Quality: Fast Growing Hierarchy Calculator High

Fast-Growing Hierarchy (FGH) Calculator – High-Quality Specification

1. Overview

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.

1. Parse ordinal notation – from standard forms (Cantor normal form) or a custom string syntax.
2. Handle successor & limit ordinals – detect if an ordinal is 0, successor, or limit.
3. Compute fundamental sequences – given ( \lambda ) and ( n ), return ( \lambda[n] ).
4. Recursively evaluate ( f_\alpha(n) ) – with tail recursion or memoization to avoid explosion.
5. Display results – for small ( n ) exactly, for larger ( n ) in scientific notation or as an iterated exponential tower.
6. Performance & safety – guard against stack overflow, set limits on recursion depth.

1. Hardcoded fundamental sequences – The choice of fundamental sequences (e.g., for (\omega^2), (\varepsilon_0), (\zeta_0)) varies across conventions. A rigid calculator cannot adapt to Wainer, Bachmann, or Buchholz hierarchies.
2. No ordinal notation – It only accepts integer subscripts. Real FGH requires notations like (\psi(\Omega^\omega)).
3. Recursion depth limits – Even (f_3(5)) explodes recursion; a naive recursive implementation crashes the stack.
4. Misinterpretation of limit ordinals – Treating (\omega^\omega) as a numeral, not a function.
5. No step-by-step expansion – You see only the final (often incorrect) output, not the reduction path.

References

1. Wainer, S. S. (1970). "A classification of ordinal recursive functions".
2. Gallier, J. H. (1991). "What's so special about Kruskal's theorem and the ordinal $\Gamma_0$?"
3. Conway, J. H. (1996). The Book of Numbers (Specifically Arrow Notation sections).

Googology Wiki Tools: The community often hosts Javascript-based calculators specifically tuned for FGH and Hardy hierarchies. fast growing hierarchy calculator high quality

3. Implementation Details (Pseudo-Code)

class FGHCalculator:
    def __init__(self, ordinal_alpha):
        self.alpha = ordinal_alpha