The definition is problematic - it appears you are using # as both the symbol, but also to stand in for an arbitrary expression. Also,
#
#1 m = # #m and
# n #l m = # n #l-1 n...n #l-1 n (m n's) seems to conflict with
# n#m = # (# n#m-1).
In addition, your definition for # n #^...^@ m only works when @ is an integer, but you need it to be an arbitrary expression. I believe this is harder than you think. Deedlit11 (talk) 13:44, April 8, 2013 (UTC)
I've tried to extend the Cascading-E notation to the further operations, and got some results:
First, we define Ea#^^##b = Ea(...(#^^#)^^#)^^#)^^#)...)^^#)b (with b #'s). Then we can naturally get to:
Ea(#^^##)^#b = Ea#^^##*#^^##...#^^##*#^^##b (b #^^##'s)
Ea(#^^##)^^#b = Ea(#^^##)^(#^^##)...(#^^##)^(#^^##)b (b #^^##'s)
Ea(#^^##)^^##b = Ea(((#^^##)^^#)^^#)...)^^#)b (with ## and b-1 #'s followed by it).
Ea#^^###b = Ea(...(#^^##)^^##)^^##)...)^^##)b (with b ##'s).
Ea#^^####b = Ea(...(#^^###)^^###)^^###)...)^^###)b (with b ###'s).
Ea#^^#^#b = Ea#^^###...###b (b #'s after ^^)
Ea#^^#^##b = Ea#^^#*#^#*#^#...#^#*#^#b (b #^#'s)
Ea#^^#^#^#b = Ea#^^#^###...###b (b #'s after ^)
Ea#^^#^^#b = Ea#^^#^#^#...#^#^#b (b #'s after ^^)
Ea#^^^#b = Ea#^^#^^#...#^^#^^#b (b #'s)
Ea#^^^##b = Ea(...(#^^^#)^^^#)^^^#)...)^^^# (b #'s)
Ea#^^^^#b = Ea#^^^#^^^#...#^^^#^^^#b (b #'s)
Ea#^^^^##b = Ea(...(#^^^^#)^^^^#)^^^^#)...)^^^^# (b #'s)
I can clarify it if something is unclear.
Ikosarakt1 (talk ^ contribs) 15:46, April 8, 2013 (UTC)
- I have checked your definitions; they are consistent with plugging the Veblen hierarchy into ordinal E#. FB100Z • talk • contribs 21:13, April 8, 2013 (UTC)
- Instead of Ea#^^##b = Ea(...(#^^#)^^#)^^#)^^#)...)^^#)b, shouldn't it resolve into ^'s instead? Perhaps Ea#^##b = Ea(#^^#)^(#^^#)^(#^^#)...(#^^#)a. That seems more natural, given how exponentiation and tetration work. Deedlit11 (talk) 03:49, April 9, 2013 (UTC)
- Yes, that's a good idea. However, I created some comparisons between extended E^ and Bowers' structures and conclude that #^^^# doesn't seem to correspond to X^^^X:
#^^# ~ X^^X
#^^#*# ~ X^^X+1
#^^#*## ~ X^^X+2
#^^#*#^# ~ X^^X+X
#^^#*#^#*#^# ~ X^^X+X*2
#^^#*#^## ~ X^^X+X^2
#^^#*#^#^# ~ X^^X+X^X
#^^#*#^^# ~ (X^^X)*2
#^^#*#^^#*#^^# ~ (X^^X)*3
(#^^#)^# ~ (X^^X)*X
(#^^#)^#^# ~ (X^^X)*(X^X)
(#^^#)^(#^^#) ~ (X^^X)^2
(#^^#)^(#^^#)^(#^^#) ~ (X^^X)^3
(#^^#)^(#^^#)^(#^^#)^(#^^#) ~ (X^^X)^4
#^^## ~ (X^^X)^X
(#^^##)^(#^^##) ~ ((X^^X)^X)^2 = (X^^X)^(X*2)
(#^^##)^(#^^##)^(#^^##) ~ (X^^X)^(X*3)
(#^^##)^(#^^##)^(#^^##) ~ (X^^X)^(X*4)
#^^### ~ (X^^X)^(X^2)
#^^#### ~ (X^^X)^(X^3)
#^^#^# ~ (X^^X)^(X^X)
#^^#^#^# ~ (X^^X)^(X^X^X)
#^^#^^# ~ X^^(X+1)
#^^#^^## ~ (X^^(X+1))^2
#^^#^^#^^# ~ (X^^(X+1))^(X^^X)
#^^#^^#^^#^^# ~ X^^(X+2)
#^^^# ~ X^^(X*2) Ikosarakt1 (talk ^ contribs) 09:38, April 9, 2013 (UTC)
Great and Terrible Tethrathoth is equal to E100#^^##100#2. AarexTiao 17:27, April 8, 2013 (UTC)
I still think we should clarify definition from article to make it more consistent. This is a bit confusing, because, for example, for #^^## 7-th rule says it is equal to #^#...#^# with ## exponents, but it isn't clear for me what it means. LittlePeng9 (talk) 20:04, April 8, 2013 (UTC)
Well, now # as a remainder replaced to *, because @ has been already used for the other work. Ikosarakt1 (talk ^ contribs) 16:33, April 9, 2013 (UTC)
Should Nested Cascading-E be slowly removed from the wiki?[]
It's outdated, seeing as xE^ has been created and is far more powerful. WikiRigbyDude (talk) 12:10, April 27, 2014 (UTC)
Nothing in this wiki should be removed!(unless it's sourceless or is ill-defined)
Even if one notation is not created or used in the best way,it will remain an intelectual property of it's owner,as long as it follows the rules it shall not be removed.
Also:dumb from me to keep forgeting to sign my writing.Boboris02 (talk) 16:39, October 26, 2016 (UTC)
Question[]
Why is this notation considered "incomplete"? ARsygo (talk) 08:52, 7 April 2021 (UTC)
- I think that the first section in this talk page explains an issue.
- p-adic 14:09, 7 April 2021 (UTC)