This article is a summary of a YouTube video "TREE vs Graham's Number - Numberphile" by Numberphile

The Fast Growing Hierarchies: Exploring Incredibly Large Numbers

TLDRThe fast growing hierarchies are a set of sequences that grow faster than any other sequence, including Graham's number and the TREE sequence. These hierarchies are built by repeatedly applying increasingly powerful operations, such as exponentiation and tetration. The fastest growing hierarchy, labeled as f_omega, grows along the diagonal of the hierarchy graph, resulting in incredibly large numbers. These hierarchies provide a measure for how fast functions grow and offer a way to explore the limits of number sizes.

Key insights

🔢The fast growing hierarchies are sequences of numbers that grow faster than any other known sequence.

⬆️Growing functions can be measured using the fast growing hierarchies, with increasing operations like exponentiation and tetration.

🌌The fastest growing hierarchy, labeled as f_omega, grows along the diagonal of the hierarchy graph, resulting in incredibly large numbers.

Q&A

How do the fast growing hierarchies compare to Graham's number and the TREE sequence?

The fast growing hierarchies grow faster than both Graham's number and the TREE sequence. They are able to generate incredibly large numbers that surpass the bounds of Graham's number and TREE.

What operations are involved in building the fast growing hierarchies?

The hierarchies are built by repeatedly applying increasingly powerful operations such as exponentiation, tetration, and other higher order operations.

What is f_omega and how does it compare to other functions in the hierarchy?

f_omega is the fastest growing function in the hierarchy. It grows along the diagonal of the hierarchy graph and generates numbers that are larger than any other function in the hierarchy.

Timestamped Summary

00:00Introduction to the concept of fast growing hierarchies and their significance in number growth.

05:54Building the hierarchy through successive operations like exponentiation and tetration.

15:03Introducing the f_omega function that grows faster than any other function in the hierarchy.