The Fast-Growing Hierarchy (FGH) is a family of functions ( f_\alpha: \mathbbN \to \mathbbN ), indexed by ordinals ( \alpha ), that rigorously defines the concept of "very fast growth" in proof theory and computability theory.
A high-quality FGH calculator goes beyond simple recursion—it must handle limit ordinals, fundamental sequences, and large countable ordinals up to (and beyond) the Bachmann–Howard ordinal.
Example in natural language:
[ \beginaligned f_0(n) &= n + 1 \ f_\alpha+1(n) &= f_\alpha^n(n) \quad \text(iteration) \ f_\lambda(n) &= f_\lambda[n](n) \quad \text(for limit ordinal \lambda \text) \endaligned ] fast growing hierarchy calculator high quality
Base Rule (Successor Ordinals): [ f_0(n) = n + 1 ] Exact big-integer result (for small outputs)
To build a high-quality Fast-Growing Hierarchy calculator, one must abandon standard arithmetic in favor of symbolic algebra. By defining a grammar for ordinals and mapping recursive steps to known hyper-operations, the calculator can provide meaningful output for numbers that would otherwise require more atoms than exist in the observable universe to write down in decimal form. n))) # incorrect
def f(a, n):
return n+1 if a==0 else (n if a==1 else f(a-1, f(a-1, n))) # incorrect; see proper iteration
Use recursion with caching of ( f_\alpha(n) ) for small ( \alpha, n ).
What does "high quality" actually mean in this context? Let us break down the indispensable features.