If BEAF is in Functions category, should this and Hyper-E notations be there too? —Preceding unsigned comment added by LittlePeng9 (talk • contribs)
Seems like a good idea. Deedlit11 (talk) 10:01, April 6, 2013 (UTC)
I think the limit growth rate of Cascading-E notation can be far beyond ε0.At the end of this page , Saibian came up with something like E100#^^##100, E100#^^^#100, E100#^^^^#100, E100#^^^^^#100, E100{#,#,#}100 (using BEAF), E100{#,#(1)2}100, E100{#,#(2)2}100, E100{#,#((((0,1)1)1)1)2}100, E100{10^^^100&#}100, E100{10^^^^100&#}100, E100{#&#&#&#&#&#&#&#}100, etc. That makes it as powerful as BEAF.
However, that isn't the actual growth rate of Cascading-E notation because we use BEAF in it. How about the growth rate without using BEAF? Here is some of my ideas.
Ea#^^#b can be written as Ea f(#) b, where f(x)=x^^x=E(x)1#1#2. Further, f(#) can be any recusive function whose "base number" is #, such as f(x)=E(x)1#1#1#1#1#1#2=x^^^^^^x, f(x)=E(x)x##x, f(x)=E(x)x##x#1#2, f(x)=E(x)x##x##x, f(x)=E(x)x###x, f(x)=E(x)x#######x, f(x)=E(x)x#^#x, f(x)=E(x)x#^##x, f(x)=E(x)x#^#^#x, f(x)=E(x)x#^#^#^#^#^#^#^#x, etc.
Then f(x)=E(x)x g(#) x, where g(#) can be any recusive function whose "base number" is #, such as g(x)=E(x)x#x#x#x, g(x)=E(x)x##x, g(x)=E(x)x###x#x, g(x)=E(x)x#^#x, g(x)=E(x)x#^#^#^#^#^#x, and g(x)=E(x)x h(#) x, where h(#) can be any recusive function whose "base number" is #.
I think, Ea f(#) b has comparable growth rate to f(n)&n, or g(n)&n&n, or h(n)&n&n&n. Just imagine this function:
En f1(#) n, where f1(x)=E(x)x f2(#) x, where f2(x)=E(x)x f3(#) x, ... where fm(x)=x^x. The m here is Cascading-E notation nests. It has comparable growth rate to n^n&n&n&n...&n (with m &'s). If m=n, we'll get the actual limit growth rate of Cascading-E notation --- the LVO. {hyp<hyp··cos>cos} (talk) 04:05, July 13, 2013 (UTC)
- I believe it's good extension, but we need more formalization on it. Also, it's highly dependent on what you defined increment to f(n). If Ea f(n+1) b = Ea f(n) a f(n+1) b-1, then your idea will be about notation up to \(\Gamma_0\) range (and I already wrote about such extension it in my blog post). If, however, Ea f(n)+1 b = Ea f(n) a f(n)+1 b, then it really can reach LVO. The difference is big, as in the first case we increment the argument of function in order to get "new type" of hyperion-array, and in another way, we increment the result of function. Ikosarakt1 (talk ^ contribs) 10:16, July 13, 2013 (UTC)
- The rule is that @a f(#)*# b=@a f(#) a f(#)*# (b-1), where @ represent any sequence in Cascading-E notation. {hyp<hyp··cos>cos} (talk) 15:31, July 13, 2013 (UTC)
Extended Cascading-E Notation[]
Ea#^^##b = Ea(#^^#)^(#^^#)...(#^^#)^(#^^#)a (b-1 ^'s)
Ea#^^#^#b = Ea#^^##...##a (b #'s after ^^'s)
Ea#^^#^^#b = Ea#^^#^#...#^# (b-1 ^'s after ^^'s)
Ea#^^^#b = Ea#^^#...#^^#a (b-1 ^^'s)
Ea#^^^##b = Ea(#^^^#)^^^#b
Tethrathoth = E100#^^#100
Tethrathothtethrathothex = E100#^^#100#2
Tetrithrathoth = E100#^^#*#3
Teterthrathoth = E100#^^#*#4
Tetppegathrathoth = E100#^^#*#5
Tetgoldthrathoth = E100#^^#*#100
Tethroothrathoth = E100#^^#*##100
Deutero-tethrathoth = E100#^^#*#^^#100
Hecato-tethrathoth = E100(#^^#)^#100
Todthrathoth = E100(#^^#)^(#^^#)100
Todthrathor = E100(#^^#)^(#^^#)^(#^^#)100
Territhrathoth = E100#^^##100
Treathrathoth = E100#^^###100
Tiganthrathoth = E100#^^#^#4
Torgethrathoth = E100#^^#^#5
Tugolthrathoth = E100#^^#^#100
Tethrathor = E100#^^#^^#100
Penthrathoth = E100#^^^#100
Hexthrathoth = E100#^^^^#100 AarexTiao 01:44, January 6, 2014 (UTC)
LECEN[]
Sbiis Saibian changed the definiton of tethrathoth-ex-terrible tethrathoth to E100#^^#>#(1+E100#^^#>100). So this might be the beginning of LECEN. Wythagoras (talk) 17:32, January 16, 2014 (UTC)
- Actually, LECEN (Limited Extension Cascading-E Notation) is the notation that goes up to the Great and Terrible Tethrathoth, what Saibian already revealed. The new notation you probably meant is "Extended Cascading-E Notation" (xE^), which will go even farther than that. So hype! -- ☁ I want more clouds! ⛅ 02:22, January 17, 2014 (UTC)
- The next next level has been unlocked. -- ☁ I want more clouds! ⛅ 05:58, January 31, 2014 (UTC)
- I haven't expected this... Ikosarakt1 (talk ^ contribs) 21:34, January 31, 2014 (UTC)
- The next next level has been unlocked. -- ☁ I want more clouds! ⛅ 05:58, January 31, 2014 (UTC)
@Littlepeng9 Yup. E#, xE#, E^, xE^, and ExE all should be "Functions" along with being "Notations". UAN under construction 08:28, February 16, 2015 (UTC)