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
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Whether you like it or not, this is actually happening in my country (America), and also most of the other Western countries, contrary to what some butthurt conservative added in as the above definition. Let's face it- whether they want to admit it or not, EVERYONE on the Religious Reich knows that this is happening, and yet many of them are pretending that it is not, and instead repeatedly, desperately insisting that atheism is "declining". BULLSHIT. Statistics have shown that we actually ARE growing. And also, if we were declining, then all of the conservatives wouldn't be all freaked out about us. I find it quite funny whenever I see them trying to flog a horse that they keep insisting is dead and decaying. :)
Whether you like it or not, this is actually happening in my country (America), and also most of the other Western countries, contrary to what some butthurt conservative added in as the above definition. Let's face it- whether they want to admit it or not, EVERYONE on the Religious Reich knows that this is happening, and yet many of them are pretending that it is not, and instead repeatedly, desperately insisting that atheism is "declining". BULLSHIT. Statistics have shown that we actually ARE growing. And also, if we were declining, then all of the conservatives wouldn't be all freaked out about us. I find it quite funny whenever I see them trying to flog a horse that they keep insisting is dead and decaying. :)
Some Fundie: Atheism is declining! Woo-hoo! We're finally overcoming the oppression! Which means that we must put an end to this growing threat, before it turns America into a communistic despotism!
Person who uses common sense: You're full of shit! Atheism is growing.
Some Fundie: *triggered* LIAAAAAAAAAAR! STAHP PEWSECUTING MEEEEE!!!!
Person who uses common sense: Just look at the statistics if you don't be-
Some Fundie: "LA, LA, LA LA LA! I'M RIGHT! YOU'RE WRONG!"
Person who uses common sense: *FACEPALM*
Person who uses common sense: You're full of shit! Atheism is growing.
Some Fundie: *triggered* LIAAAAAAAAAAR! STAHP PEWSECUTING MEEEEE!!!!
Person who uses common sense: Just look at the statistics if you don't be-
Some Fundie: "LA, LA, LA LA LA! I'M RIGHT! YOU'RE WRONG!"
Person who uses common sense: *FACEPALM*
by R0-84N November 12, 2018
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Get the gloing up mug.When a girl takes a shit on the floor or in a high chair with a hole cut out so you can watch the tail grow. A wet tail is the best.
My girl ate a whole block of swiss cheese and a piece if bananna bread. Then she said she felt like growing a tail.
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