Quest To Find The Largest Number

Summary

In this video, the host delves into the concept of the largest number that can be written down, steering clear of infinity and exploring the practical limitations of space. Starting with a simple SMS text message, the video examines increasingly complex methods to define large numbers, using factorials, Gram's number, programming languages like Python and Haskell, and finally Lambda Calculus. The host also discusses the limitations of computation, including the halting problem, and introduces fast-growing ordinals like the Buckle’s ordinal to conclude. The exploration highlights just how rapidly numbers can grow, even within a restricted text space.

Takeaways

• 😀 The largest number we can write down depends on the space available, and even a 160-character SMS can hold surprisingly large numbers.
• 😀 We can start by writing a very large number using a simple power tower, which grows fast but is still small compared to more complex numbers.
• 😀 Gram's number is often used as an example of a massive number, but we can take it further by using operations like factorials to make it even larger.
• 😀 Instead of just writing the name of a number (like Gram's number), we need to find a way to actually generate it within the space limit.
• 😀 Programming languages such as Python can generate large numbers, but more compact programming languages like Haskell might allow for even more efficient number creation.
• 😀 The challenge lies in finding programming languages that can fit large numbers into the constraints of space (like 140 bytes for an SMS).
• 😀 Lambda Calculus (LC) is a universal and compact system that can generate huge numbers using only one function and no predefined data structures.
• 😀 By using Church numerals in Lambda Calculus, we can define numbers where the number of function applications represents the value.
• 😀 Binary Lambda Calculus (BLC) can compress Lambda Calculus even further, allowing numbers to be defined with fewer symbols and bits, fitting more into limited space.
• 😀 The halting problem presents a limitation in computing the exact value of some large numbers because it’s impossible to algorithmically determine whether certain Lambda expressions will stop or run forever.
• 😀 The Buckholz ordinal is a massive, fast-growing ordinal function that fits within the character limit, and while extremely large, it doesn't make a significant difference in ranking when used with other massive numbers.

Q & A

What is the main challenge in determining the largest number that can be written down?

The main challenge is that there is no simple answer to the question. The largest number depends on the method of representation and the space available, and even the concept of infinity complicates the search for an absolute largest number.

Why is the idea of using a number like '10 to the power of 160' mentioned?

'10 to the power of 160' is used as a basic starting point when considering large numbers that can fit within the 160-character limit of an SMS text message. It serves as a simple way to approach large numbers but is far from the largest possible number.

What is a Power Tower, and how does it help in constructing larger numbers?

A Power Tower is a stack of exponents, where each level represents an exponentiation of the previous one. It grows much faster than regular exponents, allowing for the creation of much larger numbers in a compact form.

How does the use of factorials increase the size of a number?

Factorials, denoted as 'n!', grow extremely fast. For example, '159 factorial' is a much larger number than typical exponential growth, significantly increasing the size of the number written.

What is Graham's number, and why is it relevant to this discussion?

Graham's number is a famously large number used in mathematics, specifically in Ramsey theory. It is so large that it cannot be fully written out, even if every particle in the universe were used to write digits. It's used here as a benchmark for extremely large numbers.

Why is it not enough to just reference large numbers like Graham's number?

Referencing large numbers like Graham's number doesn't fit the rules of the problem because it would rely on using external definitions or assumptions, which goes beyond the constraints of a single text message with a set character limit.

What role do programming languages like Python or Haskell play in representing large numbers?

Programming languages are useful for defining algorithms that can generate very large numbers. Python and Haskell are mentioned as languages that can be used to write programs that theoretically generate numbers like Graham's number, though the programs themselves are limited by the size of the code.

What is Lambda Calculus, and how does it help in constructing large numbers?

Lambda Calculus is a simple, universal programming language that uses only one function. It allows the creation of numbers by applying a function multiple times, and its simplicity makes it a powerful tool for expressing large numbers in a very compact form.

What is Binary Lambda Calculus (BLC), and why is it more efficient than standard Lambda Calculus?

Binary Lambda Calculus (BLC) is a compressed version of Lambda Calculus that uses bit patterns to represent functions. This allows for much more compact representations of large numbers, fitting them within limited character spaces like SMS messages.

How does the concept of Omega, and its extensions, contribute to defining large numbers?

Omega (Ω) is used to represent a fast-growing ordinal number. By repeatedly applying Omega to itself (e.g., Omega + Omega, Omega squared, etc.), we generate larger and larger numbers that grow extremely fast, far surpassing typical finite numbers and even traditional operations like exponentiation.