Fast Growing Hierarchy Calculator High: Quality

Normalization (Cantor normal form, then beyond) ensures comparability.

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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 fast growing hierarchy calculator high quality

| Feature | Benefit | |---------|---------| | | Input w^2 * 3 + w * 5 + 7 | | Step-by-step trace | Show f_w(3) = f_3(3) = f_2(f_2(f_2(3))) = ... | | Growth class label | Output "Primitive recursive" (α<ω), "Ackermann" (α=ω), "ε₀" | | Large number approximation | Use Knuth up-arrows, Conway chains, or Hardy hierarchy | | Caching (memoization) | Avoid recomputing f_α(n) for same (α,n) | | Graphical tree display | Show recursion tree of fundamental sequences |

Abstract A fast-growing hierarchy is a structured family of ordinal-indexed functions that exhibit rapidly increasing growth rates. These hierarchies formalize the notion of iterated growth beyond primitive-recursive and elementary functions and connect proof theory, ordinal analysis, and computability. This paper explains definitions, canonical examples (Grzegorczyk, Wainer/Hardy, Löb–Wainer), ordinal indexing, comparison methods, and computational/analytic applications. A worked example and references conclude. Learn more Can't delete the links right now

For ( \alpha < \varepsilon_0 ):

The index is a successor of a limit ordinal ( Apply Successor Rule: Evaluate the Innermost Layer: To find , the calculator uses the fundamental sequence for Compute the Baseline: def f(a, n): return n+1 if a==0 else

Such a tool is invaluable for googologists, logic students, and anyone curious about the limits of computability and proof theory. Implementations exist online (e.g., Googology Wiki tools, GitHub repos), but few achieve both correctness and user‑friendliness. A well‑designed FGH calculator is a beautiful intersection of theoretical computer science and software engineering.

Are you focusing on a (like Cantor Normal Form or Veblen)?

For educational purposes, seeing the final output is rarely enough. High-quality calculators offer an "expansion mode." This feature reveals the mathematical reduction process, demonstrating exactly how a limit ordinal drops to its fundamental sequence, and how successor steps nest the functions. 4. Custom Fundamental Sequence Selection

A high‑quality FGH calculator can be extended: