This is meant to be both educational for those learning FGH and useful for someone wanting to implement their own calculator.
Standard recursion $f_\alpha+1(n) = f_\alpha(f_\alpha(...f_\alpha(n)...))$ is computationally infeasible.
Better to implement explicitly for all forms up to ε₀.
[Step 1] f_φ(ω,0)(4) = f_φ(ω,0)[4](4) [Step 2] φ(ω,0)[4] = φ(4,0) [Step 3] f_φ(4,0)(4) = f_φ(4,0)[4](4) ...
: The most comprehensive source, providing detailed explanations, comparisons, and often JS-based calculators for specific levels of the hierarchy [1]. fast growing hierarchy calculator high quality
Computing values in the FGH—even for modest ordinals—is a recursive explosion of staggering proportions. A single application of the successor rule can produce numbers far beyond conventional storage and require recursion depths that would crash naive implementations. Moreover, the choice of fundamental sequences for limit ordinals is not unique and can affect the growth rate. A high-quality calculator must therefore:
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 (This means applying the function fαf sub alpha recursively times, nested within itself).
High-quality tools (like those found on the ) allow users to input complex ordinals using proper mathematical syntax (e.g., omega^omega^omega + omega*5 ) [1]. 3. Arbitrary Precision/Large Number Management
In the world of everyday mathematics, we deal with numbers like 10, 1,000, or even a billion. These are tame, comprehensible quantities. But for googologists—mathematicians and hobbyists who study the growth of enormous numbers—these values are barely a starting point. To describe numbers so large that they dwarf a Googolplex (10^(10^100)), we need a system of extreme precision and power. This is meant to be both educational for
: Such a calculator can serve as an educational tool, helping students understand the concepts of growth rates and computability.
An ordinary calculator handles floating-point arithmetic up to roughly 1030810 to the 308th power
is physically impossible to output in standard decimal notation. A high-quality calculator bypasses this constraint by providing a . Instead of outputting standard digits, it converts the FGH expression into other large number notations, such as: Knuth's Up-Arrow Notation Conway Chained Arrow Notation Bowers Explicit Array Notation (BEAN) 3. Architecture of a Fast-Growing Hierarchy Calculator
The hierarchy is defined recursively using three fundamental rules: f0(n)=n+1f sub 0 of n equals n plus 1 Successor Ordinals: A single application of the successor rule can
def fund(ord, n): if ord == 0: return 0 if is_successor(ord): return predecessor(ord) # limit case if ord == ω: return n if ord == ω^(a+1): return ω^a * n if ord == ω^λ where λ limit: return ω^(fund(λ, n)) if ord is sum: # α + β α = first_term(ord) β = rest(ord) if α is limit: return fund(α, n) + β else: # α is successor return (α - 1) + ω^α * (n-1) + β? # careful: need standard rules
: A specialized tool for calculating FGH values using Buchholz's function notation. It allows you to input ordinals like to see how they expand.
to simulate the lower levels of the hierarchy. Which of these would be most useful for your research ?
Fast-growing Hierarchy Calculator Prototype by gooflang - Snap!
To build your own content or simple calculator script, use these recursive rules: Buchholz function
Instead of calculating the raw value, the engine performs . It treats the hierarchy as a string rewriting system.