cyclopentane's definitions
by cyclopentane October 30, 2022
Get the Cardinalmug. Doing a line of cocaine with your special other from opposite ends, then kissing when your noses touch.
by cyclopentane July 11, 2022
Get the The Alpine Kissmug. why does everyone hate sinusoids so much?
anyways, sin(x) is the height of a point on the unit sphere which is x radians away from 0 radians
anyways, sin(x) is the height of a point on the unit sphere which is x radians away from 0 radians
by cyclopentane June 27, 2023
Get the sinmug. Alpha male gigachad who invented set theory, was called a liar through his whole life, discovered an even stronger notion of infinity, went insane, died, and then everyone realised how true everything he said was.
by cyclopentane June 8, 2023
Get the Georg Cantormug. 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. Finnish Memes (often abbreviated FMs) are in an alternate reality bubble within our own world, they appear to be what people would find funny in a world similiar, but different from ours.
by cyclopentane August 4, 2022
Get the Finnish Memesmug. (n.) Content related to a piece of media that isn't a direct part of it. Think mini-episodes, standalone animations, offshoot series, etc.
by cyclopentane December 3, 2024
Get the extraserialmug.