While studying numbers beyond the bounds of reality seems abstract, the Fast-Growing Hierarchy is an essential tool in several theoretical fields. 1. Computability Theory
This public link is valid for 7 days and shares a thread, including any personal information you added. This link or copies made by others cannot be deleted. If you share with third parties, their policies apply. Can’t copy the link right now. Try again later.
A proper FGH calculator would let you explore this madness with a few keystrokes.
), the calculator uses a fundamental sequence to choose a specific integer path: 2. The Computational Challenge Because the nesting depth scales with the input , calculating even fast growing hierarchy calculator
Limit ordinals do not have a single, universally mandatory fundamental sequence. Different calculators may use slightly different standard sequences, resulting in different values for the exact same input at limit levels.
That is, apply (f_\alpha) (n) times to (n).
Provide a concise report describing a fast-growing hierarchy calculator: definition, supported functions, algorithmic approach, limitations, example outputs, and implementation outline. While studying numbers beyond the bounds of reality
f_ε_0(2) with ε_0[n] = ω↑↑(n+1)
Once these sequences are fixed, the definition of (f_\alpha) becomes deterministic and, in theory, computable.
For hobbyists and researchers in googology, the FGH is the ultimate yardstick. When a new large number is proposed (such as TREE(3) or SSCG(3)), an FGH calculator or theoretical analysis is used to find its index. For instance, TREE(3) requires ordinals far surpassing ϵ0epsilon sub 0 , scaling up to the Small Veblen Ordinal. Summary of Growth Rates Ordinal Index ( Common Mathematical Equivalent / Notation Growth Class Exponential Knuth's Up-Arrow ( Tetrational Ackermann Function Diagonalized / Non-Primitive Recursive ϵ0epsilon sub 0 Goodstein Sequences Beyond Peano Arithmetic Γ0cap gamma sub 0 Feferman-Schütte Ordinal Feasible Proof Theory Limit If you want to explore further, Learn how scales against the hierarchy. This link or copies made by others cannot be deleted
Are you interested in learning about the for comparison? Share public link
It is used to determine the termination of complex algorithms. If a proof's complexity can be mapped to an ordinal below ϵ0epsilon sub 0 , it can be proven sound within Peano arithmetic.
The system is defined by three simple rules, starting with the most basic operation:
iterate helper must detect overflow and convert to descriptor when exceeding limits.
# Successor Ordinal: f_alpha+1(n) = f_alpha^n(n) if isinstance(alpha, int) and alpha >= 0: # Iterate the function 'n' times result = n for _ in range(n): result = self._f(alpha - 1, result) return result