Fast Growing Hierarchy Calculator Link
Different definitions yield different results. You must choose:
Because the FGH is defined by induction on ordinals, it terminates for all inputs, but proving termination for a given implementation may require verifying that the fundamental sequences are well‑founded and that the recursion always reduces the ordinal component. This is a subtle point, as some naive implementations can easily produce non‑terminating loops if the ordinal representation is not handled correctly.
This comprehensive guide serves as an analytical calculator breakdown. It explains the mechanics, notations, and calculations behind the Fast-Growing Hierarchy. What is the Fast-Growing Hierarchy? fast growing hierarchy calculator
(a number so large it cannot be stored in the physical universe). Mapping Famous Large Numbers to FGH
The Fast-Growing Hierarchy (FGH) is a family of functions used in mathematics and computer science to classify the growth rates of functions. It is the gold standard for measuring the size of large numbers, from the merely huge (like $10^100$) to the incomprehensibly large (like Graham’s Number and TREE(3)). Different definitions yield different results
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.
), it uses a system called a "fundamental sequence" to choose a finite level based on the input variable. Note: Here, selects the -th element of the sequence assigned to the limit ordinal . For the first limit ordinal , the sequence is simply How Growth Scales: Level by Level This comprehensive guide serves as an analytical calculator
The Fast Growing Hierarchy Calculator stands out from other similar tools due to its ease of use, extensive documentation, and high performance. However, some tools may offer additional features, such as:
function evaluate_FGH(ordinal, input_n): if ordinal == 0: return input_n + 1 elif is_successor(ordinal): previous_ordinal = ordinal - 1 current_value = input_n for i from 1 to input_n: current_value = evaluate_FGH(previous_ordinal, current_value) return current_value elif is_limit(ordinal): resolved_ordinal = get_fundamental_sequence(ordinal, input_n) return evaluate_FGH(resolved_ordinal, input_n) Use code with caution.
The most prominent online calculator is the . This JavaScript tool allows you to input a natural number (n) and a countable ordinal (\alpha) expressed in the normal form for the Extended Buchholz function, a powerful system of fundamental sequences that reaches far beyond the small Veblen ordinal. It is one of the few calculators that can handle ordinals beyond (\varepsilon_0). Another notable tool is the Ordinal Expander in JavaScript (ordex) , which is designed to expand ordinals and compute their fundamental sequences, which is the core operation for any FGH calculator.
Demystifying Large Numbers: The Ultimate Guide to the Fast-Growing Hierarchy
