Okay, thanks for your analysis Deedlit, I think my number is very unclear a this point, I just had to write all that before I forgot about the idea. I think the levels could be made a lot more powerful very easily, but the main power of the number is the recursions that come up later, so I'm not even sure the Levels have to be that strong.
Deedlit wrote:Not sure what you mean by this. Taking the xkcdth step of level xkcd is not as good as taking the final number of level xkcd - that is the whole point of having a terminating process.
Okay, let me clarify. You have Level xkcd, then, you feed some number into it, again, I don't know if the number you feed it matters much, because at some point functions are so strong that they grow to the same size whether you feed them g64 or 2
But just in case, let's define a function. In my above post I said that "provided the number is increased until all of them are nines", which means that if a step returned 1000, this step returns 9999. This is necessary because the levels run on 9's, a 8 on there makes the whole of it much less powerful, but I think I can do better:
Returns x times 9. So, 9(9), returns 999999, and 9(1000) returns a string of 1000 9's. Now I define the initial seed as:
A string of xkcd 9's. This would be feed into Level 1 instead of just a 9. Just in case this matters at the end.
Now, when I say "take the xkcdth step", I mean, you put ۹ into level xkcd, you take the last number (the uncompressed one, a 1 with a bunch of zeroes) and call it x. You run 9(x), to get a string of 9's that is the length of that number, you call it, huh, ۹. Then you feed ۹ into Level xkcd, you take the last number, call it x, run 9(x), to get a string of 9's of that length which is ۹. You feed ۹ into Level xkcd. You continue on and on until you get ۹[xkcd]. Let's call it ۶.
Now, for the "xkcdth step of the level xkcd-1", you take ۶ into level xkcd-1, you take the last number x, call 9(x), get ۶, feed it into level xkcd-1, and repeat another xkcd times. You get ۶
Now that I (think) I have some clarified recursion, I notice that the levels are getting weaker while the seed is getting bigger. That may not be optimal, maybe the number would be larger if I kept feeding it to level xkcd all the time, but did it one time less.
No, actually, I can keep increasing the levels and decreasing the times, so the number of nines I add at every recursion is greater than the last time, instead of less, and I ensure that it halts.
So, lets redefine "xkcdth step of the level xkcd-1" to be: you take ۶ into level 2xkcd, you take the last number x, call 9(x), get ۶, feed it into level 2xkcd, and repeat another xkcd-1
times. You get ۶. Yes, this sounds better. You continue until you get ۶[2xkcd], this will be ۹
(yeah,I don't think people have used colors in the thread to define numbers, I think I'll use that for an extra set of recursive recursions at the end).
Now, for "xkcdth step of the level xkcd-2", use ۹
as the seed, feed it into level xkcd2
, call 9(x), get ۹
, and repeat (by repeat I don't mean you feed ۹
into itself that many times, you have to take ۹
, for the next repetition, and then ۹
, for the next repetition) another xkcd-2 times. At the end, you get ۶
"xkcdth step of the level xkcd-3", is like that, but it's feed to the level xkcdxkcd
xkcd-3 times. "xkcdth step of the level xkcd-4" is feed on level (...((xkcd)^xkcd)^xkcd)^...(xkcd times)...^xkcd)^xkcd xkcd-4 times. "xkcdth step of the level xkcd-5" feeds on level xkcd↑↑↑...(xkcd times)↑↑↑xkcd xkcd-5 times. "xkcdth step of the level xkcd-6" feeds on level xkcd↑↑↑...(xkcd times)↑↑↑xkcd↑↑↑...(xkcd times)↑↑↑xkcd↑↑↑...(xkcd times)↑↑...[xkcd times]...↑↑↑xkcd xkcd-6 times.
Note that I'm doing these progressions in the same way that I did the Level progressions, so this isn't also well defined, but eventually, you'll get to "the xkcdth step of the level 6", which is to be defined as the last number 9(x) as seed, that is feed on the level xkcdsomethinghuge that would take xkcd paragraphs to explain, or something
6 times. Decreasing like this until the last number 9(x) is used as seed on a higher and higher and higher and higher and higher level, 5, 4, 3, 2, and finally, once, on (higher and higher)*5 9(x)'s, until everything collapses on a number. That number is x. Use 9(x). This number will be ۹
That's for the first part of my confusing paragraph.
"Step xkcdth-1 of the level xkcd", would then no longer start at level xkcd, it would start at the last level used to produce ۹, which I'll call level ߖ. And while Level ߖ would increase as,ߖ ߖ2, 2
, ߖ ^ ߖ ^ ߖ^ ( times ߖ ,( ߖ↑↑↑... Man, the ߖ character sucks when writing, I don't know where a character will end up next, but it follows the progression of above; 9(x) is no longer used, instead, at this point, 9(9(x)) is used, but it grows as below. And the times of repetitions would no longer be xkcd times, they would be 9(ߖ) times.
So, put ۹ into level ߖ, get x, call 9(9(x)), get ۹, put it into level ߖ, call 9(9(x)) get ۹... repeat until you get to ۹[9(*)], this is ۶.
"Step xkcdth-1 of the level xkcd-1": Put ۶ into level 2ߖ, get x, call 9(9(9(x))), get ۶, put it into level ߖ2, repeat 9(*)-1 times call 9(9(9(x))) get ۶. Repeat that until you get to ۶[9(**)-1], this is ۹
"Step xkcdth-1 of the level xkcd-2": Put ۹
 into level ߖߖ
, get x, repeat 9(*)-2 times call 9(9(9(9(x)))), get ۹
, put it into level ߖߖ
, repeat 9(*)-2 times call 9(9(9(9(x)))) get ۹
... repeat until you get to ۹
[9(**)], this is ۶
Okay, you should get the drift by now about that goes on, and this only scratches the surface.
First, I think I need to define the progression of the levels, what's wrong with this progression?
x, 2x, xx
, (((x^)x^)...x times...^x)^x)^x, x↑↑↑...x times...↑↑↑x, x↑↑↑...x times...↑↑↑x↑↑↑...x times...↑↑↑x↑↑↑...x times... [x times]
It's meant to be a progression faster than A(), or something, so every step things get a lot more enormous than last time, surely the should be a way to formalize this growth unambiguously?
Anyway, I know that even if I made a clear definition of the idea and produced a definite number, it wouldn't dare to dream catching up to the numbers posted up to page 17, I believe, but maybe something can be done about that...