Fast Growing Hierarchy
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.
The Urban Dictionary Mug
Heavenly Mug This mug has been sent from the heavens. I'm too broke to buy it. But one day... I will. I will be mugged, dammit!
Why am I here? I don't know how I got here, but I can't stop writing weird things on the cup...😅 Help me. I have a test to study for. A family. Also, if I wasn't broke I would buy 10,000 of these mugs. They look highly entertaining. Love this website, and I probably will fail the test. 🙃
I took time designing it but wasn't sure, online tools being what they are, that what I was seeing was for sure what I'd get. Very much appreciated the customer service communication which verified that what I'd designed was what I wanted, and the shipping was quick too.
Item came on time as promised
Came within a week and it's exactly what I ordered, my friend will love it!
Sus cup I bought the sus mug for the sus king Daequan
Good quality, packaging shipped well, arrived quickly.
My mug came in broken but Urban Dictionary replaced it at no extra charge!
Excellent mug excellent service
this is the best for coffee and hot coco especially if you make the hot coco in it then pour it on your significant other and do body shots 😋🥱
I can pass away peacefully. This mug is everything I’ve ever needed and more. Fat thank you, Urban Dictionary. <3
I was really excited to receive this mug and when it did come it was perfect quality. My only complaint is that the color I choose was green teal but it came in yellow.
As always, easy to order and not-too-long of a wait for the finished product to arrive. It’s well-printed, and very sturdy. A great gag present for wedding party members.
The mug , color and saying are perfect! PMEO is what I say at work everyday. It has become a favorite saying for my coworkers when things go haywire!
Quality and style are outstanding relative to price point.
Love it! It is my favorite mug. Easy to hold because of its shape and weight. Now my go-to mug.
My wife is truly beautiful and this cup was perfect for her.
Awesome mugs! My GF laughed her ass off, lol!!!
t-this mug changed my life. At first i was a loner but then i bought this mug and i became HIM. I thank this mug everyday for its blessings
Cute, good quality, *****!
Review Details
Pro Customization
Create unique products with your own words and definitions
Live Preview
Personalize Your Design
Debug: Product Metadata
| Key | Value (click to copy) |
|---|---|
Copied! |
Return Policy
Made Just For You
Each product is custom-printed with your unique text, making it truly one-of-a-kind.
Defect-Free Guarantee
If your product arrives with printing defects, damage, or quality issues, we'll send you a free replacement.
Custom Orders
Due to the personalized nature of your order, we don't accept returns for change of mind or sizing issues.
Questions about your order? Contact our support team for assistance.