uhhh so I'm quite new to googology and i created ss or something
I honestly have no idea on how do smart people define sequence systems but yeah anyway
So we have the sequence as an input of a function that I'll call Z[S] where S is, well, a sequence. The ss stops when S = ∅. And the output of the function is the amount of rows needed for this sequence to terminate.
Case 1: last term of S = 0: remove this 0 from S.
Case 2: last term of S ≠ 0: remove the last term from S and change the new last term to itself - 1 and same with the row number'th last element and copy row number of last terms of S row number of times (if row number > length of S then copy all the elements).
Well idk that's it ig. If someone is actually not lazy then can you tell me if this terminates at all and what is it's growth rate? Or may…
My NUMBERS function
NUMBERS(x)=x^x^x… (x amount of x’s) ^x^x
Example: NUMBERS(10)=10^10^10^10^10^10^10^10^10^10
NUMBERS(x,y)=NUMBERS(x^NUMBERS(x^NUMBERS(x^… (y amount of NUMBERS(x^’s) NUMBERS(x^NUMBERS(x))))…)))
Example: NUMBERS(4,5)=NUMBERS(4^NUMBERS(4^NUMBERS(4^NUMBERS(4^NUMBERS(4)))))
List:
- 1
- 4
- 7.6 trillion
- Literally infinity
NUMBERS(x,y,z)=NUMBERS(NUMBERS(x,y)^NUMBERS(NUMBERS(x,y)^NUMBERS(NUMBERS(x,y)^… (z amount of NUMBERS(NUMBERS(x,y)^’s) NUMBERS(NUMBERS(x,y)^NUMBERS(x,y))))…)))
Example: NUMBERS(2,3,4)=NUMBERS(NUMBERS(2,3)^NUMBERS(NUMBERS(2,3)^NUMBERS(NUMBERS(2,3)^NUMBERS(NUMBERS(2,3)))))
NUMBERS(x,y,z,w)=NUMBERS(NUMBERS(x,y,z)^NUMBERS(NUMBERS(x,y,z)^NUMBERS(NUMBERS(x,y,z)^… (w amount of NUMBERS(NUMBERS(x,y,z)^’s) NUMBERS(NUMBERS(x,y,z)^NUMBERS(x,y,z))))…)))
Exampl…
My Own Names for Numbers
- (The names are combinations other words related to it)
- Also, all numbers above 19 use prefixes for numbers 2-999 (and sometimes 1).
- The Prefixes:
- \(1\) = \(Sin/Sio/Sno\)
- \(2\) = \(Dul/Duo\)
- \(3\) = \(Tri/Tre/Tes\)
- \(4\) = \(Qur/Qu/Qr\)
- \(5\) = \(Pe/Pen/Peh\)
- \(6\) = \(Hea/He/Hen\)
- \(7\) = \(Sen/Se\)
- \(8\) = \(Oct/Oc/Ot\)
- \(9\) = \(Eni/En/Ei\)
- \(10\) = \(Den/Dec/De\)
- \(100\) = \(Cen/Cet/Cu\)
- The rest are combinations.
- None for \(0\).
Sorry20221108
I have (perhaps re-)looked at some of the allegations and the evidences related to the situations surrounding p-adic. I will try to compile and provide an analysis on them. However, note that I have probably not looked at everything, nor that I have an unbiased opinion. The purpose of this blog post is to provide my (current at the time of writing) stance on the issues.
Added after writing the body: this blog post ended up focusing on Plain'N'Simple's situation. I am too tired to look at others now...
Added 2022/11/08: I am sorry, but I do not think I could continue this any deeper or at more breadth. While I wanted to digest the situation, I immediately realized that I am completely incapable of doing so. I am overwhelmed by the task, and n…
Rewriting the definition of Subspecies primitive psi function, Hyper primitive psi function and TSS-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/%E4%BA%9C%E5%8E%9F%E5%A7%8B%CF%88%E9%96%A2%E6%95%B0%E3%81%A8%E3%83%8F%E3%82%A4%E3%83%91%E3%83%BC%E5%8E%9F%E5%A7%8B%CF%88%E9%96%A2%E6%95%B0%E3%81%A8TSS-%CF%88%E9%96%A2%E6%95%B0%E3%81%AE%E5%AE%9A%E7%BE%A9%E3%81%AE%E6%9B%B8%E3%81%8D%E7%9B%B4%E3%81%97
- 1 Overview
- 2 Subspecies primitive psi
function
- 2.1 Notation
- 2.2 Ascension function
- 2.3 Cofinality
- 2.4 Fundamental sequence
- 2.5 FGH
- 2.6 Standard form
- 2.7 Naming
- 3 Hyper primitive psi
function
- 3.1 Notation
- 3.2 Depth function
- 3.3 Cut function
- 3.4 Ascension function
- 3.5 Cofinality
- 3.6 Fundamental sequence
- 3.7 FGH
- 3.8 Standard form
- 3.9 Naming
- 4 TSS-ψ function
- 4.1 Notation
- 4.2 Ascension function
- 4.3 Search function
- 4.4 Cof…
An attempt to well-define OCF as a number notation/post-SDO/Einklad
I will make a list of test cases so that you can have a quick view of the whole strength. I call it Einklad which is the combination of two German words, eins(one) and Kladde(rough book).
The syntax sugå is from the main article, plus these:
An attempt to well-define OCF as a number notation/post-SDO
In this article, I will release a new version of NIECF. It is strongly recommended to use VR (Virtual Reality) device for the optimal user experience.
- 1 Inquiry Form
- 2 The Letters
- 2.1 Symbols
- 2.2 Modification
- 2.3 But what about the song?
- 2.4 What about the Alphablocks?
- 3 Valid expression
- 4 Expansion
You might find some problems while doing the analysis. Feel free to ask via the inquiry form then:
(English) https://forms.gle/JECFSRjZWqMEarAp9
(Japanese) https://forms.gle/zSjG8gViedDPaQwP9
(Ukrainian) https://forms.gle/RJVjbMZ8JH8gZgr88
This notation is so strange that we need two new letters to the English alphabet.
†
This letter is named †ord and represents the SW sound. This letter is so strange that the capital and small letters looks exactly the same.
Example …
Move to Miraheze
I will move this wiki to Miraheze because Fandom always deny the SimpleMathJax extension and Miraheze doesn't have ads
Thing
Hey all. Sorry for not being active, I've kinda been growing away from GWiki recently. But I do still want to share my cool Googology ideas in the future so I'll try to keep my vow of being more active.
Hint: I may be making an OCF up tot he diagonalizer of \(f\)-shrewdness. I'm not sure what it is.
Game Freak's Number
Game Freak's number (1,530) is a numeral that comes after 1,529, but before 1,531, and is named after the company, Game Freak, that designed Pokemon. It is also represented as 255*6. This number represents the highest theoretical Base Stat Total (BST) a Pokemon can have from Generation 2 and onwards (As its 6 base stats being HP, Attack, Defense, Special Attack, Special Defense, and Speed all have a maximum of 255).
Rubiks cube possibilities
The Rubik’s Cube was invented in 1974 by Ernõ Rubik, a Hungarian architecture professor. Rubik later used the Cube as a learning exercise to teach his students about 3-dimensional spaces. Little did he know his “Magic Cube” (as he originally named it) would become one of world’s most famous puzzles of all time!.
now i am gonna show all the possibilities of the cubes(all the information comes from here)
2x2
The 2x2x2 Rubik’s cube (called the Pocket Cube) has 3674160 combinations.This is a manageable number. If you fiddle with the 2x2x2 cube randomly, eight hours a day continuously, you’ll solve it by pure chance roughly two or three times per year. Assuming your cube – or your wrist – doesn’t break in the meantime. Mind you, four months to solv…
Attempt at Transfinite FGH
I don't think this way of defining TFGH has been tried before (please tell me if it has!), so I wanted to just try it to see where it goes.
Denote the class \(\{x:x\in On\land (cf(x)=\omega\lor cf(x)=\omega_1\})\) as \(lim\), where cf(x) is the cofinality of x.
Essentially a very weird form of transfinite induction will be used to generate more ordinals with TFGH, and so we start with
- for \(n\in On\), if \(n
OFP 3-var ψ
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/%E3%82%AA%E3%83%A1%E3%82%AC%E4%B8%8D%E5%8B%95%E7%82%B93%E5%A4%89%E6%95%B0%CF%88
- 1 Overview
- 2 OFP 3-var ψ
- 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 Analysis
We define OFP 3-var ψ. This notation is useful when Kumakuma 3-var ψ is too difficult or strong to use for an analysis past EBO. Unlike Kumakuma 3-var ψ where \(\psi_1(0,0)\) is compatible with the least weakly inaccessible cardinal, \(\psi_1(0,0)\) in OFP 3-var ψ is compatible with the least OFP because of the setting \(\textrm{dom}(\psi_1(0,0)) = \omega\). We expect that the limit of this notation is…
Sequence system magic
A sequence system base is a triple \(([], \leq, \Lambda)\) where \([]\) is a function \(\mathbb{N}^{< \omega} \times \mathbb{N} \to S\), \(\leq\) is a linear order on \(\mathbb{N}^{< \omega}\), and \(\Lambda\) is a function \(\mathbb{N} \to \mathbb{N}^{< \omega}\), satisfying the following:
- \(S[n] \leq S\) for all \(n \in \mathbb{N}\) and \(S \in \mathbb{N}^{< \omega}\).
- \((\mathcal{S}, \leq)\) is a woset, where \(\mathcal{S}\) is the closure of \(\mathrm{ran}(\Lambda)\) under \([]\).
We say that \((\mathcal{S}, \leq)\) is the sequence system generated by \(([], \leq, \Lambda)\). If \(\leq\) is the lexicographical order, then we say \((\mathcal{S}, \leq)\) is lexicographical.
A Buchholz-style ordinal notation base is a quintuple \((\mathcal{T}…
Ordinal Increaser
This ordinal function \(\kappa(\alpha)\) simply returns a result larger than \(\alpha\).
\(\kappa(\alpha) = \beta\) where \(\beta\) is the largest ordinal satisfy the condition \(f_\alpha(3) = g_\beta(3)\).
- \(\kappa(0) = \omega+1\)
- \(\kappa(1) = \omega2\)
- \(\kappa(2) = \omega^{\omega2+2}\)
- \(\kappa(3) > \varepsilon^{\omega^\omega}\)
- \(\kappa(\omega+1) > \Gamma_0\)
Extending the extended Veblen hierarchy
This is my attempt to understand and extend the extended Veblen hierarchy. There is some lead-up, so please be patient. Also, I hope the formatting works. Please comment and let me know how this relates to known material.
I will use ordinals smaller than the second uncountable cardinal, \( \Omega_2 =\omega_2 =\aleph_2 \). These are the ordinals of cardinality at most \( \Omega = \Omega_1 =\omega_1 =\aleph_1 \). The countable ordinals are those smaller than \( \Omega \). All ordinals smaller than \( \Omega_2 \) are of one of the following types:
- The smallest ordinal, \( 0 \)
- Successor ordinals: those of the form \( \alpha+1 \) for some ordinal \( \alpha \)
- Limit ordinals of cofinality \( \omega \): These are ordinals \( \alpha \) that are the …
Quick-growing hierarchy
The quick-growing hierarchy \(Q\) is strong and can outgrow fast-growing hierarchy \(f\) easily.
First define \(f[q](n)\) where \(n\) is a positive integer and \(q\) is non-ordered array of \([x]_\alpha\), where \(x\) is a positive integer and \(\alpha\) is an ordinal.
In \(q\):
- \([0]_\alpha\) can be removed from \(q\).
- \([x]_\alpha[y]_\alpha\) is prohibited.
The \(f[q](n)\) is described as follows, read from top to bottom:
- \(f[](n) = n+1\)
- \(f[q[x]_0](n) = f^n[q[x-1]_0](n)\)
- \(f[q[x]_\alpha[0]_{\alpha-1}](n) = f[q[x-1]_\alpha[n]_{\alpha-1}](n)\) if \(cof(\alpha) = 1\); If there are multiple conditions applicable to this rule, only choose a condition whose ordinal \(\alpha\) is the smallest of all conditions
- \(f[q[x]_\alpha[0]_{\alpha[n]*}](n) = f[…
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Names for uncountable ordinals
Quite a number of articles describe various countable ordinals. These are precisely the ordinals less than the first uncountable ordinal, \( \omega_1 \). In particular, there are various FSs that 'name' certain countable ordinals by giving them as a supremum of \( \varphi[n] \). Since any countable union of countable ordinals is countable, this always gives an ordinal smaller than \( \omega_1 \).
Are there similar procedures for naming ordinals between \( \omega_1 \) and \( \omega_2 \) (the second uncountable cardinal)?
For example, there is a first ordinal such that \( \omega_1^\alpha =\alpha \). This is analogous to \( \epsilon_0 \). The collection of these fixed points smaller than \( \omega_2 \) is then enumerated by ordinals smaller tha…
A question
Define:
- \(p(0) = \omega+2\)
- \(p(n+1)\) is the smallest possible ordinal satisfy the condition \(p(n+1)[2] = p(n)\) (includes FS)
So the question is: Is there a highest integer \(x\) for \(p(x)\) to be valid? If so, what is \(x\) and \(p(x)\)?
Inaccessible ψ experimental 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/%E8%A9%A6%E8%A3%BD%E5%88%B0%E9%81%94%E4%B8%8D%E8%83%BD%CF%88%E9%96%A2%E6%95%B0
- 1 Overview
- 2 Inaccessible ψ
experimental 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 Strong inaccessible ψ
experimental 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
We define Inaccessible ψ experimental function. This notation is an extension of Ordinal Notation Associated to Extended Buchholz's OCF.
I at first tried to make the notation reaches TEBO(the ordinal which is the countable limit of Transfini…
Proposal for notation of inverted power tower
I found it cumbersome to express inverted tower, so I proposed to use inverted Knuth's arrows for this purpose:
Ω↓↓ω = ΩΩ..ΩΩ (with ω copies of Ω's)
I↓↓ω = II..II (with ω copies of I's)
ω↓↓ω = ωω..ωω (with ω copies of ω's)
and so on.
When the last term is different, we can express it this way:
[ε↓↓n]↓ε0 = εε..εε0 (with n copies of ε's and one ε0 at the bottom)
SCP cosmology
there are many posts about it, i even have one on powerlisting wiki, so i wanted to bring one here.
one universe in scp has uncountable vector in uncountable spaces this in a atomical structure,Universes have no concept of space.
there is platonic concepts and Jungian archetypes,and departments that studie philosophy,theology,metaphysics,pataphysics...etc.
there is planes that works in alephs(Cardinal)and contains all concepts.
every universe Sees each other has fiction,this is proven with the existence of the humes,that determine the amount of reality.
Since Hume Levels are used to Measure how "Real" or "Fictional" something is, A Universe with a Lesser Hume Level would be more Fictional and a Universe with a Higher Hume Level would be more Re…
Naruto Canonicity
- 1 Introduction
- 1.1 Naruto Games
- 1.1.1 Naruto generations
- 1.1.2 Naruto shippuden ultimate ninja storm revolution
- 1.1.3 Naruto ninja tribe
- 1.1.4 Naruto ultimate ninja online
- 1.1.5 Road to ninja
- 1.1.6 Naruto novels
- 1.1.7 Manga and anime
- 1.1 Naruto Games
Hello everybody,here i am gonna talk about the canon of naruto and that other canons can be parallel universes/dimensions to the original series.
Naruto shipuden ultimate ninja storm generations:
One of the biggest changes to the game came with the Story mode. The RPG overworld that players spent so much time in during the last two games has been scrapped. Instead, players will be constantly kept in battle, as they play through a variety of stories, across several different and important characters. Naruto and Sasuke will obvious…
Ben 10 Cosmology
- 1 Introduction
- 1.1 Universe
- 1.2 one
Timeline=Multiverse.
- 1.2.1 The Forge of creation
- 1.3 Conclusion
this is a blog for the Ben 10 Cosmology,credits to Eternity and Мұхаммед Жексенбай,they are the ones who know about the cosmology of ben 10 and is scale.
it is said by maltruant that the timestream is infinite(Timestream is where all past,present and future timelines interacts)Professor Paradox said that there are an infinite number of timelines(paradox say Ad infinitum to the timelines)
Ad infinitum:
In context, it usually means "continue forever, without limit" and this can be used to describe a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever," among other uses. It may also be used in a man…
How to BAN?
Me trying to understand BAN by writing it down and trying to climb. Some things might be wrong, I'll make corrections as I learn them. If you see something wrong comment so I can fix it.
I will use parenthesis instead of brackets because brackets keep messing up the latex. \( \)
So we just reached \( (10,10,10,10,10,10,10,10,10,10) \). We need a way to go beyond linear arrays. Let's add another row, with \( [2] \), which is the second-tier seperator. The comma is technically \( [1] \), the first-tier seperator.
We can rewrite \( (10,10,10,10,10,10,10,10,10,10)=(10,10[2]2) \). There is a 2 in the second row. What does it do? Well, \( (a,b[2]2)=(\underbrace{a,a,a...a,a,a}_b) \). Wow, that is crazy strong. This is around growth rate \( f_{\ome…
Adventure Time Cosmology
- 1 Introduction
- 1.1 The canonicity of adventure time
- 2 Cosmology
- 2.1 the multiverse
- 2.2 The antiverse/The nothingness
- 2.3 The Primordial monsters and Golb
- 2.4 The void monsters
- 2.5 Love and The catalyst
comet
- 2.5.1 Adventure time power ranking:
The Adventure Time series is about a boy named Finn and his magical shape-shifting dog named Jake, who encounter many strange adventures in the Land of Ooo. Throughout the series the world grows revealing more lore and adventure for the two companions.
Ryan north have a account in Tumbler:
He is the creator and author of Dinosaur Comics, and has written for the comic series of Adventure Time and Marvel Comics' The Unbeatable Squirrel Girl. His works have won multiple Eisner Awards and Harvey Awards and made New York Times B…
Link Blog to multiple sources(powerscaling)
- 1 Introduction
- 1.1 sonic The Hedgehog
- 1.2 Cartoon Multiverse
- 1.3 super mario bros
- 1.4 undertale/Others
- 1.5 Spongebob Squarepants
- 1.6 Ninjago
- 1.7 SCP
- 1.8 Naruto
- 1.9 yugi oh
- 1.10 Crash Bandicoot
- 1.11 Star Wars
- 1.12 Dragon ball
- 1.13 Marvel
- 1.14 Ben 10
- 1.15 Cosmological stuff
something happen in Powerlisting wiki,and he can't edit my own blogs,so i decided to translate the most important blogs to this wiki and edited them here.
Peoples Profiles
percival/Percival's knights/Davion/Jinx/Мұхаммед Жексенбай/Lui Perez/Bill Decipher.
The Sega multiverse/Sega verse and Contents.
Many World Interpretation in Sonic the Hedgehog/Neoplatonic Sega and Platonic Archie/Super Substances in Sonic the Hedgehog/Set Theory and levels of infinity in sonic the hedgehog/Archetypes Tier in Fiction.
The …
Googology Logo
Here's something I made, a neat logo that could represent this community.
PDF file
PNG previewSimplified icon for XeTeX
god this whole blog post is a technical mess; Fandom editor is a mess
Augmented T from LNGN
I think this may be considered a naive extension to LNGN, but I want to define it anyway to make sure I understand the concepts.
Take ZFL and ZFCHL as defined in LNGN. Denote by 11).
This is probably larger than LNGN, however I think it is a naive extension, and this number is also likely ill-defined. So pbot, if this is ill-defined (which it probably is), please tell me why.
Formalisation of Trakaplex's illions
To solve illions in the form 10(3*10↑↑a)b(↓↓c)+3, let's define \(f(x) = 3\times10^x\). So, 10(3*10↑↑a)b(↓↓c)+3 = \(10^{f^{a-c-1}(f^{c+1}(b)+3)}\) where \(f^n\) denotes function iteration.
What the heck is that???
What the heck is The List of Exclamation Mark Number's??? Why does LEXEA-Adventure ever post that??? Why do people always not read the policy??? It so easy! It will only take you 10 minutes, but you guys still never read it. Why????
The Exclamation Mark's Numbers (1! to 10,000,000!)
The List of x!'s, so you ready for these numbers if you want to, cause get ready for fun or something:
- 1! = 1
- 2! = 2
- 3! = 6
- 4! = 24
- 5! = 120
- 6! = 720
- 7! = 5,040
- 8! = 40,320
- 9! = 362,880
- 10! = 3,628,800
- 15! = 1,307,674,368,000 / \(1.3076^{12}\)
\(2.5 \times 10^{99} - 18\)
Quantum Oblivion
Quantum symbol: a symbol that can be represented by qubits
Q(m,n,o) quantum system: a system that can be described using no more than m symbols with no more than n types of symbols, and a symbol must be represented using no more than o qubits
Quantum Oblivion = largest finite number that can be uniquely defined using no more than a tarintar symbols in a Q(tarintar,tarintar,tarintar) quantum system
How to OCF?
Me trying to understand Buchholz's OCF by writing it down and trying to climb. Some things might be wrong, I'll make corrections as I learn them. If you see something wrong comment so I can fix it.
This is part 3 of climbing ordinals. In part 2, we used Veblen's \( \varphi \) and reached the \( LVO \). Now we need something stronger to keep going. We need an OCF.
There's a lot of different OCFs, but the one we're talking about is Buchholz's \( \psi \). An ordinal is expressed like \( \psi(\alpha) \). Let's set the ground rule, which is that \( \psi(0)=1 \).
\( \psi(0)=1 \)
The next \( \psi \) number is the smallest ordinal that cannot be made with a finite number of additions and the previous \( \psi \) number. And also, we can't have an infin…
Some New Foo!
MEGA-ER SCALE STEP INCOMING! Make a new function denoted ‘The Great End of Time (λ1)’. (SO creative, I know.) And now, let’s define some internal functions (only apply inside this step). Also, X refers to X before this step started,
- Alphafunc (α): Create a new system called TimeNotation. This notation contains all the things that can be done to every step/ function that has been iterated so far. But first, we have to make an list. Create a list of every step that has been created, every list that has been made, every permutation of those lists and every individual function in each of them. Then add every way you can reach the same result with every combination of whole number inputs from 1 to the value of every function in the list before this…
S2.0 Systematic Way for Nesting ordinal with Hyper operator (NOHO)
This serves as a correction of my previous blog.
Define hyperoperators using symbol , where x denotes a natural number, e.g. = + ; = * ; = ^ or ↑ ; = ↑↑ and so on.
For normal hyper-operations, where a and b are natural numbers:
- a*(b+1) = a(b1) = a(ab) = a+(a*b)
- a^(b+1) = a(b1) = a(ab) = a*(a^b)
- a↑↑(b+1) = a(b1) = a(ab) =a↑(a↑↑b)
- a↑↑↑(b+1) = a(b1) = a(ab) =a↑↑(a↑↑↑b)
- and so on.
Hence, we can systemize the ordinal hyper-operations into:
- A(C+1) = A.3+2)))))))[2] = ...
Extended Feferman's theta function
I create an extension of the Feferman's theta function that goes beyond the Takeuti-Feferman-Buchholz ordinal, and will eventually reach the omega fixed point level.
- 1 Definition
- 2 Properties
- 3 Comparison between
extended Feferman's and extended Buchholz's
- 3.1 Up to Bachmann-Howard ordinal
- 3.2 Up to Buchholz's ordinal
- 3.3 Up to Takeuti-Feferman-Buchholz ordinal
- 3.4 Up to the countable limit of Extended Buchholz's function
- 4 Notes
For any ordinals \(\alpha\) and \(\beta\), a family \((C_n(\alpha,\beta))_{n \in \omega}\) of sets of ordinals, a set \(C(\alpha,\beta)\) of ordinals, and an ordinal \(\theta_{\alpha}(\beta)\) are defined by the following mutual recursion: \begin{eqnarray*} C_0(\alpha, \beta) &=& \beta \cup \{0, \Omega, \Omega_2, ..., \Omega_{\ome…
A new method for defining ordinal collapsing functions
I have found a simpler way for defining ordinal collapsing
functions. Take hyp cos's I OCF, which is defined in this page as
follows:
\(\begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\
C_{n+1}(\alpha,\beta) &=&
\{\gamma+\delta,\Omega_\gamma,I_\gamma,\psi_\pi(\eta)|\gamma,\delta,\pi,\eta\in
C_n(\alpha,\beta)\wedge\eta
tree vs. TREE: continuing Hyp cos' work
Source:
User blog:Hyp cos/tree function and TREE(3)
Quoting Hyp cos:
"If we get (()()) and at most n vertices, the next tree is ((...(())...)) with n+1 vertices. At the end we get () and at most 2n+1 vertices, at the same time H_w(n) = 2n, so we call (()()) has level w."
Comment: It will reduce in 2n+1-n = n+1 steps, see later.
For starting n and tree ((()())) (path of length 1 appended to the root of tree with height 2), I delete this path edge by edge first. Now I have n+1 max vertex and tree of height 2, so I can transform this into path of length n+2, then P_{n+1}... all the way down to P_1. Everything will take 2n+3-n = n+3 steps.
In general, for connecting these numbers and the result of TREE(m,n) we formulate the following lemma:
Lemma 1. …
Fast growing function (FGF)
Welcome to fast growing function! (FGF). This function was inspired by Fast growing hierarchy (FGH). I dont know what else to put so i'll just show u the function.
- Σ0 (x) = x+1.
- Σ0 (100) = 100+1 = 101.
- Σ0 (1) = 1+0 = 1.
- U then move onto Σ1 (x).
- Σ1 (100) = 100*100 = 10,000.
- That means Σ1 (x) = x*x = x2.
- Then Σ2 (x) equals xx.
- That means Σ2 (100) equals 100100 equals 10200.
- Then Σ3 (100) equals 100^^100.
- Σ4 (100) equals 100^^^100.
- Σx (100) equals 100{x-1}100 equals {100, 100, x-1}.
- U then reach Σα (x).
- Σα (x) equals Σx (x).
- Σα+1 (x) equals Σα (Σα (Σα (x))).
- That means Σα+n (x) equals Σα+(n-1) (Σα+(n-1) (Σα+(n-1) (x))).
- Σα2 (x) equals Σα2-1 (Σα2-1 (Σα2-1 (x)))
- That means Σα^2 (x) equals Σα^2-1 (Σα^2-1 (Σα^2-1 (x))).
- Σα^α (x) equals Σα^α-1 (Σα^α-1 (Σα^α-1 (x)))
- U can keep nesting Σα^α^α^α... (x)
- U …
extended UNOCF (by me)
this and base UNAN is ill-defined
\(\psi_D(0) = C(0) = \Omega\)
\(\psi_D(1) = C(1) = \Omega_2\)
\(\psi_D(2) = C(2) = \Omega_3\)
\(\psi_D(D) = C(1,0) = I\)
\(\psi_D(D+1) = C(1,1) = I_2\)
\(\psi_D(D+2) = C(1,2) = I_3\)
\(\psi_D(D+\omega) = C(1,\omega) = I_\omega\)
\(\psi_D(D2) = C(2,0)\)
\(\psi_D(D3) = C(3,0)\)
\(\psi_D(D^2) = C(1,0,0)\)
\(\psi_D(D^2+1) = C(1,0,1)\)
\(\psi_D(D^2+D) = C(1,1,0)\)
\(\psi_D(D^2 2) = C(2,0,0)\)
\(\psi_D(D^3) = C(1,0,0,0)\)
\(\psi_D(D^4) = C(1,0,0,0,0)\)
\(\psi_D(D^\omega) = C(1,0,0\cdots0,0,0)\)
\(\psi_D(D^D) = C(1;0) = M\)
\(\psi_D(D^D+1) = C(1;1)\)
\(\psi_D(D^D+D) = C(1;0,0)\)
\(\psi_D(D^D+D^2) = C(1;0,0,0)\)
\(\psi_D(D^D 2) = C(2;0)\)
\(\psi_D(D^D 3) = C(3;0)\)
\(\psi_D(D^{D+1}) = C(1,0;0)\)
\(\psi_D(D^{D+1}+1) = C(1,0;1)\)
\(\psi_D(D…
Extending Schütte's notation up to BHO
I originally wanted to write a lot, but I decided against it. I'll describe this in detail in my book. Here, I'll just give the basic idea. For the sake of visual simplicity, assume there is an OCF \(\varphi\) with the properties shown below. The limit of this OCF is going to be the Bachmann-Howard ordinal (BHO).
The limit of Veblen's 2-variable \(\varphi\) function is the ordinal \(\Gamma_0\), which can be denoted as \(\varphi(1,0,0)\) using the extended Veblen function. This can also be written as \(\varphi(\Omega^2)\) using the aforementioned OCF. The limit of the extended Veblen function is the Small Veblen ordinal (SVO), which can be denoted as
(I've used "at" instead of @ due to technical limitations. If you know how to do matrices in …