Fast Growing Hierarchy Calculator -

Building a calculator for this hierarchy requires bridging the gap between standard arithmetic and ordinal arithmetic.

This isn't a tool but an incredible live example of the manual process. A community member provided a step-by-step expansion of $f_\omega^3(2)$, showing the nested iterations required to evaluate the function by hand. It's a fantastic demonstration of the underlying mechanics.

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. fast growing hierarchy calculator

Just don’t expect it to finish before the heat death of the universe.

When finite indexation runs out, calculators switch to transfinite ordinals: The function matches its index to the input. , which translates to a massive tower of exponents. Level : This level iterates the fωf sub omega Building a calculator for this hierarchy requires bridging

The hierarchy is defined systematically starting from a basic successor function. For any non-negative integer , the functions are constructed using three fundamental rules: 1. The Base Case At the absolute bottom of the hierarchy ( ), the function simply increments the input by one. f0(n)=n+1f sub 0 of n equals n plus 1 2. The Successor Stage For any step where the index is a successor ordinal ( ), the function iterates the previous function level

An FGH calculator helps contextualize famous googology bounds by pinpointing their location within the hierarchy: Large Number Approximate FGH Index Description ( 1010010 to the 100th power Lower than Easily computed at the exponential level. Skewes' Number A massive power tower. Graham's Number Nests inside the first transfinite steps. TREE(3) Requires the Small Veblen Ordinal level. Rayo's Number Beyond the standard FGH Extends past all recursive ordinal bounds. Algorithmic Logic of an FGH Calculator It's a fantastic demonstration of the underlying mechanics

The fast-growing hierarchy has far-reaching implications in various fields, including:

The first few functions of the hierarchy are already familiar:

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.

What specific ordinal, like ε₀ or Γ₀, are you trying to evaluate? Are you looking to compare FGH to the ?