Past few days I have tried to show you some big numbers and math to go even bigger. It's pretty much all the preparation I could give before getting into the actual topic the Graham's number.
Before we start I would like to reiterate a few things: our insanely huge number that expands all of the multiverse, the Knuth's up-arrow notation and how adding just a single arrow launces the number to a completely new magnitudes. And finally the fact that mere 3↑↑↑↑3 is already much bigger than that multiverse-spanning meganumber.
With Graham's number we start with 3↑↑↑↑3 which is called g1. Then we construct the next level, the g2 by putting g1 number of arrows between those number threes! Blown away yet? Well... we don't stop there. To get to the Graham's number we keep doing this a total of 64 times, always using the result of previous iteration as a number of arrows for the next one. Yeah, it's BIG.
The fact that the number we start with is already something that is impossible to rationally comprehend truly makes you question your sanity gazing at these immerse numbers. Surely we need to be approaching infinity with such things?
Not even close. Let me introduce you to TREE(3). Compared to the magnitude of it Graham's number could as well be zero. How about SSCG(3) then? And the craziest thing is these numbers aren't just made up as a sort of "come up with biggest number imaginable". They actually have some obscure, but real uses.For the mathematicians just showing off, there is the Rayo's number.
Now the infinity, that's another thing. What if I told you there is different orders of magnitude of infinities as well?