User blog:Zeuxippus/Colon Function System

Colon Function System
This is my first attempt at an actual googological notation system that has stood the test of, uh, working at all. I've been reading about existing notations and other sorts of ways to create large numbers and I think this is probably the best way I've come up with. So it's probably riddled with inconsistencies and errors, but if you need anything clarified or edited, just ask! Still a WIP.

Basic Colon Function System - limit ε_0
Rule 1. n[0] = n^n = n^^2

Rule 2. n[x◆] = n[x-1◆][x-1◆] ... [x-1◆][x-1◆] w/ x pairs of brackets where ◆ is any number of colons

Rule 3. n[n◆x] = n[0◆x+1]

Rule 4. n[x◆n] = n[x[0:]]

Note: Brackets-within-brackets, as referenced in Rule 4, act just like 1st-level brackets, except Rule 2 applies first. So for example 3[0[0:]] is equal to 3[0:::], and 3[1[0:]] is equal to 3[0[0:]][0[0:]][0[0:]], but 3[0[1:]] is NOT equal to      3[0:[0:]]. Instead it is equal to 3[0[0:][0:][0:]], which is itself equal to 3[0:::[0:][0:]] and 3[3::[0:][0:]].

Extended Colon Function System - limit ζ0 (sketchy)
Rule 5. n[x:0:] = n[0[0[0 ... [0:] ... ]]] with x+1 pairs of brackets

Rule 6. n[x:z+1] = n[x[x[x ... [x:z:] ... z:]z:]z:] with x+1 pairs of brackets

Hypercolon Function System - limit Γ0 (sketchy)
Rule 7. the colon is a hypercolon of order 1, the semicolon  is of order 2, and the interpunct (·) is of order 3

Rule 8. n[0:○h] = n[0○h-10[0○h-10[0○h-10 ... [0○h-10:] ... ]]] with x+1 pairs of brackets where ○h is a hypercolon of order h (greater than 1)

Rule 9. Rules 2 - 4 can be applied to HFS so that following the ◆ there is a positive number of hypercolons of order >1. Additionally, rules 5 and 6 can be applied so that immediately preceding the outermost closing bracket there is a positive number of hypercolons of order >1.

Rule 10. If multiple hypercolons of order >1 (a.k.a. HOG1's) are present at the end of a function they behave like colons and stack accordingly.

Rule 11. A hypercolon of order h can be represented by the symbol {h}.

Examples/Googolisms:
First of all we use Rule 1 to define our first number. It is obviously an extremely large number - more than the number of atoms in the universe, in fact - but to a googologist it is infinitesimal. Don't worry, we'll get to MUCH larger numbers later.

Itol - 100[0] = 100^100 = 10^200 = gargoogol = guppychime

From there, we can repeat rule 1 - let's say an itolplex is itol[0]. We know that itol is 100[0], so we can say that an itolplex is 100[0][0].

Itolplex - 100[0][0] = 100^100^100 = 100^itol

The same concept applies to the itolduplex:

Itolduplex - 100[0][0][0] = 100^100^100^100 = 100^^4

Now we move on to Rule 2. If we want to keep increasing the size of our numbers past stacks of 100's, we need to nest all of those [0]'s together. And as Rule 2 states:

Ditol - 100[1] = 100[0][0] ... [0][0] w/ 100 pairs of brackets = 100^^100 = 100^^^2 - this is comparable to a giggol or a grangol

Ditolplex - 100[1][1] = 100^^100^^100 = 100^^^3

And to represent stacks of [1]'s:

Tritol - 100[2] = 100^^^100 = 100^^^^2 - comparable to a gaggol or a greagol

And [2]'s:

Quadritol - 100[3] = 100^^^^100 = 100^^^^^2 - comparable to a geegol or a gigangol

So to extend our system past lots of up-arrows, we need to invoke Rule 3 and add one more colon to the end of the function's notation.

Ibol - 100[0:] = 100[100] - comparable to a gugold or a boogol - this works because when the number outside the brackets is equal to the very first number inside them, a colon is added immediately after the latter number and it is set to 0

Ibolplex - 100[0:][0:] = ibol[0:] - this is not just 100[100][100]. Instead it is ibol[ibol], which is a hell of a lot larger. This is the power of a function that lies around fω on the Fast-Growing Hierarchy

Dibol - 100[1:] = 100[0:][0:] ... [0:][0:] w/ 100 [0:]'s - remember, with each successive [0:], you take the number outside of the brackets and place it within the next set of brackets, replacing the 0:, evaluate that, and move on to the next [0:]. This number is larger than Graham's Number, which is often considered to be the largest number that really "means" something

Tribol - 100[2:] = 100[1:][1:] ... [1:][1:] w/ 100 [1:]'s - this number is not just equal to 100[0:][0:] ... [0:][0:] with 10,000 [0:]'s, at it may seem - instead, it is equal to dibol[1:][1:] ... [1:][1:] w/ 99 ones

Quadribol - 100[3:] = 100[2:][2:] ... [2:][2:] w/ 100 [2:]'s - the same rules apply as above

Itrol - 100[0::] = 100[100:] - this should be self-explanatory - 100[99:][99:] ... [99:][99:] w/ 100 [99:]'s

Itrolplex - 100[0::][0::] - things get dicey here but it basically follows the same principle as above - it's equal to itrol[0::], or itrol[itrol:], and you can imagine how many ridiculous layers of nesting that entails, so now we are just one level of recursion above the ibol numbers but already leaps and bounds above their size

Ditrol - 100[1::] - the same kind of idea here as in itrolplex, but instead of two layers of insane nesting there are a hundred

Tritrol  - 100[2::]

Iquadrol - 100[0:::]

Diquadrol - 100[1:::]

Iquintol - 100[0::::]

Ixtol - 100[0:::::] - so now these colons are getting annoying to write out in full. So we'll have to make an addition to the syntax of the system to progress further - an obvious next step seems to be to make the number of colons a value that can be changed, and so here is where Rule 4 comes into play

(MORE COMING SOON)