The fast growing hierarchy (shortened to FGH) is a method of defining large numbers. It takes in two inputs.
We define f(0,n) = n+1. For example: f(0,3) = 4. Next step is iteration. f(1,n) is f(0,f(0...f(0,n)...)) where f(0,...) is iterated n times. For example, f(1,2) = f(0,f(0,2)) = 4. Same rules for f(m,n).
Now let's define what ordinals are. Very simplified, they're a kind of infinity.
Consider this: |||....|
This has infinite sticks, but there's a 1st stick, 2nd stick... the last stick is the ωth stick. You can have ω+1, ω+2, ω+3 etc too. For our purposes, a limit ordinal is an ordinal that has no finite part at the end (so ω+3 is not a limit ordinal but ω×3 is.).
So how can we use this within FGH? We need to define a fundamental sequence (FS). An FS is the steps we take to reach a new limit ordinal. So the FS for ω is 0,1,2... and for ω×2 it's ω,ω+1,ω+2...
We can write this as: ωn = n, ω×2n = ω+n, ω^2n = ω×n and so on. There are more ordinals, but it'll do for our purposes.
This is not the only system for an FS. There's more, but I cannot fit it in an entry.
Now consider an ordinal α. Now FGH can be defined concretely:
for f(α,n):
if α is 0, it is n+1.
if α is not a limit ordinal, it is f(α-1,f(α-1...f(α-1,n)...)) where f(α-1,...) is iterated n times.
if α is a limit ordinal, it is f(αn,n).
Let's do an example: f(ω,3) = f(3,3) = f(2,f(2,f(2,3))). I know that f(2,n) = n×2^n, so it's 1.804356 × 10^15151336, which is HUGE! Imagine how large f(ω,10) is.
We define f(0,n) = n+1. For example: f(0,3) = 4. Next step is iteration. f(1,n) is f(0,f(0...f(0,n)...)) where f(0,...) is iterated n times. For example, f(1,2) = f(0,f(0,2)) = 4. Same rules for f(m,n).
Now let's define what ordinals are. Very simplified, they're a kind of infinity.
Consider this: |||....|
This has infinite sticks, but there's a 1st stick, 2nd stick... the last stick is the ωth stick. You can have ω+1, ω+2, ω+3 etc too. For our purposes, a limit ordinal is an ordinal that has no finite part at the end (so ω+3 is not a limit ordinal but ω×3 is.).
So how can we use this within FGH? We need to define a fundamental sequence (FS). An FS is the steps we take to reach a new limit ordinal. So the FS for ω is 0,1,2... and for ω×2 it's ω,ω+1,ω+2...
We can write this as: ωn = n, ω×2n = ω+n, ω^2n = ω×n and so on. There are more ordinals, but it'll do for our purposes.
This is not the only system for an FS. There's more, but I cannot fit it in an entry.
Now consider an ordinal α. Now FGH can be defined concretely:
for f(α,n):
if α is 0, it is n+1.
if α is not a limit ordinal, it is f(α-1,f(α-1...f(α-1,n)...)) where f(α-1,...) is iterated n times.
if α is a limit ordinal, it is f(αn,n).
Let's do an example: f(ω,3) = f(3,3) = f(2,f(2,f(2,3))). I know that f(2,n) = n×2^n, so it's 1.804356 × 10^15151336, which is HUGE! Imagine how large f(ω,10) is.
by cyclopentane December 1, 2022
Get the Fast Growing Hierarchymug. by Alicatt1616 October 5, 2019
Get the colorado fast breakmug. by Kinch_Dedalus June 25, 2014
Get the fast-slow foodmug. by Zde829 December 26, 2021
Get the Gotta Go Fastmug. A knee jerk reaction to requirements or features requested by a customer which have not been designed nor any realistic time line been set for their implementation, other than expressing the sentiment that said features are "Fast Followers". This syndrome is characterized by the upending of any manageable process that establishes a specification of, a time line for, and allocation resources for implementation.
Dev1: "Hey, did you guys here what the product manager just promised the customer? We're not even done with the original features."
Dev2: "Yeah, and we have no way of knowing of when we're going to do that because there's not even any specifications."
Dev1:"I wish product would attempt to avoid Fast Follower Syndrome."
Dev2:"Yeah we're definitely going to be working over Christmas again."
Dev2: "Yeah, and we have no way of knowing of when we're going to do that because there's not even any specifications."
Dev1:"I wish product would attempt to avoid Fast Follower Syndrome."
Dev2:"Yeah we're definitely going to be working over Christmas again."
by hel112570 February 13, 2023
Get the Fast Follower Syndromemug. Girl: you never wanna hangout you’re a terrible friend
Guy: you accused me of being a bad friend, you fast ass bitch
Guy: you accused me of being a bad friend, you fast ass bitch
by Dannydavito December 28, 2021
Get the Fast ass bitchmug. by Willy S. December 24, 2007
Get the fast food steermug.