Fast Growing Hierarchy Calculator [2026]
Fast-Growing Hierarchy Calculator — Report
7. Implementation pseudocode (core evaluator)
Provide a concise evaluator outline — pseudo-Python:
def f(alpha, n, limits):
# limits: max_steps, max_bits
key = (alpha.serialize(), n)
if key in cache: return cache[key]
if alpha.is_zero(): return n+1
if alpha.is_successor():
beta = alpha.predecessor()
# compute iterate of f_beta, repeated n times starting at n
val = iterate(lambda x: f(beta, x, limits), n, n, limits)
cache[key] = val; return val
# alpha is limit
beta = alpha.fundamental(n)
val = f(beta, n, limits)
cache[key] = val; return val
iterate helper must detect overflow and convert to descriptor when exceeding limits. fast growing hierarchy calculator
Challenge 3: Output Size
If you did compute ( f_\omega+1(4) ) as an integer, you’d need more than ( 10^100 ) bits of memory—physically impossible. Hence any honest FGH calculator never expands to a full integer; it stays in a compressed symbolic form unless the result is tiny. Fast-Growing Hierarchy Calculator — Report 7
The Explosion of Growth
- f_1(n) = ( 2n )
- f_2(n) = ( n \cdot 2^n ) (Roughly exponential)
- f_3(n) is roughly tetrational (power towers)
- f_4(n) involves pentation
By the time you reach f_ω(n), you are at the limit of primitive recursive functions (Ackermann function territory). By f_ε₀(n), you surpass the proof-theoretic strength of Peano arithmetic. iterate helper must detect overflow and convert to
The core problem: Performing ( f_3(4) ) by hand is tedious. Performing ( f_ω+1(3) ) without a calculator is virtually impossible for a human. This is why we need a Fast Growing Hierarchy calculator.
