User:Vel!/pu/Other things

The basic notation rules
Almost all recursive functions have these three rules: All of these rules are needed - without the recursive rule, the function would just be useless. Without the terminating rule, the recursive rule would continue infinitely. Without the base rule, almost everything would be undefined.
 * Base rule: provides something all of the notation is based around and will eventually simplify into.
 * Examples:
 * Base rule of BEAF: {} = 1, {a} = a, {a,b} = ab
 * Base rule of arrow notation: a↑b = ab
 * Terminating rule: Allows decompositions to terminate. Could possibly be merged with the base rule.
 * Examples:
 * Terminating rule of BEAF: {a,1 #} = a
 * Terminating rule of arrow notation: a↑n1 = a
 * Recursive rule: Does recursion.
 * Examples:
 * Recursive rule of BEAF: {a,b,c #} = {a,{a,b-1,c #},c-1 #}
 * Recursive rule of arrow notation: a↑nb = a↑n-1(a↑n(b-1))

You can have multiple of each type of rule. To continue past primitive recursion, we need another rule: Sometimes a cropping rule is also included, which allows you to remove unneeded parts - such as {# 1} = {#}.
 * Diagonalizing rule: Diagonalizes over a subsystem of the function to make it more powerful. Sometimes is merged with a copy of the recursive rule to recurse diagonalization right away (as in BEAF)
 * Examples:
 * UNAN diagonalizing rule: a[0,0,...,0,n #]b = a[0,0,...,b,n-1 #]a
 * FGH diagonalizing rule: $$f_{\alpha}(n) = f_{\alpha[n]}(n)$$ if $$\alpha$$ is a limit ordinal

Compounded exponential notation
This notation is in the form a[...] where [...] is an array of entries, each of which is either a number or a plus sign. An example of a valid expression is 10[+,1,2], which solves into the number time-plusoogol. The number outside of the array is called the base.

The rules are as follows:


 * [terminating rule] If the base is 1, set the expression to 1
 * [base rule] If the array is empty, set the expression to the base
 * [cropping rule] If the last entry in the array is 1, remove it
 * [recursive rule #1] If there are any plus signs in the array, then go to the first plus, and:
 * If the first plus is the first entry of the array, it is replaced with the expression but with the base decreased by one.
 * If the first plus is preceded by a one, set it to the base.
 * Otherwise, replace the plus with the expression but with the number before it decreased by 1.
 * [diagonalizing rule] If there are no pluses in the array and the first entry is one, decrease the first non-one entry by one, and replace all previous entries with pluses
 * [recursive rule #2] Otherwise, decrease the first entry and multiply by the base.

This should get us to $$\omega^{\omega}$$.

Numbers:


 * Dekgol = 10[10]
 * Lekgol = 10[50]
 * Googol = 10[100]
 * Demgol = 10[500]
 * Kilgol = 10[1000]
 * Myrgol = 10[10000]
 * Lakgol = 10[100000]
 * Meglogol = 10[1000000]
 * Giglogol = 10[1000000000]
 * Dekgolplex = 10[10 [10]]
 * Colossogol = 10[50000000000000000]
 * Lekgolplex = 10[10 [50]]
 * Googolplex = 10[10 [100]]
 * Demgolplex = 10[10 [500]]
 * Kilgolplex = 10[10 [1000]]
 * Myrgolplex = 10[10 [10000]]
 * Lakgolplex = 10[10 [100000]]
 * Meglogolplex = 10[10 [1000000]]
 * Giglogolplex = 10[10 [1000000000]]
 * Colossogolplex = 10[10 [50000000000000000]]
 * Googolduplex = 10[10 [10 [100]] ]
 * Googoltriplex = 10[10 [10 [10 [100]] ] ]

a[b] is the same as ab. The brackets are unnecessary.


 * Plusoogol = 10[+] = 109 [+]  = 109 8 [+]  =

109 8 7 6 5 4 3 2

That is, the exponential factorial of 10, 10!2 in HAN. It also equals 10[1,2]


 * Plusoogolplusex = 10[10 [1,2],2]
 * Plusoogolduplusex = 10[10 [10 [1,2],2] ,2]
 * Plusoogolduplusex = 10[10 [10 [10 [1,2],2] ,2] ,2]
 * Plitroogol = 10[1,3] = 10[+,2]
 * Plitroogolplitrex = 10[10 [1,3],3]
 * Plaquoogol = 10[1,4] = 10[+,3]
 * Plaquoogolplaquex = 10[10 [1,4],4]
 * Plaptoogol = 10[1,5] = 10[+,4]
 * Plaptoogolplaptex = 10[10 [1,5],5]
 * Plaxoogol = 10[1,6] = 10[+,5]
 * Plaxoogolplaxex = 10[10 [1,6],6]
 * Plastoogol = 10[1,7] = 10[+,6]
 * Plastoogolplastex = 10[10 [1,7],7]
 * Plaktoogol = 10[1,8] = 10[+,7]
 * Plaktoogolplaktex = 10[10 [1,8],8]
 * Planoogol = 10[1,9] = 10[+,8]
 * Planoogolplanex = 10[10 [1,9],9]
 * Pladkoogol = 10[1,10] = 10[+,9]
 * Pladkoogolpladkex = 10[10 [1,10],10]
 * Timoogol (pronounce time-oogol, not tim-oogol) = 10[+,+] = 10[1,1,2]
 * Paratimoogol = 10[1,+] - slightly smaller than timoogol

HAN numbers
Why not?
 * Faxul = 200!
 * Diafaxul = 200!2
 * Triafaxul = 200!3
 * Quadrafaxul = 200!4
 * Quintafaxul = 200!5
 * Sextafaxul = 200!6
 * Septafaxul = 200!7
 * Octuafaxul = 200!8
 * Nonuafaxul = 200!9
 * Decimafaxul = 200!10
 * Expofaxul = 200!1
 * Dia-expofaxul = 200!21
 * Tria-expofaxul = 200!31
 * Quadra-expofaxul = 200!41
 * Quinta-expofaxul = 200!51
 * Sexta-expofaxul = 200!61
 * Septa-expofaxul = 200!71
 * Octua-expofaxul = 200!81
 * Nonua-expofaxul = 200!91
 * Decima-expofaxul = 200!101
 * Tetrofaxul = 200!2
 * Dia-tetrofaxul = 200!22
 * Tria-tetrofaxul = 200!32
 * Quadra-tetrofaxul = 200!42
 * Quinta-tetrofaxul = 200!52
 * Sexta-tetrofaxul = 200!62
 * Septa-tetrofaxul = 200!72
 * Octua-tetrofaxul = 200!82
 * Nonua-tetrofaxul = 200!92
 * Decima-tetrofaxul = 200!102
 * Pentofaxul = 200!3
 * Hexofaxul = 200!4
 * Heptofaxul = 200!5
 * Octofaxul = 200!6
 * Ennofaxul = 200!7
 * Dekofaxul = 200!8
 * Neofaxul = 200!9
 * Goliafaxul = 200!10
 * Hyperfaxul = 200![1]
 * Dia-hyperfaxul = 200![1]2
 * Tria-hyperfaxul = 200![1]3
 * Quadra-hyperfaxul = 200![1]4
 * Quinta-hyperfaxul = 200![1]5
 * Sexta-hyperfaxul = 200![1]6
 * Septa-hyperfaxul = 200![1]7
 * Octua-hyperfaxul = 200![1]8
 * Nonua-hyperfaxul = 200![1]9
 * Decima-hyperfaxul = 200![1]10
 * Hyperduaxul = 200![2]
 * Dia-hyperduaxul = 200![2]2
 * Tria-hyperduaxul = 200![2]3
 * Quadra-yperduaxul = 200![2]4
 * Quinta-hyperduaxul = 200![2]5
 * Sexta-hyperduaxul = 200![2]6
 * Septa-hyperduaxul = 200![2]7
 * Octua-hyperduaxul = 200![2]8
 * Nonua-hyperduaxul = 200![2]9
 * Decima-hyperduaxul = 200![2]10
 * Hypertriaxul = 200![3]
 * Hyperquaxul = 200![4]
 * Hyperquintuxul = 200![5]
 * Hypersexul = 200![6]
 * Hyperseptuxul = 200![7]
 * Hyperoctuxul = 200![8]
 * Hypernonuxul = 200![9]
 * Hyperdexul = 200![10]
 * Giaxul = 200![200] = 200![1,2]
 * Dia-giaxul = 200![1,2]2 = giaxul![giaxul]
 * Tria-giaxul = 200![1,2]3
 * Giaxufaxdu = 200![2,2]
 * Giaxufaxtri = 200![3,2]
 * Giaxufaxtet = 200![4,2]
 * Giaxufaxpent = 200![5,2]
 * Giaxufaxhex = 200![6,2]
 * Giaxufaxhept = 200![7,2]
 * Giaxufaxoct = 200![8,2]
 * Giaxufaxenn = 200![9,2]
 * Giaxufaxdek = 200![10,2]
 * Triaxul = 200![200,2] = 200![1,3]
 * Dia-triaxul = 200![1,3]2
 * Tria-triaxul = 200![1,3]3
 * Triaxufaxdu = 200![2,3]
 * Triaxufaxtri = 200![3,3]
 * Triaxufaxtet = 200![4,3]
 * Triaxufaxpent = 200![5,3]
 * Teteraxul = 200![1,4]
 * Peptaxul = 200![1,5]
 * Ectaxul = 200![1,6]
 * Eptaxul = 200![1,7]
 * Octaxul = 200![1,8]
 * Entaxul = 200![1,9]
 * Dekaxul = 200![1,10]
 * Galaxul = 200![1,200] = 200![1,[1]] = 200![1,1,2]
 * Dia-galaxul = 200![1,1,2]2
 * Galactoduaxul = 200![2,1,2]
 * Galactotriaxul = 200![3,1,2]
 * Flaxul = 200! [[1],1,2]
 * Fladixul = 200! [[2],1,2]
 * Flatrixul = 200! [[3],1,2]
 * Flaquaxul = 200! [[4],1,2]
 * Flaquixul = 200! [[5],1,2]
 * Flasexul = 200! [[6],1,2]
 * Flaseptaxul = 200! [[7],1,2]
 * Floctaxul = 200! [[8],1,2]
 * Flanonaxul = 200! [[9],1,2]
 * Fladekaxul = 200! [[10],1,2]
 * Galactogiaxul = 200![1,2,2] = 200! [[1,2],1,2]
 * Galactogiantoduaxul = 200![2,2,2]
 * Galactogiantotriaxul = 200![3,2,2]
 * Giantoflaxul = 200! [[1],2,2]
 * Giantofladixul = 200! [[2],2,2]
 * Giantoflatrixul = 200! [[3],2,2]
 * Galactotriaxul = 200![1,3,2]
 * Galactotetertaxul = 200![1,4,2]
 * Waxul = 200![1,1,3]