The Exponent Functions
Section I
The Ultrex Group
Let's go through the first function...the Ultrex. Ep (i'll be calling him that) has a overly complicated definition for Ultrex:
n u 1 = n^n
n u b = umm...ahh...
However, we can look at this example.
2 u 4= 2^2^(2^2)^(2^2^(2^2))^(2^2^(2^2)^(2^2^(2^2)))
=2^2^4^65536^(2^2^2^2^17)=2^2^4^2^((2^4)*2^2^2^2^17)
2^2^4^2^2^(2^2^2^17+2^2)
~2^2^2^2^2^2^2^2^17
2 u 3, aka 2^2^4^65536, translates to 2^2^2^2^17.
Ep claims that those inclined to Tetration would be rapidly left behind by this function... yet his own barely reaches n^^2^b, which is not even a recursive step from n^^b.
However, he adds more definitions to the function u. For example... (apply RTL)
n uu b = n u n u b
n 99u b =n u n 98u b and n 1u b is n u b
n u b = n (n u b)u b
Here is when Ep's recursion fails...
n ku b < n (n u (b+k-1))u b
n (u) b = n (n (n u b)u b)u b
This is horrendously built upon! Instead of going up, Ep is going sideways! Instead of trying to apply a primitive recursion on u, Ep is going for recursive steps! To add on that, ku literally builds upon u itself. Luckily, Ep does do the primitive recursion at [u]. Then... Ep makes the building upon u mistake again...and then it becomes complete nonsense...
Let's now go through HOU. Firstly... (RTL!)
n hu 1 = n^(n)n
and the number of arrows for the next increment is n u 2...
then n u 3...
This creates a function upper bounded by n^((n u b)+1)(n u b).
nhu' = n hu n hu n hu... n hu's... hu n hu n
which is one primitive recursion
The hu function has a growth rate of w. nhu' is w+1. (we assume fgh)
Again, arrows and extensions are nonsense.
Next, to SHOT!
n s 1 = n^(n)n
n s k = n s (k-1)^(n s (k-1))(n s (k-1)) and remove all parentheses
Now, we're just gonna ignore the {} [] extensions and go to the s^ thingy.
2 s 2 with s^ is 2 s^^ 2 s^^^^ 4,
2 s^^ 2 s^^^ 2 s^^^ 2 s^^^ 2
2 s^^ 2 s^^^ 2 s^^^ 2 s^^2 s^^2...
but...
oh.
So, SHOT is literally HOU+.
LHOT...?
3 l 1 = 3^^3 s^^ 3
3^^3 s^3 s^3^3
3^^(3 s 3 s 3^3)
3 s 3 s 3^3
which basically... umm...
WE DON'T GET WHAT YOU MEAN BY SPECIAL ARROWS, EP.
The UCG function.
Let's go through the first function...the UCG function.
ucg(n)=n>n>(n>n)>(n>n>(n>n)...>(copy of it behind)
which means ucg(n)<cg(n+1)...
oh.
Let's go through the Superconway function, then.
The... uhh... umm...
The Popble function
The popble function at last! Let's see the steps...
N is ultrexed N times to bound N.
This number is then placed before and after as many Knuth upward arrows.
The resulting number is then used as the base,bound, and number of iterations in the Ultra Conway-Guy Function,using Conway right-arrows..
The then resulting number is placed within that many Moser polygons each of that many sides.
This resulting number is then used as every term of a Bowers Exploding Array measuring as many in each of as many dimensions.
The entire cycle of operations repeats N times.
The first step has at most n^^^^n power, then fgh w, fgh w^2, fgh w again and fgh w^w^w...
The popble function is a salad function! None of the steps other than step 5 even nudges the value by anything! If the 5th step wasn't present, this function would not even score a w^3 in FGH. Not even a w^2+2.
Section II
Epstein's -illion enumeration system
Now, we come to a point where Epstein's systems at least aren't complete nonsense or ill defined.
His rules are:
I) Systematic
II) Compatible with short/long scale illions
III) Not Random
IV) Having words work (???)
His popble is supposed to represent polygon, point and blast... but has forgotten polyiblast... Popble literally sounds like a normal english word, while polyiblast sounds so much better. Already IV is violated, but at this point I'm not surprised as we all know Ep is a complete beginner in googology.
Here's the older versions of popble and what they are comparable to, upper and lower bounds.
R will stand for the resulting number while n is the input
First Ver.
1) n^n
2) R[2]R using SEAN
3) CG(R) (Conway-Guy function)
4) R[R+1] using R[1]@ = R@^R@, R[n]@=R[n-1][n-1]..R [n-1]'s..[n-1]@
5) {R,R(R)2} using BEAF
6) Repeat steps 1-5 n times (ignore this step once done, and replace n in step 1 with R)
Comparable to f_(w^w^w)+1(n) (this is FGH)
2016 2nd Ver.
1) At most n^^^^n, n nu n (ultrexed n times)
2) R[2]R
3) CG(R)
4) R[R+1]
5) {R,R(R)2}
6) Repeat steps 1-5 n times
Comparable to f_(w^w^w)+1(n)
In- in 5 years, there has been MINIMAL improvement! Let's just get on, we don't really care about rule 5.
Now, firstly, Ep does pretty well, with some standard prefixes attached to the illion.
(in this case, it's 1,000,000^(prefix value), aka 10^6*(pv))
And attaching more prefixes, and more, and more, adds them together. So far so good.
Ep also uses different prefixes for multiplication and addition, which is great! Next!
Ah, the hyphen! Let's go through some examples...
Quintillio-illion is a Quettillion based upon SI prefixes. Of course, we have:
Decaquintillio-illio-illion. 1000000^1000000^10^31.
Ep is correct that this is just short of the first Skewes' number.
And Yottillio-illio-illion, roughly E24#4, is larger than E964#3, an upper bound for the second Skewes'.
The hyphen is just a small connector to separate illions.
Now, the apostrophe.
It works like this:
(n'k)-illio-illio-illio...-illion = (k^^n)-illio-illio-illio...-illion
Of course, n can be in the form (n'k)-illio-illio-illio...-illion as well...
'da' works like:
n da b ' @ = (b^n)' @
'de' works like:
n de b da c = b^^n da c
Ditto for di and do. Now, du...
n du c @ = c^^^... n ^'s ...^^^n @
This is a hell lotta better than all the di and do's. This also means, that Ep's illio, system has BIG gaps.
'sha' also works similarly to du, like:
n sha c @ = (c s n) @ (SHOT operator)
Now, here is an epic fail for the system: pa. Totally arbitary, which breaks rule three.
Pa works like this:
n pa c @ = (c popbled n times) @.
And as you can see, SHOT is at w+2 level, but due to Bowers' exploding arrays in popble, pa is (w^w^w)+2. EPIC FAIL FOR ALL THE NUMBERS IN BETWEEN...
Time for re, which I hope can at least have the fgh ordinal of (w^w^w)+w...
b re k re c (op1) m (op2) is one structure for 're'.
This becomes (k(op1)bc)(op1)m(op2) based upon the example.
n rere b re c (op) is re (b (op) nb^2) re c (op)
n re-re is nb^b, while n re're-re is nb^^(nb^b).
Guess what... THIS DOESN'T EVEN RAISE THE ORDINAL BY A MEASURABLE AMOUNT!
ARGGGGGGGG-
Anyways, let's go on with pre...
oh wait, pre is just re but second order.
Section III
Titled and Epstein numbers.
Let's calculate the ISN first.
We have an number n at most at (w^w^w)+w level.(n= f_(w^w^w)+w Popbling has no effect so let's leave it there first.
Firstly, use n as a upper bound. This is already extremely generous. I'm not going to read it and leave it as there. (all the functions have no effect!)
Guess what... ALL the titled numbers are bounded by n!
Let's continue with the Alphabet numbers.
n^^n npp times n... umm...
Let's just bound 'a' with our previous n.
Now, the salad function described in the page I will call 'the disappointing function' or d.
To jump to the next number, A (non optimal), just apply d^a(a).
And finally, not all numbers are bounded by n with Epstein devising a function.
We now have (w^w^w)+w+1.
And E(k) is at (w^w^w)+w2, E(n,1) is (w^w^w)+w2+1.
Another epic fail: Ep literally asks us to 'continue' on his E function series. This is being:
1) Lazy
Not only that, this means his E function is ill defined, thus ALL his numbers beyond that point is ill defined.
Name: Louis Epstein
Notation making: F
Function making: C-
Understanding of growth rates: F-
Understanding of notation: D
-illion system present: T
Usefulness in representing numbers smaller than 10^^^10: A+
Usefulness in representing numbers in form {a,b,1,2}: B
Usefulness in representing numbers for Linear and Dimensional BEAF arrays: F-
Overall illion system Grade: B (good)
Total score: D+ (barely pass)