REVIEW & SUMMARY FOR SECTION II
We've finally finished section 2 of this site, where we looked at all the popular large number notations. We learned about Knuth's up-arrows, the strong and weak hyper-operators, the Ackermann function, Steinhaus-Moser notation, some large number naming systems, Graham's number, Conway's chain arrow notation, and extensions to Conway's notation. Already the second last topic we went over (Conway's chain arrow notation) is about as far as popular discussions of large numbers go (with the exception of the occasional mention of the busy beaver function, but let's not get ahead of ourselves). But even though we've now finished going through the majority of the popular large numbers, we've barely scratched the surface of what googology has to offer - we haven't even gotten started with learning about the googological work of Jonathan Bowers (besides his -illions).
The most important point to make before we go on to section 3 is that googology has developed MUCH FURTHER than the popular large numbers like Graham's number might suggest. Many of the popular large numbers weren't so much made for the purpose of studying large numbers (i.e. googology) than to give people an idea of how big infinity is. I'm certain that Knuth, Conway, and Steinhaus could easily have devised MUCH larger numbers than their respective googological notations do if they really tried. However, they didn't do so because most mathematicians treat very large numbers as merely an "amusing limerick to mathematics", as Sbiis Saibian said in his article on Graham's number, rather than as an actual field of study. A notable exception is Harvey Friedman, who has studied various sequences in mathematics that lead to very large numbers (most famously TREE(3)), many of which are MUCH larger than numbers like Graham's number.
So in section 3, we will finally get around to learning about the lesser-known large numbers that come from treating googology as an actual field of study. Section 3 will be split into 3 sub-sections:
Section 3.1 - large number notations up to the order of ww in the fast-growing hierarchy
Section 3.2 - large number notations on the order of ww to epsilon-zero in the fast-growing hierarchy
Section 3.3 - fast-growing sequences that lead to large numbers
So are you ready for some real googology? Amateur hour is over - welcome to section 3 of this site.