Problem with expression in BEAF
The expression is X^^((X^)X^^^(-2+X) 2) using prime 2. Currently I can't find the expansion of it. This shows how much difficult it is to define hexational arrays.
Hyperfactorial numbers
There are a few hyperfactorial numbers on the list ==Factorial numbers
- Faxul, \(200!\)
- Kilofaxul, \(200!^2\)
- Megafaxul, \(200!^3\)
- Gigafaxul, \(200!^4\)
- Terafaxul, \(200!^5\)
- Petafaxul, \(200!^6\)
- Exafaxul, \(200!^7\)
- Zettafaxul, \(200!^8\)
- Yottafaxul, \(200!^9\)
- Brontafaxul, \(200!^{10}\)
- Geopafaxul, \(200!^{11}\)
- Geopakilofaxul, \(200!^{12}\)
- Amosafaxul, \(200!^{21}\)
- Haprafaxul, \(200!^{31}\)
- Kyrafaxul, \(200!^{41}\)
- Pijafaxul, \(200!^{51}\)
- Saganafaxul, \(200!^{61}\)
- Pectrafaxul, \(200!^{71}\)
- Nisabafaxul, \(200!^{81}\)
- Zotzafaxul, \(200!^{91}\)
- Alphafaxul, \(200!^{101}\)
- Betafaxul, \(200!^{201}\)
- Gammafaxul, \(200!^{301}\)
- Deltafaxul, \(200!^{401}\)
- Thetafaxul, \(200!^{501}\)
- Iotafaxul, \(200!^{601}\)
- Kappafaxul, \(200!^{701}\)
- Lambdafaxul, \(200!^{801}\)
- Sigmafa…
A thingi
\(\mathfrak{J}(\alpha)\) does this: \(\mathfrak{J}(\alpha)\) is stationary over \(\alpha\) the same way Mahlo cardinals are over Inaccessible cardinals. Just another thingi related with User blog:RomaronzoTHEThingy/A thinge. That is all I have to say.
Milliquinsexagintasescentillion
A milliquinsexagintasescentillion is equal to \(10^{4,998}\), or 1 followed by 4,998 zeros. Its full decimal expansion is shown below.
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0…
A thinge
\(\mathfrak{w}_{\alpha}\) =
Make a function: \(f(0)\) = Inaccessible cardinal, \(f(1)\) = Mahlo cardinal, \(f(2)\) = What the Mahlo cardinal does to the Inaccessible cardinal but done to the Mahlo cardinal, repeat the pattern for \(f(3)\), and \(\mathfrak{w}_{\alpha}\) = \(f(\alpha)\)
OCFFCOOCFFCO
WHEN THE DENTIST LICKS YOUR TEETH CLEAN‼‼‼‼‼‼‼
\(\textrm{WHEN THE DENTIST LICKS YOUR TEETH CLEAN‼‼‼‼‼‼‼}\)
\(\mathfrak{WHEN THE DENTIST LICKS YOUR TEETH CLEAN‼‼‼‼‼‼‼}\)
\(\psi_0(0)\) = 0
\(\psi_0(\Omega)\) = \(\omega\)
\(\psi_0(\Omega+\alpha)\) = \(\omega+\alpha\)
\(\psi_0(\Omega \times 2)\) = \(\omega \times 2\)
\(\psi_0(\Omega \times \alpha)\) = \(\omega \times \alpha\)
etc.
\(\psi_0(I)\) = \(\vartheta(\Omega^{\Omega})\)
\(\psi_0(I+\alpha)\) = \(\vartheta(\Omega\uparrow\uparrow\alpha)\)
\(\psi_0(I \times \alpha)\) = \(\vartheta(\Omega\uparrow\uparrow\uparrow\alpha)\)
\(\psi_0(I^\alpha)\) = \(\vartheta(\Omega_{\alpha})\)
\(\psi_0(I^\Omega)\) would literally be \(\vartheta(\Omega_{\Omega})\)
\(\psi_{\alpha}(\Omega)\) = \(\psi_{\alpha}(\alpha+1)\)
\(\psi_…
New Notation For Schmittyillion Plus Tier Myrillion Illions
I updated the notation for Schmittyillion:
10^{(10100↑↑3*10)4,182,001(10100-1[down])}+3
Sorry if it says "down". Couldn't paste down arrows. I am working on tier myrillion illions, one of it being called uvaldillion.
- 10^{(1030,003↑↑3*10)558,390,117(1030,003-1[down])}+3
The tier micrillion illions are all named after places in Florida:
- Prudillion, 10^{(103,000,003↑↑3*10)3,000(103,000,003-1[down])}+3
- Tallahasillion, 10^{(103,000,003↑↑3*10)3,000,000(103,000,003-1[down])}+3
- Miamillion, 10^{(103,000,003↑↑3*10)3,000,000,000(103,000,003-1[down])}+3
- Orlandillion, 10^{(103,000,003↑↑3*10)3*1012(103,000,003-1[down])}+3
- Jacksonvillion, 10^{(103,000,003↑↑3*10)3*1015(103,000,003-1[down])}+3
- Tampillion, 10^{(103,000,003↑↑3*10)3*1018(103,000,003-1[down])}+3
- Lauderillion, 10^{(103,000,003↑↑3*10)3…}
A10.2 - Nest Ordinals with OCFs
U(γ) (where γ is any countable ordinal) was defined in Blog A3.1, section Ordinals + CEN. It was designed to combine the power of CEN with ordinal expansion to a level much higher than ordinals nested with Knuth's up arrow.
- General form is U_{OCF}(Expressions), where OCF (Ordinal Collapsing Function) can be Veblen's 𝝋, Buchholz's 𝜓, Rathjen's 𝜓, Taranovsky's C, etc. Here I only used Taranovsky's C as example, as it is the largest ordinal for computable functions.
- The example below used Taranovsky's C, which is defined in here.
- For Tar. C, C(1,0) = ⍵; for nested U system, U_{C}(1,0) = U(Σ).
- For Tar. C, C(𝛀_{1},0) = ε_{0} = ⍵^{⍵⍵...}; for nested U system, U_{C}(𝛀_{1},0) = (U(Σ))^{(U(Σ))(U(Σ))...}.
- The largest ordinal for Tar. C is C(C(...C(𝛀_{n},2,0)...,0),0); for nested U …
My attempt at making a UNOCF-style OCF (very bad and ill-defined)
WIP
Below OFP, it works like EBOCF
Examples:
\(\psi_0(\Omega) = \varepsilon_0\)
\(\psi_0(\Omega^\omega) = \varphi(\omega,0)\)
\(\psi_0(\Omega^\Omega) = \varphi(1,0,0)\)
\(\psi_0(\Omega^{\Omega^2}) = \varphi(1,0,0,0)\)
\(\psi_0(\Omega^{\Omega^\omega}) = \varphi\begin{pmatrix}1 \\ \omega \end{pmatrix} = SVO\)
\(\psi_0(\Omega^{\Omega^\Omega}) = LVO\)
\(\psi_0(\Omega_2) = BHO\)
\(\psi_1(0) = \Omega\)
\(\psi_1(\Omega_2) = \varepsilon_{\Omega+1}\)
\(\psi_2(0) = \Omega_2\)
\(\psi_2(\Omega_3) = \varepsilon_{\Omega_2+1}\)
\(\psi_\omega(\Omega_{\omega+1}) = \varepsilon_{\Omega_{\omega}+1}\)
etc.
\(\psi_I(0) = \Phi(1,0) = OFP\)
\(\psi_I(1) = \Phi(1,1)\)
\(\psi_I(2) = \Phi(1,2)\)
\(\psi_I(\Omega) = \Phi(1,\Omega)\)
\(\psi_I(OFP) = \Phi(1,\Phi(1,0))\)
\(\psi_I(I) = \Phi(2,…
Simplex-growing hierarchy
So a few weeks ago I did the factorial-growing hierarchy. But what if we added rather than multiplied? The additive analog of the factorial is triangular numbers. The sum of triangular numbers is called a tetrahedral number. We can go further into higher dimensions, creating what we can call simplex numbers. So to extend to "above-dimensional spaces", we create the simplex-growing hierarchy.
- \(0\Delta_\alpha = 0\)
- \(n\Delta_0 = 1\)
- \(n\Delta_{\alpha+1} = \sum_{k=0}^n k\Delta_\alpha\)
- \(n\Delta_\alpha = \sum_{k=0}^n k\Delta_{\alpha[k]}\) iff \(\alpha\) is a limit ordinal
Use the first rule that applies. 0Δ_{0} is 0 from rule 1, rather than 1 from rule 2. Like the other heirarchies, it depends on the definition of fundamental sequences. Here we wil…
To moderators - 1
Please. I address to all moderators of Googology Wikia.
I ask you all to delete my blog post "Proof of well-definedness of BEAF using formal strings and operations" when it will be empty. I will bring all of it to separate sections of my another blog post, "BEAF using formal strings". And another: please rename latter to "BEAF using formal strings and operations (with proof of well-definedness)" because I analyzed all sets of rules and it seems to me that all problems are solved.
Something
\(\reverse lunate sigma_{\alpha}\) = \(\lunate
sigma_{\psi_{\alpha}(\Omega_{\omega})}\)
Much more is coming soon, this is WIP!
Just an idea...
This is an idea for how Sbiis Saibian's Cascading-E notation could get bigger.
100 is used as an example.
When the dentist licks your teeth clean.
Thus Sprach Zarathustra = E100{#,#,#/2}100
Also E100{#,#/2}100 = E100{#&#&#&#&#&#&#&#&#&#& ... &#&#&#&#&#&#&#&#&#&#}100 100 times aka Sprach Zarathustra.
Now, let's use BEAF!
Remember, this is just an idea.
Yog-sothoth = E100{#,#(1)2/2}100
Yog-yog-sothoth = E100{#,#(2)2/2}100
n-ex Yog-sothoth = E100{#,#(n)2/2}100
Duyus Zarathustra = E100{#&&#}100
Sprach Duyus Zarathustra = E100{#,#//2}100
Truyus Zarathustra = E100{#&&&#}100
Sprach Truyus Zarathustra = E100{#,#///2}100
Tetruyus Zarathustra = E100{#&&&&#}100
Sprach Tetruyus Zarathustra = E100{#,#////2}100
Giganticuloth = E100{L,X}_{#,#}100
Pseudomonarchia giganticulo…
Emphasizing Schmittyillion
A lot of you seem to have been confused by the notation. Pretty much it is like this... how the suzillion in Tier 14 looks (10^{^{13}3\times10^{3,000}+3}\). Except in schmittyillion, the 13 on the left is 10^{100}-1, which marks it as a tier googol illion, so it would look like (10^{^{googol-1}3\times10^{4,182,001}+3}\). Easliy explainable. It is practically a googol-1 3*10's base exponents. I tried to put it in much easier notation on paper using up and down arrows, but regretted it.
Unfortunately, I am working on a larger illion called schmiganticillion as a combination of "schmitty" and "gigantic". This new illion would be in tier number schmittyillion plus a googol, with a schmittyillion+googol-1 3*10 plexes. This is crazy to even think about it,…
A L P H A S M inaccessible cardinal
Cardinal beyond Alphasm's imagination.
\(\tilde A_{\alpha}\) = The \(\alpha\)-th Alphasm inaccessible cardinal.
Alphasm
When the dentist licks your teeth clean.
Boogolquadriplex
Thr bppgplquadriplex is a number that is (10,(10,(10,(10,(10,100,10),10,)10,),10),10) in Seangem1242's Array notation, where there are a boogoltriplex ↑'s between the Two tens. This term was coined by User: Seangem1242
A function
When I say "a function", I don't mean an A function.
B(a,b) = \(I^b(a)\) (Iota function)
B(a,b,c) and beyond works like BEAF.
This is probably THE WORST function ever in the history of
googology (well idk but probably pretty close)
When the dentist licks you teeth clean.
THE DENTIST LICKS YOUR TEETH CLEAN
WHEN THE DENTIST LICKS YOUR TEETH CLEAN WHEN THE DENTIST LICKS YOUR TEETH CLEAN WHEN THE DENTIST LICKS YOUR TEETH CLEAN WHEN THE DENTIST LICKS YOUR TEETH CLEAN WHEN THE DENTIST LICKS YOUR TEETH CLEAN
WHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN THE DENTIST LICKS YOUR TEETH CLEANWHEN…
Okay... Umm
Lmao. The picture showed that there was a source... #unquadragintamilliunquinquagintacentillion
This was the second time when "quadrimilliquattuordecicentillion" was put in the deletion log. As I have said, even Bing now puts it in the search suggestions. Please explain this.
I literally turned my profile into what I coined...
Check out all what I have coined here! At User:Trakaplex! From pastavecillion to onliyetapunillion! My main wiki is the testing.
Some random OCF thingy
\(\acute A(\Omega+\alpha)\) = \(\varepsilon_{\omega+\alpha}\)
\(\acute A(\Omega\times 2)\) = \(\zeta_0\)
\(\acute A(\Omega\times \alpha)\) = \(\zeta_1\uparrow\uparrow\alpha\)
\(\acute A(\Omega^2)\) = \(\zeta_{\omega+1}\)
\(\acute A(\Omega^\alpha)\) = \(\eta_\alpha\)
\(\acute A(\Omega^{\Omega^\alpha})\) = \(\eta_{\eta_{\varepsilon_\alpha}}\)
\(\acute A(\Omega^{\Omega^\Omega})\) = \(\eta_{\eta_{\varepsilon_{\omega+1}}}\)
\(\acute A(\Omega^{\Omega^{\Omega^\alpha}})\) = \(\eta_{\eta_{\eta_\alpha}}\)
\(\acute A(\Omega^{\Omega^{\Omega^\Omega}})\) = \(\eta_{\eta_{\eta_{\omega+1}}}\)
\(\acute A(\Omega\uparrow\uparrow\Omega)\) would be like \(\acute A(\Omega^\Omega)\) but using \(\varphi(4,\textrm{stuff})\) instead of \(\eta_{\textrm{stuff}}\)
\(\acute A(\Om…
some random stuff i came up with
Kilollion = 1006.93167, based on how smaller units of measurement in the metric system have bigger -illions using their names, and doing the opposite of bigger -illions. Kilollion is 10^3.003, being a 1000th of an -illion.
Decallion, a 10th of an -illion, equal to 1995.26231.
- 1000=1K
- 1000000=1M
- 1000000000=1B
- 1000000000000=1T
- 15 zeros = Qd
- 18 zeros = Qt
- 21 zeros = Hx (i call it hextillion because hex sounds a bit more appropriate, the other word for hextillion... i'll leave to you.)
- 24 zeros = Hp
- 27 zeros = Oc
- 30 zeros = Nn
- 33 = Dc
- 36 = MDc
- 39 = BDc
- 42 = TDCc
- 45 = QdDc
- 48 = QtDc
- 51 = HxDc
- 54 = HpDc
- 57 = OcDc
- 60 = NnDc
- 63 = Vg
it's important to note that you can use lower level abbreviations before higher level abbreviations to indicate stuff like the illions inbe…
Whatdoicallthis
ϐ( = \(\omega\)
ϐ) = \(\Omega\)
ϐ() = ϐ(() = \(\Omega_\omega\)
ϐ(\(\alpha)\) = \(\Omega_\alpha\)
ϐ[ = \(\varepsilon_0\)
ϐ] = \(\zeta_0\)
ϐ[\(\alpha\)] = \(\varphi(\alpha,0)\)
ϐ[] = ϐ[[]
ϐ{ = \(\vartheta(\Omega)\)
ϐ} = \(\vartheta(\Omega_2)\)
ϐ{\(\alpha\} = \vartheta(\Omega_\alpha)\)
}+ϐ({()})
ϐ{,.\(\alpha\)} = \(\vartheta(\Omega^\alpha)\)
Whatamievendoing
WHENTHEDENTISTLICKSYOURTEETHCLEAN
ϴ
NOTE:ϴ IS NOT TO BE CONFUSED WITH θ.
ϴ(\(\Omega^\alpha\)) = \(\vartheta(\Omega_{\Omega_{\Omega_\cdots}})\) \(\alpha\) times.
ϴ(\(\Omega^\Omega\)) would work similar to \(\vartheta(\Omega^\Omega)\)
That's really all.
OCFs/Ordinal notations
\(\newcommand{\"}{\!"}\)Ordinal notations associated to the OCFs in User_blog:C7X/OCFs. (Order-preserving) bijections haven't been proven between these and set-theoretic ordinals
Define a set of strings \(T\):
- \(``0\",``\Omega\"\in T\)
- For \(a,b\in T\), then \(a+b\in T\)
- For \(a\in T\), then \(\omega^a,\psi(a)\in T\)
The members of \(T\) are called "terms".
Define a set \(APT\subset T\):
- \(APT:=\{a\in T:\nexists(a_0,a_1\in T)(a\textrm{ is of the form }a_0+a_1)\}\)
The members of \(APT\) are called "additive principal terms".
Define a relation \(
Fastest Way to Beat Kirby-Paris hydra
Kirby-Paris hydra sequence is the number of steps that beat kirby-Paris hydra totally. But this way used to cut hydra's head is not an optimal way. The strategy used is always to cut right-most inner brackets as you can see in the right image.
I did a study on the fastest way to beat Kirby-Paris hydra. The best strategy should be:
- Always cut most inner brackets / deepest level nodes
- Among all brackets from above rule, cut the one that has most "brothers".
I name the sequence as optimal_hydra(n). For example, to beat hydra_3 with above stargety:
I would be happy to see if someone can analysis the growth rate of optimal_hydra(n).
Minimalist ordinal notation
Minimalist ordinal notation (MON) is an ordinal notation using only two symbols.
- 0 = \(0\)
- wαβ = \(\omega^\alpha+\beta\) where α β are strings representing the ordinals \(\alpha\) and \(\beta\)
- 0 = \(0\)
- w00 = \(1\)
- w0w00 = \(2\)
- w0w0w00 = \(3\)
- w0w0w0w00 = \(4\)
- w0w0w0w0w00 = \(5\)
- ww000 = \(\omega\)
- ww00w00 = \(\omega+1\)
- ww00w0w00 = \(\omega+2\)
- ww00w0w0w00 = \(\omega+3\)
- ww00ww000 = \(\omega\cdot2\)
- ww00ww00w00 = \(\omega\cdot2+1\)
- ww00ww00ww000 = \(\omega\cdot3\)
- ww0w000 = \(\omega^2\)
- ww0w00w00 = \(\omega^2+1\)
- ww0w00w0w00 = \(\omega^2+2\)
- ww0w00ww000 = \(\omega^2+\omega\)
- ww0w00ww00w00 = \(\omega^2+\omega+1\)
- ww0w00ww00ww000 = \(\omega^2+\omega\cdot2\)
- ww0w00ww00ww00w00 = \(\omega^2+\omega\cdot2+1\)
- ww0w00ww00ww000 = \(\omega^2+\omega\cdot3\)
- ww0w00ww0w000 …
aaaa
start with A, which is 1. B is 2. we can conject BA by multiplying, but the highest one in value takes the conjuction's value to the power of it's value, so BA conjunction creates (2x1)^2=4. define the next letter as 1 greater than the conjunction of all previous ones together. 4+1=5=C
note: you can't conjunct conjunctions, you can only conjunct all the letters at once. we can't get stuff like D(CAB(DB)), we also cannot
conjunction of CA = (5x1)^5=3,125
CB = (5x2)^5=100,000
A=1 is useless in multiplication and only useful for declaration of conjunction to get that exponentiation so we can declare the value of D to be 100,001.
DA=100,001^100,001=some big number that'd take god knows how long to even read the digits of
DB=200,002^100,001
DC=500,005…
eternal notation for fat infinities
so i've devised this array function that creates cardinals/ordinals i guess
1. you can choose between a cardinal and an ordinal by prefixing the array with "C" or "O". for example: C[0, 1] = ].
3-ary Psi calculator
Japanese version of the post
- 1 The site
- 2 Expandable notations
- 3 Notation
- 4 Fundamental sequence expansion feature
- 5 Other feature
I made an online calculator for the expansion of 3-ary Psi.
https://koteitan.github.io/ordex/
The notations which can be expanded with the site are the followings:
- W. Buchholz, "A new system of proof-theoretic ordinal functions", Annals of Pure and Applied Logic, Vol.32, pp195--207, 1986
- P進大好きbot, "Ordinal Notation Associated to Extended Buchholz's_OCF", Googology Wiki, User Blog, Dec. 2019
- Kanrokoti, "3 variables ψ which is larger than EBOCF", Googology Wiki, User Blog, Aug. 2020
The notation can be expressed by the five charactors "0", "(", ")", ",", "+" and the elements of the following \(T\) can be recognized. \begin{eqnarr…
Chitan notation (first arrays level)
Expression of this level are of the type :
[k] <A> |n, with k and n natural numbers and A being array(s). Let's explore their roles !
- The k (or king) number is the only term that won't change in the expression.
- A is the content of at least one array.
- The n (or follower) number is the principal term of the expression : it will be the final output after the computing process
[4] |1 = [4] |4 = [4] |4↑↑↑4
Verse-Related Numbers
(WIP PAGE)
All Dimensions Wiki
FM(α) = The size of a verse with a finality index of α (size units don't matter in finalityverses)
some sort of hydra function
I will just make a list of things. You can basically use any function, this is adding by one.
[1][1] = 2
[1][1,1] = [2][1] = 3
[2][2] = [3][1,1] = 5
[2][2,1] = [3][2] = [4][1,1,1] = 7
[2][2,2] = [3][2,1,1] = [4][2,1] = [5][2] = [6][1,1,1,1,1] = 11
[2][2,2,2] = [3][2,2,1,1] = [5][2,2] = [6][2,1,1,1,1,1] = [11][2] = [12][1,1,1,1,1,1,1,1,1,1,1] = 23
[2][3] = [3][2,2] = [4][2,1,1,1] = [7][2] = [8][1,1,1,1,1,1,1] = 15
[2][3,2] = [3][3,1,1] = [5][3] = [6][2,2,2,2,2] = [7][2,2,2,2,1,1,1,1,1,1] = [14][2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1] = [28][2,2,1,1,1 . . . 1,1,1] = [56][2,1 . . . 1] = [112][1 . . . 1] = 223
[2][3,2,1] = [3][3,2] = [3][3,1,1,1] = [6][3] = 447
[2][3,2,2] = [3][3,2,1,1] = [5][3,2] = [6][3,1,1,1,1,1] = [11][3] = 14335
(AD) Kanji Starter Pack 80 TM Contents
etc.
That's not all! Be creative and make more words by combining characters:
一円玉天気(ICHIENDAMA TENKI): Stable sunny weather.
山火事(YAMAKAZI): A googological term used in Y sequence. *事 is included in Grade 8.
WIP.
Some relatively smol numbers
Hello. My job here is to coin numbers and functions that haven't
been named. They won't be very big ones at all because I am too
smooth-brained to try and play the 'bigger number game'
undoubtedly people will come up with bigger (they already have but
shush). I dunno what to say now other than here's some
numbers.
Sorry if I can't into math.
These functions start off with a simple concept. Let's have a
list of things. How original. There's a variable
we'll put into it, \(x\). We'll expand these functions more and
more until this stuff becomes completely insane. And maybe it grows
kinda fast, I dunno.
First, let's define our first function, \(TL1(x)\). We have a list of objects of length x. In addition to that, each object in the list has x traits, with …
AAAA 5: more functions
I will soon move this "series" to a user post.
This is kinda like a set theory.
?_{a} = Variable a
~_{a} = a'th Hyperoperator
1)
If I am wrong, please make a comment.
Qwerty Array Notation
(WIP PAGE)
[a,0] = -a
[a] = [a,1] = a+1
[a,2] = a²
[a,b] = a b-ated to a for b>1
[a,b,2] = [a,[a,[a,...a],a],a]...] b times
In general:
[#,a,b+1] = [#,[#,[#,...],b],b],]...,b],b] [#,a,b] times. Also [#,1] = [#].
[a,b[1]2] = [a,a,a...] b times.
Gozol = [10,100]
Grand Gozol = [10,100,2]
Bigrand Gozol = [10,100,3]
Trigrand Gozol = [10,100,4]
Tergrand Gozol = [10,100,5]
Peygrand Gozol = [10,100,6]
Hexgrand Gozol = [10,100,7]
etc.
Dozol = [10,10,100]
Gocrop = [10,100,1,2]
Grand Dozol = [10,10,100,2]
Grand Gocrop = [10,100,1,1,2]
Bigrand Gocrop = [10,100,1,1,1,2]
Bigrand Dozol = [10,10,100,1,1,2]
etc.
Gognoohaha = [10,100[1]2]
The same rules for [#] can be applied to [#[1]2].
The naming system can also be applied to [#[1]2]
For example:
Bigrand Dognoohaha = [10,10,100,1,1,…
uhhh coining random googological numbers because why the hell not
ok i have like 0 understanding when it comes to coining terms or prefixes but ill try (also im sure some of these numbers are already coined but idk,)
(n)-bunch = 10^5 * n
(n)-swarm = 10^10 * n
(n)-fest = 10^15 * n
(n)-mash = 10^30 * n
(n)-mass = 10^50 * n
(n)-mob = 10^75 * n
(n)-duo = n^2
(n)-trio = n^3
(n)-quadrio = n^4
(n)-quintio = n^5
(n)-plex = 10^n
1e115 (or 10^115) = Googolfest
1e120 (or 10^120) = Googolfestbunch
1e125 = Googolfestswarm
1e130 = Googolmash
1e135 = Googolmashbunch
1e140 = Googolmashswarm
1e145 = Googolmashfest
1e150 = Googolmass
1e155 = Googolmassbunch
1e160 = Googolmassswarm
1e165 = Googolmassfest
1e170 = Googolmassmash
1e175 = Googolmob
1e180 = Googolmobbunch
1e185 = Googolmobswarm
1e190 = Googolmobfest
1e195 = Googolmobmash
1e200 = Googolduo…
Luxius' Theta Notation, based off of the other Theta Array Notation.
I said I would make some revisions.
So, let's make some numbers. Note that the addition is defined very loosely here.
0(0) = 1
0(0,0) = 2
0(0,0,0) = 3
0(1) = w
0(0,1) = w+1
0(0,0,1) = 0(0,1)+1 = 0(1)+(w+1) = w2+1
0(0,0,0,1) = 0(0,0,1)+1 = 0(0,1)+(w2+2)+(w+1) = w3+3
0(1,1) = w^w+w
0(0,1,1) = w^w+w+1
0(0,0,1,1) = 0(0,1,1)+1 = 0(1,1)+(w^w+w+1) = (w^w)2+w^w+w+2
0(0,0,0,1,1) = 0(0,0,1,1)+1 = 0(0,1,1)+((w^w)2+w^w+w+1) = 0(1,1) = ((w^w)3+w^w+w2+3)
0(1,1,1) = (w^(w+1))+w
0(0,1,1,1) = 0(1,1,1)+1 = (w^(w+1))+w+1
0(0,0,1,1,1) = 0(0,1,1,1)+1 = 0(1,1,1)+(w^(w+1))+w+1 = (w^(w+1))2+w2+2
0(1,1,1,1) = (w^(w+2))+w^2+w
0(0,1,1,1,1) = (w^(w+2))+w^2+w+1
0(0,0,1,1,1,1) = 0(0,1,1,1,1)+1 = 0(1,1,1,1)+(w^(w+2))+w^2+w+1 = (w^(w+2))2+(w^2)2+w2+1
0(2) = e0+(w^(w2))+w^w
AAAA 4: Array Notation
Here are some simple rules.
() = 0
(1) = 1
Okay, let's get going.
(a) = a*2
(a,1) = ((a))
(2,1) = ((2)) = (4) = 8
(a,2) = (((a)))
(a,b) = a -> (a) b+2 times = a * 2^(b+1)
(a,0,1) = (a,(a))
(3,0,1) = (3,(3)) = (3,6) = (((((((3))))))) = 3 * 2^7 = 384
(a,0,2) = (a,(a,(a)))
(a,0,b) = a -> (x,(a)) and x = a, b times. (The reason why the x is there is so it doesn't turn into (((a,(a)),(a,(a)).
(a,1,1) = (a,0,a)
(a,1,2) = (a,0,(a,0,a))
(a,1,b) = a -> (x,0,a) and x = a, b times.
(a,c,b) = a -> (x,c-1,a)
⃤! ⃢M ⃢o ⃢r ⃢e ⃢ ⃢⃢ c ⃢o ⃢m ⃢i ⃢n ⃢g ⃢ s ⃢o ⃢o ⃢n ⃢ ⃤!
Factorial-growing hierarchy
As you probably know, the factorial is defined as n! = 1∙2∙3∙...∙n. Sloane's superfactorial is defined as n$ = 1!∙2!∙3!∙...∙n!, extending on the idea. So you could imagine a third factorial defined as 1$∙2$∙3$∙...∙n$. Here is the natural conclusion: the factorial-growing hierarchy (!GH)
- \(0!_\alpha = 1\)
- \(n!_0 = n\)
- \(n!_{\alpha+1} = \prod_{k=0}^n k!_\alpha\)
- \(n!_\alpha = \prod_{k=0}^n k!_{\alpha[k]}\) iff \(\alpha\) is a limit ordinal
Use the first rule that applies. 0!_{0} is 1 from rule 1, rather than 0 from rule 2. Like the other heirarchies, it depends on the definition of fundamental sequences. Here we will be using standard Wainer Heirarchy fundamental sequences.
- n!_{0} = n (except when n = 0)
- n!_{1} = n! ~ n^{n}
- n!_{2} = n$ ~ n^{n2}
- n!_{3} ~ n^{n3} (our "third factor…
Iterated fundamental sequence patterns
I came up with this idea in January 2019, but then I did not reach even ε_{0} using it. But yesterday I had some new thoughts about it.
(In this blog "α[n]" designates n-th element of fundamental sequence of α, not "base-booster").
- 1 Expansions
- 2 Comparison
- 3 Termination algorythm
- 4 Type of ordinal
- 5 Fundamental sequence
- 6 Fixed maximal length of
sequence
- 6.1 Maximal length 1
- 6.2 Maximal length 2
- 6.3 Maximal length 3
- 6.4 Maximal length n
- 7 Patterns
In my programs, generating lists of ordinals in ascending order, a list initially has only one ordinal α, which is the largest ordinal in the list, then new ordinals can be added to the list using "small expansions".
Small expansion of ordinal β in the list is find least β[n] larger than all ordinals in the list less than β…
Tree Array Notation
This is an array notation using the TREE sequence. It works like BEAF but with the TREE function.
Examples of how to write it:
TREE(10,100)
\(10 \uparrow\uparrow 100 TREE\&\ 10 \)
10^{100}TREE&&10
TREE\(\{\{L100,10\}_{10,10}\text{&}L,10\}_{10,10}\)
TREE(n,2) = TREE^{n}(n)
TREE(n,3) = If TREE(n,2) = ƒ(n), then TREE(n,3) = ƒ^{n}(n)
Coming soon! (very soon) (uh oh now the word "soon" sounds so weird! NOOO‼‼‼‼)
Some clues past allowed limit of BEAF (+ corrections)
Note: this is not official.
"Allowed limit of BEAF" is {X,X,2(1)2}, which is expected to be at LVO.
I try to formalize BEAF even beyond. If my approach and estimate are correct then {X,X(1)X} ~ X_2+1 & X should be at BHO.
But start with base 2.
As we know, any array starting with 2 and having more than 2 entries (three entries, if linear) should be equal to 4. That means that even I keep adding & signs, it will still be the same array.
The golden array of BEAF is {2,2(1)2,2}. This is X+2 & 2, X*2 & 2, X^2 & 2, X^^2 & 2, X^^^2 & 2... {X,X,X} & 2, {X,X,1,2} & 2... and so on.
X_3 & X_2 & X & 2
X_3 & X_2 = {X_2, X_2}
{X_2,X_2} & X = X_2^X_2 & X = X_2^2 & X = {X,X(1)X,X}
{X,X(1)X,X} [2] = {X,X(1)X,2}
{X,X(1)X,2} [2] = {X,X(1)2,2}
{X,X(1)2,2} [2] = {X,X(1)…
Buff function
This summer I experimented with "Buff function".
In my "base-booster" system I used 7 rules:
- No booster rule
- Empty booster rule
- Successor booster rule
- Regular rule
- Reduction rule
- Main rule
- Cascade rule
(Earlier I mentioned 10 rules, but I considered rules for "c" and for empty string as 2 different rules instead of single No booster rule; Special reduction rule, which turned out to be only a special case of Reduction rule, but still useful for optimization; and Limit rule, but technically it is not rule for base-booster strings, since it is rule for special string "Limit").
The most problematic rule was Cascade rule, for example, I noticed that for Cascade rule cases cofinality is not always ω, so I had to do cascade transformation to find cofinalit…