I've tried to create super-hyper-operators**(see attribution at bottom of article)** to express numbers so large that they're meaningless in this universe, even as expressions of mass, information, energy, power, time, probability, volume etc.
The idea is twofold: 1) to help me fantasize more scientifically accurately aabout imaginary alternate universes and 2) to help me appreciate how big infinity really is.
You know that 10^10 is 1 with 10 zeroes after it. 10^10,000 would be 1 with 10,000 zeroes after it. 10^100 is known as a Googol. 10^10^100 is 1 with a Googol of zeroes after it, known as a Googolplex. In comparison, 10^10,000 is 10^10^4, so a googolplex would be 10^10^96 times greater. That's assuming that you add the second exponent if you multiply, I'm not sure if you would have to exponentify the numerator to multiply the exponent so as to add the second exponent. Which just shows you even more, how massive the number is.
But then I tried inventing a new system for showing even larger numbers.
The next thing I tried to do was create a symbol to represent the next level of hyper operation. I used an up arrow but the keypad doesn't have that. 10x10=10^2. 10x10x10=10^3, and so on, we all know. 10^10=10,000,000,000. 10^10^10= 10^10,000,000,000. A massive number. 1 followed by ten billion zeroes. In contrast, a googol is 1 followed by 100 zeroes. But 1 followed by 10^10 zeroes is still nothing compared to 1 followed by 10^100 zeroes, which is a googolplex.
Now, 10^10=10>2. 10^10^10 = 10>3. 10^10^10^10 = 10>4. 10>2^2 is a googol (10^100, 10^10^2) and 10>3^2 is a googolplex (10^10^100, 10^10^10^2)
So my next big numeral to invent was 10>100 (or 10>(10^2), not to be confused with 10>10^2. The latter would have 10 "^"s with the 11th position being a ^2; the former would have 100 "^s" -- I'm doing my best to stay consistent with established mathematical symbolism).
If I wrote 10>10>10 that would not be 10>1 followed by 10 zeroes (10>(10^10) would be that), it would be 10>10 raised to the tenth power successively 10 times, or 10 to the googolplex to the googolplex to the googolplex. MASSIVE, MASSIVE NUMBERS. And then I pretended that I could comprehend the size of a universe where the average planet was 10>100 km in radius, supposed that I couldn't go any higher, and went to bed mentally exhausted.
Trying to imagine what exponential operations represent is an exercise in memory. Trying to keep it straight in your head is hard. Consider that 10^1001 is 10 times more than 10^1000. But if you had 10^1000 stars (our universe has probably 10^80 atoms in it) in your colossal universe, and added an equal amount of stars, the total would be, at twice as many stars, only 2x10^1000, whereas 10^1001 is 10x10^1000. Trying to create a universe with 10^10,000 stars is not as simple as just having 10 times as many or even a thousand times as many stars as 10^1000. A million times as many is represented by 10^1006. To get to 10^10^4 (10>2^4) in any reasonable fashion, you'd have to have circles within circles, a million of your universes within a second-order galaxy, and a million of them within a third order galaxy, and so on and so on, adding 6 to the exponent, (in this example), 16 times, and having 10,000 of the 16th order galaxy for good measure, to get to 10^10,000 (10^10^4).
There's a reason our universe is so small.
Because if this size universe is comprehensible to me, at least numerically if not visually, and considering that it's practically infinite by comparison (at 10^160, every electron in our universe now CONTAINING our universe would give you an idea) , and yet knowing that it's nowhere close to a numerically uncountable infinity, leads me to reason that this universe is not big. No, this universe is tiny, despite its massiveness in comparison to human ability to populate and travel and manipulate. Why should it be so small, yet so big?
For the lesson it teaches us about the capabilities of the source behind it. It seems so clear. The universe needs to be small enough to comprehend, and big enough to always have something new to discover in it. It must be small enough that we can see it all, and know that it's finite, yet large enough that it inspires awe and reveals our own weakness and cosmic insignificance to us. That allows us to understand the nature, and significance of the love extended by, the Creator. "What is man, that you are mindful of him, or the son of man, that thou visiteth him?" ~ Psalm 8:4.
I'm amazed by the fact that science fiction universes are so limited. Star Trek and Halo imagine galactic empires 2-500 years in the future, where technological advancements are somewhat predictable. Only Star Wars and the mmorpg Eve attempt to truly involve deep time (of continuous historiography), but this is still only tens or hundreds of thousands of years for the more developed parts of their fantasy histories. In order for Halo, Mass Effect, Warhammer 40k, Stargate, and Eve to create comprehensible universes while still giving the sense of a scope of hundreds of thousands or millions of years, they have to incorporate catastrophes, bottleneck events that reduce the galactic population and technological advancements down to comprehensible levels. They nearly always import a dark age following a humanity-wide cataclysm, borrowing from the Noachian Deluge in many cases. Isn't that interesting? What I'm waiting for is the truly visionary artists who can imagine a universe where ships are planet-sized and shoot stars as weapons, and humanity numbers in the pentillions. No one has attempted anything like this, yet.
~ Rak Chazak
credit: I got the word "hyper operator" from a comic (uses pun humor of a sexual sort in the link, FYI) on the web-comic site Saturday Morning Breakfast Cereal: http://www.smbc-comics.com/?id=2615