Goodstein's Theorem - Sequence Length As A Function of The Starting Value

Sequence Length As A Function of The Starting Value

The Goodstein function, is defined such that is the length of the Goodstein sequence that starts with n. (This is a total function since every Goodstein sequence terminates.) The extreme growth-rate of can be calibrated by relating it to various standard ordinal-indexed hierarchies of functions, such as the functions in the Hardy hierarchy, and the functions in the fast-growing hierarchy of Löb and Wainer:

  • Kirby and Paris (1982) proved that
has approximately the same growth-rate as (which is the same as that of ); more precisely, dominates for every, and dominates
(For any two functions, is said to dominate if for all sufficiently large .)
  • Cichon (1983) showed that
where is the result of putting n in hereditary base-2 notation and then replacing all 2s with ω (as was done in the proof of Goodstein's theorem).
  • Caicedo (2007) showed that if with then
.

Some examples:

n
1 2
2 4
3 6
4 3·2402653211 − 2
5 > A(4,4)
6 > A(6,6)
7 > A(8,8)
8 > A3(3,3) = A(A(61, 61), A(61, 61))
12 > fω+1(64) > Graham's number
19

(For Ackermann function and Graham's number bounds see fast-growing hierarchy#Functions in fast-growing hierarchies.)

Read more about this topic:  Goodstein's Theorem

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