# 26 definitions by **cyclopentane**

by cyclopentane November 10, 2019

by cyclopentane October 30, 2022

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

A term used to refer to black cats. Probably comes from the fact that in some lower quality pictures, black cats appear to be pitch black.

by cyclopentane August 8, 2022

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

When you are bored out of your mind and have already typed 1234567890 , 0987654321 and 1029384756 already so you first type the odd numbers then the even numbers.

by cyclopentane November 10, 2019

by cyclopentane August 9, 2023