How Many Digits Are in the Biggest Number Ever Used in a Mathematical Proof?

Mathematics often delves into the infinite, exploring numbers that stretch beyond our comprehension. While many of these numbers are theoretical constructs, some arise from concrete problems in fields like Ramsey theory. Among these, the number Grahams stands as one of the largest ever used in a mathematical proof.

Graham's Number

Graham's number originated as an upper bound for a problem in Ramsey theory, a branch of mathematics that deals with the conditions under which order must appear. This number is so colossal that even the observable universe can accommodate only a minuscule fraction of it. To put it into perspective, the number of digits in Grahams number itself is so mind-blowingly large that the universe is insufficient to contain any representation of those digits, let alone the number itself.

For instance, consider the Planck volume, which is the smallest measurable space in physics. Even if every digit in Grahams number occupied a Planck volume, the physical space required to represent it would dwarf the observable universe. Moreover, the number of digits in this representation would itself be infeasibly large, and even more so would the number of digits in that number, and so forth.

However, it is important to note that the upper bound of the problem in question has been refined since the time of Graham's number, and is now known to be considerably smaller, even though it remains an immense and unimaginably large number.

The Last Known Digits of Graham's Number

You can find the last 16,000,000 digits of Graham's number, represented in 16,000 pages of 10,000 digits each, here. These digits, while informative, barely scratch the surface of the astonishing magnitude of Graham's number itself.

Rayo's Number: Beyond Graham's

Unlike Graham's number, which originated as a solution to a problem in Ramsey theory, Rayo's number was intentionally crafted to be exceptionally large. Rayo's number is defined as the smallest number bigger than all numbers that can be defined using a finite number of English words. This number is so vast that it cannot be practically computed or even fully described within the bounds of the English language.

It is worth noting that while Rayo's number is huge and far surpasses Grahams number in any conceivable physical representation, it does not arise directly from a mathematical proof or theorem.

All Natural Numbers and Transfinite Numbers

The term "for all x in N" in mathematical proof often encompasses all natural numbers, theoretically including the largest number that can possibly exist within the mathematical framework. However, the concept of the largest number in a practical sense is often seen as transfinite—a concept that transcends the finite boundaries of our universe.

Inaccessible cardinals are one such example of these transfinite numbers. These are numbers that are so large that they cannot be constructed from smaller sets using standard set-theoretic methods like taking unions and power sets. In the absence of such methods, these cardinals are as large as, or larger than, any other number derived from smaller sets. The existence of inaccessible cardinals is guaranteed by certain axioms in set theory, such as the inaccessible cardinal axiom, which ensures the existence of an infinite hierarchy of such cardinals.

The statement "There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest" highlights the vastness of the mathematical universe. This universe, limited only by our imagination, gives mathematicians the freedom to explore numbers that stretch beyond our current understanding and the physical constraints of the universe.

From Graham's number, through Rayo's number, to the concept of inaccessible cardinals, the journey through mathematical numbers reveals a universe of unimaginable scale and complexity, pushing the boundaries of what we can conceive and imagine.