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python code

def hyper(a, b, c): # a^^...(c ^s)...^b
  if c<1: return a*b
  if b<2: return a
  return hyper(a, hyper(a, b-1, c), c-1)

def R(n, k1, k2): # replace k1 to k2 in base-k1 arrow notation of n
  if n<k1*k1:
    return n//k1*k2+n%k1
  b=2; c=1
  while hyper(k1, b, c+1) <= n: c += 1
  while hyper(k1, b+1, c) <= n: b += 1
  return hyper(k2, b, R(c, k1, k2)) + R(n - hyper(k1, b, c), k1, k2)

def G(n, b=3): # Goodstein function using base-k arrow notation
  if n < 1: return b
  return G(R(n, b, b+1)-1, b+1)

Proof (sketch) that G(n) is well-difined

For a "structure" s of base-k arrow notation, a corresponding ordinal O(s) can be defined by replacing:

  • 1 with  \omega
  • k with  \omega ^ 2
  • k \uparrow ^{s'} c with  \omega ^ {O(s') + c}

Example

base-3 arrow notation of 390: 3 \uparrow ^{2} 2 \times 12 + 3 \uparrow ^{1} 2 \times 2 + 3 \times 2 + 2

corresponding ordinal: \omega ^{\omega 2 + 2} 12 + \omega ^{\omega 1 + 2} 2 + \omega ^2 2 + \omega 2

If R(n,k_1,k_2) preserve the structure of the notation, the corresponding ordinal decreases in each step of G(n)

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