I’ve been studying the Fast-growing hierarchy and I’m trying to figure out the value of f_ε0(3). This number is undoubtedly too large to directly calculate, but I should be able to reduce it to f_n(3), where n is a natural number.
Since ε0 is a limit ordinal, you take the 3rd term in its fundamental sequence, which is ω^ω. The 3rd term in its fundamental sequence is ω^3, then ω^2*3. However, I can’t figure out how to go further from here.
I’m guessing that the 3rd term in the fundamental sequence for ω^23 is the same as the 3rd term in ω^2 (ω3) multiplied by 3, so it would be ω*9. If so, the 3rd term in the fundamental sequence for ω is just 3, making f_ε0(3) equal to f_27(3). That’s about 2 ↑(26) 3, so way too large to directly calculate.
Did you just watch the Numberphile video on TREE(Graham), too?
Of course, the fun part about this is that all of these numbers, even the ones that need transfinite ordinals to express their indices, and all perfectly finite.
I did watch that, though I was a bit disappointed that they didn’t even mention the actual position of TREE(n) on the fast-growing hierarchy (which, according to this link, is F_θ(Ω^ω*ω)(n).)
You might get a better answer at MathOverflow, or at /r/math on Reddit.