A great googolplex is equal to g(2, 1000, g(3,2,10)) in Ackermann’s Generalized Exponential notation by André Joyce.[1] 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,[2] \(g(2,b,c) = c^{b}\) 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[2] 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 | \(10\uparrow(1+10\uparrow100)\) | |
| 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}} | |
| Hyper-E notation | 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 | \(g_{\omega^{\omega^{\omega^2}+1}}(10)\) | |
Sources[]
See also[]
Originals: googol · googolplex
Cockburn's examples: gargoogolplex · fzgoogolplex · (mega)fugagoogolplex
Googol-n-plex: googolduplex · googoltriplex · -quadriplex · -quinplex · -sextiplex · -septiplex · -octiplex · -noniplex · -deciplex · -centiplex
Googol-103n-plex: googolmilliplex · -megaplex · -gigaplex · -teraplex · -petaplex · -exaplex · -zettaplex · -yottaplex · -xennaplex · -vekaplex · -mekaplex
Bowers' extensions: giggol · gaggol · geegol · boogol · biggol · troogol · goobol · more...
Saibian's extensions: googol-minutia · googolchime · googoltoll · googolgong · grangol · greagol · gigangol · gugold · graatagold · gugolthra · throogol · godgahlah · tethrathoth · more...
Miscellany: googolbang (10100!) · googolminex (10^-10100) · googolteen · googolty · great googol(plex) · gooprol · little googol/googolbit · zootzootplex