User blog comment:Simplicityaboveall/Insanely Fast-Growing Functions/@comment-28606698-20171025182658/@comment-30754445-20171029151930

I'm sorry, Simplicity, but your H function - as you've defined it - has a growth rate of no more than ω×2:

H(10) = h(0,10) = 10 [10 arrows] 10 ~ fω(10)

H(11) = h(1,10) ~ fω+1(10)

H(12) = h(2,10) ~ fω+2(10)

.

.

H(100) = h(90,10) ~ fω+90(10) ~ fω×2(90)

which is much much smaller than fω²(3), let alone fω²(10)

Indeed, with Denis' up-arrow extension we can write the following precise equation:

For n>10: H(n) = 10↑ω×2(n-10)

And if you prefer an equation without ordinals:

H(n) = { 10, 10 , n-10 , 2 } in BEAF = (2,0)|(n-10) in my own array notation.

Analyzing your other functions, I see that:

gn(k) ~ { 10, k , n , 3 } = (2,n)|k

Great function G(k) ~ { 10, 10, n , 3 } = (3,0)|k

Godoolus(k) ~ { 10, k , 1 , 4 } = (3,1)|k

Godoogol ~  { 10, 10100 , 1 , 4 } = (3,1)|10100

That's a pretty large number, but it isn't anywhere as big as you claim it is. It's comparable to N3.10 in my letter notation or to 4→4→4→4→4→4 in Conway Chained Arrows, and in the FGH it can be approximated as fω×3+1(10100).

As for the use of ordinals: They are nothing more than tools, but they are among the most powerful tools we know to generate fast-growing functions. fω²(4) (~ N3.35) is already bigger than your Godoogol, and ω² is a tiny ordinal.

The problem is that you aren't using this tool properly. You seem to have some basic misconceptions regarding how ordinals are used in googology, which is why you are oblivious to their immense power (and why you're constantly overestimating the ordinal strength of your own functions).

Trust me, once you learn to build an actual (say) Γ₀-level function using the tools of ordinal analysis, you will no longer wonder why the people here are using them. Γ₀-level recursion is downright crazy. It's completely beyond human understanding, unless we organize the process with ordinals (or something equivalent, like trees).

As for the philosophical questions: Any ordinal which googologists actually use can be represented as a finite object: number arrays, trees etc. That's exactly how we make googological notations: we take the general abstract idea represented by a given ordinal and map it to an actual tangable structure. For example, any ordinal up to ωω can be written as a simple row of numbers, and any ordinal up to the BHO can be encoded as a tree of numbers.

So you see, for the purpose of googology, it doesn't really matter whether these ordinals "really" exist or not. What matters is that they provide us with a conceptual framework needed to build these complicated structures.