Fast Growing Hierarchy Calculator High Quality -
Fast-Growing Hierarchies — A Concise Expository Paper
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.
- Exact big-integer result (for small outputs).
- Compressed towers (e.g., notation for exponentiation/tetration stacks).
- Symbolic F_α(n) nodes when expansion is infeasible.
- Limitations and practical considerations
from functools import lru_cache
1.1 Growth Rate Examples
To understand the engineering constraints, consider the magnitude of numbers involved: fast growing hierarchy calculator high quality
- Wainer hierarchy (for ordinals below ( \varepsilon_0 ))
- Veblen-based hierarchy (for ordinals below ( \Gamma_0 ))
- Buchholz hierarchy (for accessing larger ordinals)
Example: Hardcoded ε₀ calculator
def f_epsilon0(n):
"""Compute f_ε₀(n) using fundamental sequences."""
def f(a, b):
if a == 0:
return b + 1
if a == 1:
res = b
for _ in range(b):
res = f(0, res)
return res
if a == 'w':
return f(b, b) if b > 0 else b + 1
# Full implementation omitted for brevity
return 0
return f('e0', n) Exact big-integer result (for small outputs)
- Natural number inputs ( n )
- Ordinal notations up to some limit (e.g., ( \varepsilon_0 ) or beyond)
- Fundamental sequence assignments for limit ordinals
- Exact or approximate evaluation of ( f_\alpha(n) )
