Great googolplex

A great googolplex is equal to g(2, 1000, g(3,2,10)) in Ackermann’s Generalized Exponential notation by André Joyce.Michael Joseph Halm Googology archived 2008/12/23 The value has been claimed to be equal to \(10^{10^{100}+1}\) from 2008-12-16 to 2023-11-27 in this article. With this value, this number is equivalent to a googolplex multiplied by ten. However, as explained below, the value does not match the definition. Intended value may be \(10^{10^{1000}}\), although it is not very clear because of inconsistency in Joyce's googology.
== Different values ==
In the source, it is only written that g(a, b, c) = g(a - 1, g(a, b - 1, c), c), and no information about g(a, 0, c) is written, so it is doubtful if it was properly defined at the time. If we assume that g(a, 0, c) = 0, as written in the current page,Michael Joseph Halm Googology Retrieved 2023-11-27 \(g(2,b,c) = c^{b}\) and \(g(3,b,c) = c\uparrow\uparrow{b}\). If we take this definition, great googolplex is calculated as
g(2, 1000, g(3,2,10)) = g(2, 1000, 10↑↑2) = g(2, 1000, 1010) = \((10^{10})^{1000}\) = 1010000
It should be noted that in the current page the definition of great googolplex changes to
(2, g(2, g(2, 3, 10), 10), 10) = g(3, 1, 2, 2, 10)
If we only look at the left-hand side of the equation, it can be calculated as
(2, g(2, g(2, 3, 10), 10), 10) = g(2, g(2, 1000, 10), 10) = g(2, 101000, 10) = \(10^{10^{1000}}\) = 10great googol
In the talk page of this article, Sbiis Saibian said on 2014-04-16 "Joyce's googology is exceptionally broken. He consistently makes the mistake (a^b)^c = a^(b^c). Consequently Joyce mistakenly uses the definition g(2,1000,g(3,2,10)) which translates to (10^10)^1000, assuming this is the same value as 10^10^1000. To make matters worse he defines the great operator a n-great = g(2,a+1,b) where n = g(2,a,b). Following this reasoning, since googolplex = g(2,10^100,10) , great googolplex should be g(2,10^100+1,10). Hence the discrepancy. Joyce doesn't even follow his own rule when he establishes it! By a similar reckoning since googol = g(2,100,10), it should follow that great googol = g(2,101,10). Instead it's g(2,1000,10)." It explains why the definition of great googolplex changed from (10^10)^1000 to 10^10^1000. Maybe it was the intended value from the beginning.
==Approximations==
Assuming it is equal to \(10^{10^{100}+1}\)
Notation 	Lower bound 	Upper bound
Arrow notation 	\(10\uparrow(1+10\uparrow100)\) 	\(10\uparrow(1+10\uparrow100)\)
Down-arrow notation 	\(10\downarrow\downarrow101\) 	\(269\downarrow\downarrow42\)
Steinhaus-Moser Notation 	56[3][3] 	57[3][3]
Copy notation 	9[9[100]] 	1[1[101]]
H* function 	H(3H(32)) 	H(4H(32))
Taro's multivariable Ackermann function	A(3,A(3,330)) 	A(3,A(3,331))
Pound-Star Notation 	#*((1))*(0,6,8,1,1)*8 	#*((1))*(0,6,6,3,4)*7
BEAF 	{10,1+{10,100}} 	{10,1+{10,100}}
Hyper-E notation 	E(1+E100) 	E(1+E100)
Bashicu matrix system 	(0)(1)[18] 	(0)(1)[19]
Hyperfactorial array notation 	(68!)! 	(69!)!
Fast-growing hierarchy 	\(f_2(f_2(325))\) 	\(f_2(f_2(326))\)
Hardy hierarchy 	\(H_{\omega^22}(325)\) 	\(H_{\omega^22}(326)\)
Slow-growing hierarchy 	\(g_{\omega^{\omega^{\omega^2}+1}}(10)\)	\(g_{\omega^{\omega^{\omega^2}+1}}(10)\)
== Sources ==
== See also ==
Category:Numbers
Category:Class 3
Category:Googol series
Category:Powers of 10
Category:Upper class 3