AHAHAHAHAHA i win. (someone probably)
(if only it were well defined... Wait!)
I'll call it the ill-defined penguins' number. (at least i can have some variation of penguins' number for now.)
So.
Define a language L called Chick order set theory.
Chick Order Set Theory (hereafter called COST) consits of symbols, variables, terms and constants.
For constants, we have the largest number definable in a kth order set theory using k symbols in which the number also satisfies the property of being able to use said number symbols in this kth order set theory to describe the halting number a k-state, k-colour Halting Turing machine in a 2d plane allowing movement in all 8 directions (up,down,left,right,upleft.upright,downleft,downright)
As for symbols, we add a little special ones:
a→B modify a to meet requirements in b
a*B returns a if a is not in B, else returns the sum of all elements within B
a$b create a set consitting of a, with size b.
a%B(c) return c if a substituted into statelment is holds true. Else, returns 0.
L(n) reutrns the largest number definable ina nth-order set theory with n symbols.
All symbols above count as one symbol. (yes, that includes the %())
First order Branch Theory:
Express numbers via "branches", with leaves, and each leaves is a set. A branch takes the sum(?) of all the numbers in its leaves. Note that the only way to get leaves is using the W symbol, as in Wa,b is a branches with b leaves consisting of only bs. We also have a R symbol, which works like so:
aRb = a in Wa,(Wa,b)
We also have a F symbol
aFB = force a into set B
a@b = a and b
A/B A negate B
Ic(A,B) If c meets the requirements of A, return c, else, check for requirements in B, if it works, return 1, else, return 0.
So maybe
kth order branch theory:
We have sub(n)branches that can express a system of a k-1order branch theory as a whole.
Using this threory, i make a function called Branch(n), which does the following:
Take n, use COST to find the largest number definable using n symbols, call it k.
If k can be sufficiently expressed via a k-th order "branch" theory in k symbols, return k, else, try a k-1th order branch theory with k symbols, agai return k, until first order branch theory, if it still does not work, return 1.
Definable(a,b)
Checks if a is definable in bth order set theory. If so, returns a+b
Force(c)
var a = 1
var b = 1
Repeat until output = c(
Repeat until Branch(a) != 1 (
a++
)
if Definable(Branch(a),b) == a+b && a+b ==c
output == c
) else(
b++
)
)
Basically forces the numbers until it works.
COST(n) largest number definalbe using n symbols in COST.
Large number function:
f_n(a) = The largest number satisfying the following properties: a n-state, n-colour turing happens to halt within Force(COST(a)) steps, and the number of steps is able to be expressible using COST(a) symbols in COST, return that number.
The ill-defined penguins' number is
f_{10^100}(10^(100)10), where ^(n) means n uparrows.