up and cry, then come back, do it and get a Nobel peace price.

(OK, that last part was a joke but the first part was true.)

## An LPrSS extension extension

An expression in NLPrSS are sequences defined recursively as \(()\),\(\Omega_n\) for an arbitrary natural number \(n\), or a sequence thereof.

\(\&\) denotes concatenation, \(\#_n\) denotes a sequence variable where \(n\) is an arbitrary natural number (\(n\) may be omitted if there is only one).

Let \(S\) be an NLPrSS expression. Define recursively a function \(\mathrm{cof}\) taking an NLPrSS expression and giving an NLPrSS expression as follows:

- Suppose \(S=()\)
- \(\mathrm{cof}(S)=()\)

- Suppose \(S=\#\&()\)
- \(\mathrm{cof}(S)=(())\)

- Suppose \(S=\#_0\&(\#_1\&())\)
- \(\mathrm{cof}(S)=((),(()))\)

W.I.P.

## What if numbers had loops?

In this alternate universe numbers have loops that are infinitely bigger. Also here infinity does not exist.

So let's start with Loop 1.

In Loop 1 numbers start with "Big".

So it's Big One, Big Two, Big Three...Big Oblivion, Big Utter Oblivion and Big Sam's Number.

And the numbers are written like L1(1), L1(2), L1(3)...L1(Oblivion), L1(Utter Oblivion) and L1(Sam's Number).

And next is Loop 2, and the numbers start with "Huge".

So it's Huge One, Huge Two, Huge Three... and so on.

And like Loop 1, the numbers are written like L2(1), L2(2), L2(3)... and so on.

And theres Loop 3 which starts with "Giant", Loop 4 which starts with "Mega" and Loop 5 which starts with "Grand".

Then theres Loop 6 which starts with "Grand Big", Loop 7 which starts with "Gra…

## An LPrSS extension

An expression in ExLPrSS is composed of a finite hereditary sequence, that is, a sequence which can be empty, or a sequence of other finite hereditary sequences.

Let \(S\) denote an expression in ExLPrSS, \(\&\) denote concatenation, \(T\) denote the last entry of \(S\), \(b\) denote the rightmost entry of \(S\) which is lesser than the last entry of \(S\), \(\#\) denote an arbitrary sequence, \(Z^n\) denote a sequence of \(n\) \(()\)s where \(n\) is an arbitrary non-zero natural, \(B\) denote the subsequence of \(S\) which is composed of \(b\) and everything to the right of \(b\) except for the last entry, \(G\) denote the subsequence of \(B\) which is the complement of \(B\) with the last column removed, and \(+^n\) denote \(n\) concaten…

## Definition of C sequence

The form of C sequence is as follows: C(a1, a2, a3,... an), where 'a1' is the first item, 'an' is the last item, each item is separated by commasand, and the values of all items are positive integers.

Parent: To an item, the item nearest before it and less than it is its parent. The first item has no parent, other items may also have no parent.

(If a C sequence is considered as the order-difference-sequence of another C sequence, the parent of the item in the order-difference-sequence cannot be more right than the parent of the original sequence. Therefore, the parent of an item in the order-difference-sequence must meet three points: 1. It is smaller than the item, 2. It cannot be more right than the parent of the item in the original sequ…

## Ordinal Notation associated to the Successor OCF based on Weakly Compact Cardinals

Cute cake OCF! Time for an ON. Note that all the following definitions are simultaneous. \(\newcommand{\cake}{\textrm{🍰}} \newcommand{\altcake}{\textrm{🎂}} \newcommand{\othercake}{\textrm{🥮}} \newcommand{\pancakes}{\textrm{🥞}}\)

- 1 Notation
- 2 Check functions
- 3 Comparison
- 4 Standardness

Let \(\mathcal{L}_\cake\) denote a formal alphabet consisting of the symbols \(0\), \(\cake\), \(\altcake\), \(\othercake\), \(\pancakes\), parentheses and comma. We define sets \(RT_\cake \subset LT_\cake \subset T_\cake\) of formal strings over \(\mathcal{L}_\cake\) in the following recursive way:

- \(0 \in T_\cake\).
- \(\cake \in T_\cake \cap LT_\cake \cap RT_\cake\).
- For \(s \in T_\cake\), \(\pancakes(s) \in T_\cake\).
- For \(s \in T_\cake\), \(\othercake(s) \in T_\cake \…

## Using weak-b notation

Hello everyone, today I'll try to calculate some weak-b notation inputs

b[2,2] = 4

b[2,2,2] = b[1,2,b[2,1,2]] = b[1,2,b[2,4]] = b[1,2,6] = b[2,6] = 8

b[2,2,2,2] = b[1,2,2,b[2,1,2,2]] = b[1,2,2,b[2,4,2]] = b[1,2,2,b[1,4,b[2,3,2]]] = b[1,2,2,b[1,4,b[1,3,b[2,2,2]]]] = b[1,2,2,b[1,4,b[1,3,8]]] = b[1,2,2,b[1,4,b[3,8]]] = b[1,2,2,b[1,4,11]] = b[1,2,2,b[4,11]] = b[1,2,2,15] = b[2,2,15] = b[1,2,b[2,1,15]] = b[1,2,b[2,17]] = b[1,2,19] = b[2,19] = 21

b[2,2,2,2,2] = b[1,2,2,2,b[2,1,2,2,2]] = b[1,2,2,2,b[2,4,2,2]] = b[1,2,2,2,b[1,4,2,b[2,3,2,2]]] = b[1,2,2,2,b[1,4,2,b[1,3,2,b[2,2,2,2]]]] = b[1,2,2,2,b[1,4,2,b[1,3,2,21]]] = b[1,2,2,2,b[1,4,2,b[3,2,21]]] = b[1,2,2,2,b[1,4,2,b[2,2,b[3,1,21]]]] = b[1,2,2,2,b[1,4,2,b[3,24]]] = b[1,2,2,2,b[1,4,2,27]] = b[1,2,2,…