Preface

🌌 Googology: The Art of Going Big

Googology isn’t just about big numbers—it’s about how big can get bigger, and then explode beyond imagination. It’s the study of numbers so large, they stretch the limits of logic, recursion, and even language itself.

But it’s not just math—it’s a philosophy of growth.

🔢 From Googol to Graham

We start with familiar giants:

• A googol is 10^100

• A googolplex is 10^(10^100)

• Graham’s number? It’s so big it can’t be written with ordinary math—it uses up-arrow notation like 3↑↑↑↑3 to define it. (Not stopping at 3↑↑↑↑3 actually. To learn more about Graham's Number, see Graham's Number)

But googologists don’t stop there. They build notation systems that climb faster than any calculator can follow.

The ω-Series (aka w-series)

This is the ladder of operations:

• ^ = exponentiation

• ^^ = tetration

• ^^^ = pentation

• and so on...

Each step adds an arrow, and each arrow is a leap in growth speed. This ω-series is foundational to many googological constructions. When you say “w-series,” it’s shorthand for this formal structure.

🌀 Aperiotion and Recursive Speed

Now imagine a function that doesn’t just use these operations—it chooses which one to use based on its input, and then recursively applies that logic. That’s aperiotion: a meta-function that diagonalizes over the ω-series.

But here’s the twist: instead of growing at a fixed pace, aperiotion lets the speed itself grow at googological speed. It’s like building a rocket that upgrades its own engine mid-flight.

Now, if this isn't enough, I'll give a layman-friendly understatement of understatement of the even larger numbers. Remember any description of these numbers that even fits in this universe is the king of understatements and that's an understatement <--- and that's an understatement <--- and that's an understatement and so on.

Now, if aperiotion is a lot, let's just define a matrix system (we'll soon see who actually made it, and a more formal definition, but let's be friendly for now)

{a,b,0,2} is this crazy aperiotion operator.

{a,b,1,2} repeats aperiotion, and it's called expansion

{a,b,2,2} is called multiexpansion. it repeats expansion

{a,b,3,2} is called powerexpansion. it repeats multiexpansion

{a,b,4,2} is called expandotetration. it repeats powerexpansion

...

{a,b,0,3} is called aperioexpansion. it diagonalizes over the preceding sequence. It's defined as {a,a,b,2}

Of course we have {a,b,1,3}, {a,b,2,3}, {a,b,3,3}, ... that {a,b,0,4} diagonalizes over. and we get a new sequence...

{a,b,0,1,2} diagonalizes over {a,a,a}, {a,a,a,2}, {a,a,a,3}, {a,a,a,4},...

Of course we can have larger arrays...

What happens? We get larger and larger arrays. but let's be smarter

🚀 Loader’s Function: The Growth Monster

Loader’s function D(x) is designed to outpace all primitive recursive functions. Even for small inputs like D(0), the result is already bigger than 10^18. By the time you hit something like D(D(D(D(D(99))))), you’re in a realm far beyond Graham’s number.

Loader’s number isn’t just big—it’s structurally explosive, likely exceeding even the limits of 3-row BMS systems.

In fact, even for the tiniest inputs, this equation is always true:

D(x) > 15^^(2+x)

🧠 BMS: Comparing the Incomparable

The BMS format is a way to express and compare large numbers. It’s not a strict hierarchy—it’s more like a dimensional framework. Each number is expressed in rows, and the number of rows is called its level.

• 1-row BMS: Graham’s number lives here. Big, but tame.

• 2-row BMS: Begins at ε₀, where transfinite ordinals start to dominate.

• 3-row BMS: Includes ψ(Ω_ω), the limit of Buchholz collapsing functions.

• 4-row and beyond: Where Loader’s number and other monsters roam.

The content of the rows matters, but the number of rows gives a quick way to compare two numbers. More rows = deeper recursion, more abstraction, and more explosive growth.

📚 The Y-Sequence: Googology’s OEIS

The Y-sequence is the most well-defined sequential notation in googology. It’s not just a function—it’s a framework that absorbs and organizes other sequences:

• It contains the natural numbers

• The Bell series

• The Cake sequence

• And many more

It’s like googology’s own OEIS—a living encyclopedia of recursive growth patterns. The structure of the input matters more than its length: Y(1,4) can be far bigger than Y(1,3,5,7,9).

🌐 Beyond Dimensions

Eventually, googology escapes even the bounds of notation:

• DBMS: Diagonal BMS, where dimensions themselves become recursive.

• w-DBMS: Transfinite-dimensional systems.

• And beyond all this lies Rayo’s number—the largest number definable in a formal language with a fixed number of symbols. It’s not just big—it’s philosophically maximal.

✨ The Feel of Googology

Googology is like standing at the edge of a mathematical cliff and realizing the cliff itself is growing beneath you. It’s playful, poetic, and profound. It’s not just about numbers—it’s about how we think, how we define, and how we push the limits of imagination.

And the secret? Don’t go finitely slow. Let the speed itself grow at googological speed.