by Use someone else's name June 20, 2023
Get the Fast food Mommies mug.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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Get the Fast Food Restaurant mug.When your family member intentionally or indirectly screws you over in a business specifically in real estate, loans, or theft.
Joe pulled a Fernandez Fast One over Peter when he sold the house to him. He screwed over Joe with ease.
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Get the Fernandez Fast One mug.Not eating all day prior to going to dinner at Chef Mickey's all you can eat buffet at Disney World.
Since we are going to Chef Mickey's today I'll have to chef mickey's fast so that I can stuff my face later.
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