Just stumbled upon a spreadsheet ranking “well-known” googological numbers created by redditor u/Laxxius1 as a passion project. I’ve been idly interested in googology since childhood, so perusing this list felt like getting the Christmas present I’ve always wanted but kept forgetting to ask for. (It’s also just intrinsically fun if your mathematical aesthetic is wrestler, which I’m guessing is overrepresented among LWers.)
Yudkowsky’s number, which Eliezer created back in 2007 to win this contest on the xkcd forum, features at #12. (Eliezer also wrote Why isn’t googology a recognized field of math? whose sentiment I predictably agree with.) Graham’s number is all the way down at #98; TREE(3) is far larger at #49; 3^^^3 (as in torture vs dust specks) is at #145 even though it’s already an exponential tower of 3s that’s 7,625,597,484,987 layers tall. The “most finite damage with 3 MTG cards” ranks #134 at 2^^2^^7, which makes me wonder whether there are other popular games where you can easily reach very large numbers. The largest code golf-related number in the spreadsheet is loader.c at #18, Ralph Loader’s winning entry in the Bignum Bakeoff contest (“write a C program in 512 characters or less that generates the largest possible output on a theoretical machine with infinite memory”), although there are a couple of numbers purporting to be larger on code golf SE generated by ultra-short programs written in JavaScript, Python, Ruby, and binary lambda calcululs.
Rayo’s number and LNGN are defined in recursive theories. What if we use a meta-theory to create a non-recursive theory more expressive than any recursive theory, and then use that theory to create a large number? …
I don’t really like the number N. Modern mathematics isn’t made to create large numbers, so what you need to do to create a large number can be very weird. N (and probably also other extensions of Rayo’s number) is mostly a product of reverse-engineering the current philosophy of natural numbers and then to create something that barely resembles a number. So I decided to name this number to something else I don’t like: my current username. … So this number is now called ‘DaVinci’.
The #2-ranked number is LNGN, short for the quirkily-named Large Number Garden Number, which also attempts to diagonalize over set theory. It’s for now still considered by the googology community to be the largest well-defined googologism that’s not a salad number, as it hasn’t yet been proved to be ill-defined, and LNGN’s creator P進大好きbot disputes DaVinci’s superiority claim. I’m mentioning it here because of its quirky etymology, translated from the original Japanese:
Come on, friends, the large number garden is finally complete!
Let me explain the function of this garden. The first is the determination function of the address and the floor plan. When a character string is read, it automatically determines which miniature garden address it represents and in which miniature garden the floor plan of a large number garden can be reproduced.
The second is the floor plan analysis function. If you specify the address of the miniature garden and read the floor plan of the reproducible large number garden there, it will tell you the large number that the garden can produce.
The third important function is the ability to generate large numbers. Once a natural number is entered, all character strings within the upper limit of the number of characters are searched, and each is read into the address and floor plan determination function, leaving only the reproducible floor plan for each miniature garden. By enumerating them and loading them into the analysis function of the floor plan, you can obtain the large numbers that they can produce, and by putting them all together, you can create new large numbers!
Huh? Can you really get a large number with that? As usual, my ally is skeptical. But hey, here’s the floor plan for the large number garden itself. If you load this into the analysis function, it will tell you how large numbers you can generate. Huh? How many characters does this floor plan have? What’s the use of knowing such things?
Scott Aaronson wrote that “the Busy Beaver game seems like about as good a yardstick as any for gauging humanity’s progress against the uncomputable”. I think of googology the same way, just lower-status than BusyBeaverology.
Just stumbled upon a spreadsheet ranking “well-known” googological numbers created by redditor u/Laxxius1 as a passion project. I’ve been idly interested in googology since childhood, so perusing this list felt like getting the Christmas present I’ve always wanted but kept forgetting to ask for. (It’s also just intrinsically fun if your mathematical aesthetic is wrestler, which I’m guessing is overrepresented among LWers.)
Yudkowsky’s number, which Eliezer created back in 2007 to win this contest on the xkcd forum, features at #12. (Eliezer also wrote Why isn’t googology a recognized field of math? whose sentiment I predictably agree with.) Graham’s number is all the way down at #98; TREE(3) is far larger at #49; 3^^^3 (as in torture vs dust specks) is at #145 even though it’s already an exponential tower of 3s that’s 7,625,597,484,987 layers tall. The “most finite damage with 3 MTG cards” ranks #134 at 2^^2^^7, which makes me wonder whether there are other popular games where you can easily reach very large numbers. The largest code golf-related number in the spreadsheet is loader.c at #18, Ralph Loader’s winning entry in the Bignum Bakeoff contest (“write a C program in 512 characters or less that generates the largest possible output on a theoretical machine with infinite memory”), although there are a couple of numbers purporting to be larger on code golf SE generated by ultra-short programs written in JavaScript, Python, Ruby, and binary lambda calcululs.
The #1-ranked number in Laxxius1′s spreadsheet is DaVinci103′s “random idea” for beating Rayo’s number (and LNGN) attempting to diagonalize over set theory:
The #2-ranked number is LNGN, short for the quirkily-named Large Number Garden Number, which also attempts to diagonalize over set theory. It’s for now still considered by the googology community to be the largest well-defined googologism that’s not a salad number, as it hasn’t yet been proved to be ill-defined, and LNGN’s creator P進大好きbot disputes DaVinci’s superiority claim. I’m mentioning it here because of its quirky etymology, translated from the original Japanese:
Scott Aaronson wrote that “the Busy Beaver game seems like about as good a yardstick as any for gauging humanity’s progress against the uncomputable”. I think of googology the same way, just lower-status than BusyBeaverology.