09/28/2025, 07:15 PM
i made this algorithm on desmos where i extended the fast growing hierarchy to real numbers: https://www.desmos.com/calculator/i9ue4cyvan
i took the superfunction of \(f_{2}(x)\) on the integer points and used lagrange interpolation on the left side of the superfunction.
and that's how i got \(f_{3}(x)\).
and you can go through the same process to get \(f_{4}(x)\), \(f_{5}(x)\), \(f_{6}(x)\), and so on...
but i don't even know how \(f_{\omega}(x)\) can be extended.
we first have to define the fast growing hierarchy for non-integer subscripts.
\(f_{n}(1)=2\) for all n. and we can choose to keep it true for non-integer values of n.
the only catch is that we don't yet have a full understanding of the fast growing hierarchy with real numbers.
we can't just start taking derivatives, plugging in values, or anything like that.
we have to actually understand the fast growing hierarchy.
if we understand the fast growing hierarchy well enough, we can even extend it to complex numbers.
i took the superfunction of \(f_{2}(x)\) on the integer points and used lagrange interpolation on the left side of the superfunction.
and that's how i got \(f_{3}(x)\).
and you can go through the same process to get \(f_{4}(x)\), \(f_{5}(x)\), \(f_{6}(x)\), and so on...
but i don't even know how \(f_{\omega}(x)\) can be extended.
we first have to define the fast growing hierarchy for non-integer subscripts.
\(f_{n}(1)=2\) for all n. and we can choose to keep it true for non-integer values of n.
the only catch is that we don't yet have a full understanding of the fast growing hierarchy with real numbers.
we can't just start taking derivatives, plugging in values, or anything like that.
we have to actually understand the fast growing hierarchy.
if we understand the fast growing hierarchy well enough, we can even extend it to complex numbers.