User blog:Edwin Shade/A Revised Fast-Growing Hierarchy and Extended Copy Notation

There is an issue which today I've sought to resolve, and it concerns the use of $$\omega$$ and other ordinals. By definition, $$\omega$$ is the limit of the natural numbers, and hence is infinite. It causes me a bit of head-scratching then when Googologists speak of a value such as $$f_{\omega}(3)$$ as if it were a properly defined, finite number. If I am to take $$f_{\omega}(3)$$ literally, then since $$\omega$$ is greater than all natural numbers $$f_{\omega}(3)$$ should be equal to $$\infinity$$. In fact, any value of the fast-growing hierarchy where an ordinal equal to $$\omega$$ or greater is in the subscript and the input is greater than one should always be infinite if you take the definition literally.

Of course, I know that $$f_{\omega}(3)=f_{3}(3)$$, and that the $$\omega$$ is simplified down to the 3rd number in $$\omega$$'s fundamental sequence, but that is not explicitly stated just by looking at $$f_{\omega}(3)$$. To insert an additional argument denoting the nth element of the fundamental sequence of a limit ordinal $$\alpha$$, where n is the input of the fast-growing hierarchy would clarify matters, but also complicate them. So I am doing away with the use of ordinals in the subscript altogether, and instead will use a more logical, and easier understood system.

Here is my notation, which uses a variation on TechKon's copy notation in the definitions, so any definition can be explicitly defined. Note that I am using a variation on TechKon's notation, the way I denote concatenation is using double square brackets, with the string of characters to be copied on the left, separated by a comma and then a number denoting how many times the string is to be copied. Below are some examples for understanding.

2,5=22222

^&$*$,3=^&$*$^&$*$^&$*$

banana,2=bananabanana

Note that any string of characters may be concatenated even if they are not numbers. This is why I call it "Extended Copy Notation", because it can be used to copy any mathematical operation, symbol, or really anything you want to. This becomes of use later. So for now on to the notation ! It is defined as follows.

[0]n=n+1

[1]n=[0],nn

[2]n=[1],nn

$$\ldots$$

[m]n=[m-1],nn

As of yet, we have not reached what in the FGH would be called "omega level", (which to be honest has as much sense as saying a given function is at "two-level"- it's not an accurate label because $$\omega$$ is infinity as I explained before), but that is about to change. Instead of utilizing an $$\omega$$ we will introduce a symbol which I call the "replacement symbol", denoted 'r'. When a replacement symbol appears it is to be replaced with the input of my function, (the number that appears after the brackets); this is more correct and intuitively easier to understand.

[r]n=[n]n

[r+1]n=[r],nn

[r+2]n=[r+1],nn

$$\ldots$$

[r+m]n=[r+m-1],nn

[r*2]n=[r+r]n=[r+n]n

[r*m]n=[r(m-1)+r]n=[r(m-1)+n]n

[r^2]n=[r*r]n=[r*n]

[r^3]n=[(r^2)*r]n=[(r^2)*n]n

$$\ldots$$

[r^r]n=[r^n]

[r^r^r]n=[r^r^n]n

[r^^r]n=[r^^n]n

Note that [r^^r]n has an equivalent growth rate to $$f_{\epsilon_0}(n)$$. Using copy notation however, we can easily surpass this growth rate.

[ (,m r ^^r),m ]n has a growth rate equivalent to $$f_{\epsilon_m}(n)$$.

[ (,n r ^^r),n ]n has a growth rate equivalent to $$f_{\epsilon_{\omega}}(n)$$

[W.W.B.D.O.I.F.O.A.T.T.D. otherwise known as "Work Will Be Done Once I Figure Out All The Technical Details"]