Fast Growing Hierarchy Calculator High Quality Jun 2026

is the first transfinite ordinal, the function chooses its index based on the input itself: 2. Core Features of a High-Quality FGH Calculator

A web‑based calculator might have:

def fgh(alpha, n, limit_ordinal_fundamental=None): """ Compute f_alpha(n) with custom fundamental sequences. Args: alpha: int or callable for limit ordinals returning alpha[n] n: int >= 0 limit_ordinal_fundamental: function(alpha, n) -> alpha_n """ if alpha == 0: return n + 1 if isinstance(alpha, int): # successor result = n for _ in range(n): result = fgh(alpha - 1, result, limit_ordinal_fundamental) return result # limit ordinal if limit_ordinal_fundamental: alpha_n = limit_ordinal_fundamental(alpha, n) return fgh(alpha_n, n, limit_ordinal_fundamental) raise ValueError(f"No fundamental sequence for alpha") fast growing hierarchy calculator high quality

| Feature | Benefit | |---------|---------| | | Input w^2 * 3 + w * 5 + 7 | | Step-by-step trace | Show f_w(3) = f_3(3) = f_2(f_2(f_2(3))) = ... | | Growth class label | Output "Primitive recursive" (α<ω), "Ackermann" (α=ω), "ε₀" | | Large number approximation | Use Knuth up-arrows, Conway chains, or Hardy hierarchy | | Caching (memoization) | Avoid recomputing f_α(n) for same (α,n) | | Graphical tree display | Show recursion tree of fundamental sequences |

: ( \omega^\alpha_1 \cdot c_1 + \dots + \omega^\alpha_k \cdot c_k ) with ( \alpha_1 > \dots > \alpha_k ) and ( c_i ) positive integers. is the first transfinite ordinal, the function chooses

While standalone desktop applications are rare, several high-quality web-based and programmed resources exist:

| Name | Max ordinal | Notes | |------|-------------|-------| | | ε₀ | Good for learning | | M. J. H. Heule’s ordinal calculator | Γ₀ | Research quality | | Python ordinal library | ε₀ | Customizable | | Desmos FGH | ω^ω | Visual, limited | | | Growth class label | Output "Primitive

Before exploring the tools, it helps to understand the core concepts of FGH. It is a family of functions indexed by ordinals ((f_\alpha: \mathbbN \rightarrow \mathbbN)), defined by three simple rules:

fα+1(n)=fαn(n)f sub alpha plus 1 end-sub of n equals f sub alpha to the n-th power of n