User:80.98.179.160

Diagonalizing over oodle theory
For sake of this, \(E\) is shorthand for \(E_0\). \(E\geq^*f_{\vartheta^\text{CK}(\Omega^\Omega)}\), with \(\vartheta^\text{CK}(\Omega^\alpha)\) enumerating the first ordinal that isn't reachable within \(\alpha\)-ary Church-Kleene fixed point function, if \(\alpha<\Omega\), and predicatively many arguments if \(\alpha=\Omega\), and the Church-Kleene fixed points are \(\varphi^\text{CK}(\alpha,\beta,\cdots)\)-enumerated.
 * 1) \(E\) is an operator hierarchy, defined as:\(aEb\) is the largest finite ordinal expressible in \(a\) symbols in \(b\)th order oodle theory.
 * 2) And \(aE\)is\((...((aEa)E(aEa))E...E((aEa)E(aEa))...)E(...((aEa)E(aEa))E...E((aEa)E(aEa))...)\) with \(2^{aEa}\) parens and \(2^{aEa+1}\) operands in total.
 * 3) Define \(aE_\alpha\) as \(aE_{\alpha[\cdots[aE_{\alpha[a]}]}\) with \(aE_{\alpha[aE_{\alpha[a]}]}\) brackets, iff \(\alpha\) is limit.
 * 4) Define \(aE_{n+1}=aE_n...E_n\) with \(aE_n\,E_n\)s.
 * 5) Define Church-Kleene-Bachmann-Howard Oodles as \(10^{100}E_{\vartheta^\text{CK}(\varepsilon_{\Omega+1})}\), that is, with the "Church-Kleene-Bachmann-Howard ordinal".

Large \(n\)umbers
\(a_0=(10^{421290}+1)421290, a_{n+1}=(10^{a_n}+1)421290\). This goes beyond tetration level, and even tritri, but not up-arrow notation level. Also note that I use the xenna- prefix as \(10^{27}\), and xentillion for \(10^{10^{27}n+6-n}\), where \(n=6\) in long scale and \(n=3\) in short scale. Define \(b(n,m)=\frac{d(n,m)}{10}\), with base case \(m=31\), \(d(n,m)=10^{10^{\cdots^{10^m3+1}\cdots}3+1}3\) with n+2 10s, iterating only with 3+1s but NOT with 3s and default case m=31, \(f(n,m)=a_{a_{a_{a_{\cdots_{a_m}}}12}50}\) with n+2 "a"s, \(c(n,k,m)=d(\cdots(d(n,k),k)\cdots),k)\), with m "d"s, \(e(n,m)=10^{10^{\cdots^{10^{10^{m3+1}3+1}3+1}\cdots}3+1}\) with base case m=31 and m 10s. Methynillion refers to the group methyne group, methanoicillion refers to formic acid, ATP-illion refers to adenosine triphosphate. UMP-illion isn't to confuse with "umptillion".

Omega pentations and beyond
\(\omega\uparrow^3\omega=\zeta_0,\omega\uparrow^3(\omega+1)=^{\zeta_0}\!\zeta_0=\varepsilon_{\zeta_02}\), \(\omega\uparrow^{\omega+1}\omega=\Gamma_0\), \(\{\omega,\omega,1,2\}=\vartheta(\Omega^3)\).

Extra-Fast Growing Hierarchy

 * 1) \(h_0(n)=n+1\).
 * 2) \(h_{\alpha+1}(n)=h_\alpha^{h_\alpha^{\cdots^{h_\alpha(n)}\cdots}(n)}(n)\) with (\(h_\alpha^{h_\alpha^{\cdots^{h_\alpha(n)}\ddots}(n)}(n)\) with (... with \(h_\alpha^{h_\alpha^{\cdots^{h_\alpha(n)}\ddots}(n)}(n)\) \(h\)s) \(h\)s)...)\(h\)s; with \(h_\alpha(n)\) lines (1D spaces), then similarly defining planes (2D spaces), 3D, ..., defining superdimensions as dimension-number-defining lines, super-superdimensions as superdimension-number-defining, ..., trimensions as \(h_\alpha^{h_\alpha(n)}(n)+1\)-ex-superdimensions, similarly defining supertrimensions, super-supertrimensions,... as superdimensions, super-superdimensions,... but replacing every instance of "dimension" with "trimension", tetramensions, pentamensions, hexamensions,... as trimensions but replacing every instance of "dimension" with "trimension", "tetramension", "pentamension",... Continue with \(h_\alpha^{h_\alpha(n)}(n)\)th element of dimension, trimension, tetramension,... sequence. Treat these as ordinals: dimension->0, superdimension->1, ..., trimension->omega, tetramension->omega2,...
 * 3) \(h_\alpha(n)=h_{\alpha[h_{\alpha[\cdots_{h_{\alpha[n]}(n)}\cdots]}]}(n)\) with (\(h_{\alpha[h_{\alpha[\cdots_{\alpha[n]}\cdots]}(n)}(n)\) with ... with \(h_{\alpha[h_{\alpha[\cdots_{\alpha[n]}\cdots]}(n)}(n)\) \(h\)s) ...) \(h\)s) with (\(h_{\alpha[...]}(n)\) with (... (with \(h_{\alpha[n]}(n)\) \(h\)s) lines (1D spaces))...) then similarly defining 2D, 3D, ..., define superdimensions, super-superdimensions,..., trimensions, tetramensions,..., as in the successor rule, but place after each alpha a bracket containing the rest of values and remove superscripts. Continue with \(h_{\alpha[h_{\alpha[n]}(n)]}(n)\)th element of dimension, trimension,... sequence iff \(\alpha\) is limit.

Define surprimitive recursive functions to be \(<^*h_\omega\), and surrecursive ones to be \(<^*h_{\omega_1^\text{CK}}\). Then,

Tasks

 * 1) where in the FGH, HH, MGH and SGH is \(h_\omega\), the first non-surprimitive-recursive function?
 * 2) where in the FGH is \(h_{\omega_1^\text{CK}}\), the first non-surrecursive function?
 * 3) Does the FGH catch up* to the EFGH, and if yes, at what ordinal?
 * 4) If the FGH catches up* to the EFGH, does the HH catch up to the EFGH, and if yes, at what ordinal?
 * 5) If the HH catches up* to the EFGH, does the MGH catch up to the EFGH, and if yes, at what ordinal?
 * 6) If the MGH catches up* to the EFGH, does the SGH catch up to the EFGH, and if yes, at what ordinal?

&#42; using the most natural definition as in: \(\exists\alpha<\omega_1:h_\alpha\approx^*f_\alpha\).

Googolisms
Default for self-defined googolisms is 1 in function argument. Not because I don't want too large growth, but it's still large enough.

Ordinals beyond Church-Kleene fixed point

 * φCK(1,0), CHURCH-KLEENE FIXED POINT also ε0CK
 * φCK(1,1), also ε1CK
 * φCK(2,0), also ζ0CK
 * φCK(3,0), also η0CK
 * φCK(ω,0)
 * φCK(ε0,0)
 * φCK(φ(ω,0),0)
 * φCK(Γ0,0)
 * φCK(Γ1,0)
 * φCK(ϑ(Ω^3),0)
 * φCK(ϑ(Ω^ω),0)
 * φCK(ϑ(Ω^Ω),0)
 * φCK(ϑ(ε(Ω+1)),0)
 * φCK(ω1Ch,0), if ω1Ch=ω1CK is proven this equals φCK(ω1CK,0)
 * φCK(ε0CK,0)
 * φCK(ζ0CK,0)
 * φCK(η0CK,0)
 * φCK(φCK(ω,0),0)
 * φCK(φCK(ε0,0),0)
 * Γ0CK, also ϑCK(Ω^2), also φCK(1,0,0)
 * Γ1CK, also φCK(1,0,1)
 * φCK(1,1,0)
 * φCK(1,0,0,0), also ϑCK(Ω^3)
 * φCK(ω&α), also ϑCK(Ω^ω)
 * φCK(ω&α&α), ...it's possible to extend Veblen arrays just as most arrays
 * φCK(ω&α&α&...), also ϑCK(Ω^Ω)
 * ϑCK(ε(Ω+1)), What do we get when combining BHO with Church-Kleene ordinal? This ordinal.

SbS' errors

 * 1) He uses unary (as in, he writes 1+1+1+1+1 instead of 5) in describing FGH.
 * 2) He doesn't un-parenthesize the omega powers with either 1 element or only exponentiation.
 * 3) His notation uses carets instead of superscripts.
 * 4) He uses +, *, \(x^y\) as function instead of as operator.

Omega one quantum 2048
Omega one of quantum 2048 is the supremum of all possible scores on an omega by omega board when only finitary superpositions are allowed but all other rules being of quantum 2048's.

When infinitary superpositions are allowed the ordinal is omega one tilde of quantum 2048.

When is on a 3D board the ordinal is omega one of 3D quantum 2048.

When is on a 3D board and infinitary superpositions are allowed the ordinal is omega one tilde of 3D quantum 2048.

The CAN (Copy Array Notation)
The first entry is the copiand, the second the iterator, after which the first nonzero entry is the exhauster.

Rules
The symbol for  is b¶a, when b is a number, but not an expression (except for addition) For example, <9,1>=9[9#9]=9[9,9,9,9,9,9,9,9,9]. Zeroes can be omitted as in, <99,88,,,,,,,,,,,,,,,,,,,,,,,,,,,,,9999999> as we see in dimensional arrays.
 * 1) &lt;a>=a
 * =\(A_b(a)\) where A is defined as:
 * 1) \(A_1(a)\) is a[a#a] in Copy notation.
 * 2) \(A_{n+1}(x)=A_n^{A_n(x)}(x)\). Thus, 2-entry ABAN is NOT primitive recursive, a thing most arrays exhaust at 3 or 4 entries!
 * <@,0>=<@>
 * =a
 * =,c-1@> if b>0 and c>0
 * 1) \(\langle a,b,\underbrace{0,0,0,\cdots,0}_c,d@\rangle=\langle\underbrace{a,a,a,\cdots,a,a}_{c+1},\langle a,b-1,\underbrace{0,0,0,...,0}_c,d@\rangle,d-1@\rangle\)

Tasks

 * 1) Approximate n¶n in FGH, HH, MGH, SGH and 999-growing hierarchy.
 * 2) Calculate <2,3,5,7,11,13,17,19,23>.
 * 3) Is <3,3,3> larger than tritri? If yes, why, if not, why?

Rules
For example, <3,3,2[1]2>=tritri¶3.
 * 1) In general, =>. (note: the comma is same as [0] separator), if c is a b×b×b×...(c times)...×b×b hypercube of as, but the last a is replaced with , if it wasn't replaced.
 *  is a b×b×b×...(c times)...×b×b hypercube of as, but the last a is replaced with , if d>0.
 *  is a b×b×b×...(c times)...×b×b hypercube of as, but the last a is replaced with 
 * 1) Multi-entry arrays inside brackets are same as BEAF parenthesized arrays.

Tasks

 * 1) Approximate n^n¶n in FGH, HH, MGH, SGH, 999-growing hierarchy.
 * 2) Evaluate <9,999[99999]9999999>.

Tetrational arrays, and beyond
I'll define ba¶c=a^a^...(b times)...^a^a¶c, continue with pentation, hexation,... \(\{a,b,1,2\}\)¶c=a{a{...(b times)...{a{a}a}...a}a}a¶c. Then, define tetrexpandal arrays ({a,b,2,2}¶c), pentexpandal arrays ({a,b,3,2}¶c),..., explodal arrays,..., megotional arrays, and even beyond! After all operators, define =. Note: a¶b¶c is defined as: a. the ath is of a[a]-a[a-1]. Then the planes have size b, b[2]-b,...,b[b]-b[b-1], realms b, b[2],...,b[b], etc., with size a[2,k]. The tetrational spaces have then b, \&lt;b,b>, &lt;b,&lt;b,b>>... dimensions. Continue with pentational, hexational,... and with BEAF, and all values are c (except the filling numbers, which are 0).
 * 1) The 1st row is of a entries,
 * 2) the 2nd is of a[2]-a,
 * 3) the 3rd is of a[3]-a[2],

Dynamic Googolisms
The XKCD number is {A,B,C,D,E}, where A is Loader's function called to the sum of all numbers all XKCD comics contain, B is Loader's function called to the decimal expansion of the binary data of all XKCD comics, concatenated and C is Loader's function called to the decimal expansion of the binary data of all Time frames, concatenated, and D is Loader's function called to the decimal expansion of the binary data of all pictures of all What If? entries, concatenated, and E is Loader's function called to the sum of all XKCD comics' titles' lengths.