Hypergiganton – one of the largest known numbers!

I am sorry for my English.

Subscribed to a rather amusing author, and here's their latest article. I found it quite interesting. This article explores the author's attempt to construct the largest possible number. It's not a new idea and is somewhat relative, but the author's approach and the concept of Hypergiganton itself seemed fascinating to me. It's great that the article is available in English, but I read it in the original language (I hope the English translation is accurate).

Essentially, in the article (it’s only 9 pages), the author proposes a model for constructing yet another enormous number, possibly the largest at the moment (though, of course, we all understand that if you add 0.0(00)1 to it, it’s no longer the largest). But that’s fine — the author basically acknowledges this at the end. It’s a fun and engaging mathematical exercise, though it could be useful for developing abstract thinking and understanding science in general. I won’t claim it’s a perfect example (I didn’t dive deeply enough to verify if everything is accurate), but the concept itself is intriguing.

I won't torture LaTeX to include all the formulas here (after all, I attached the article for a reason). Briefly described in words: Hypergiganton is based on a progression limited by the mega-function Omega, Ω(10)^{Ω(10)}. Omega is a modification of the Ackermann function (and quite a clever one at that). The progression continues until the sum of its terms reaches this limit. The Hypergiganton itself equals the product of all the terms in this limited progression. Needless to say, it’s "a ridiculous amount"!

But what if we set N=Ω(10)^{Ω(10)}, meaning that Hypergiganton now equals the product of all terms in the progression from 1 to Ω(10)^{Ω(10)}? I can't even imagine such a number! But it would be incomparably larger than the current version (unless I’m mistaken). Actually, you could use the Busy Beaver Function instead of the limiting Omega function — that would be utterly mind-blowing.

Even Hypergiganton will never be computed! To calculate it, you’d need a super-quantum computer with resources exceeding all the matter in the universe. :mrgreen: By the way, it’s possible that this number (the original one from the article) lies on the boundary between theoretically computable known numbers and inherently non-computable ones.

 

All numbers are computable with imagination. Further the largest possible number is infinite. Or rather the "set" of all infinites, finites, and infinitesimals.

Imagination is greater than factualization. Abstraction more powerful than concrete.

I suggest to you, that if you cultivated your imagination, you could easily imagine the number Hypergiganton.

There are only two types of numbers in all existence....

Numerus Numerans = absract numbers

Numerus Numeratus = concrete numbers

The definition of concrete numbers rests, on the definition of absract numbers. If you take away absract numbers, then concrete numbers do not have a definition. Therefore abstraction is greater, therefore imagination is greater, than the concrete. Greater than the factual. Greater than the empirical.

 

Thank you for the response. I actually agree with you. The largest number equals infinity. But I think this whole game of large numbers has more of a psychological context than a scientific one.

I’m by no means saying that such exercises are useless—on the contrary, they develop mathematical skills (and Petrov's article even presents a very interesting modification of the Ackermann function). However, the key point here is that the human mind struggles to conceive abstract numbers or numbers that go beyond comparison with something physical. Yet, there is a constant drive toward the new and the peaks of the unknown.

We can easily imagine (albeit conditionally) a billion. Even the total number of atoms in the universe is still within the grasp of human understanding. But Graham's number is already something abstract. Petrov's Hypergiganton is even more abstract, although formally it is clearly defined and has a specific finite value.

My version, the "Super-Hypergiganton" (N = Ω(10)^{Ω(10)})—is an even larger number.