A: Goodstein’s argument over the natural numbers
For any given natural number , we can express the Goodstein sequence for in a form where each term of the sequence is expressed in it’s hereditary representation:
where the first term denotes the unique hereditary representation ofthe natural number in the natural number base :
and if then is defined recursively from as below.
B: The recursive definition of Goodstein’s Sequence over the natural numbers
For let the term of the Goodstein natural number sequence be expressed syntactically by its hereditary representation as:
and for each the exponent too is expressed syntactically by its hereditary representation in the base ; as also are all of its exponents and, in turn, all of their exponents, etc.
We then define the term of as:
C: The hereditary representation of
Now we note that:
(a) if then the hereditary representation of is:
(b) whilst if for all , then the hereditary representation of is:
and so its hereditary representation in the base is given by:
where and .
D: Goodstein’s argument in arithmetic
For we then consider the difference:
(a) if we have:
(b) whilst if for all we have:
(c) if in equation (6) we replace the base by the base in the term:
and the base also by the base in the term:
then we have:
(d) whilst if in equation (7}) we replace the bases similarly, then we have:
We consider now the sequence:
obtained from Goodstein’s sequence by replacing the base in each of the terms by the base for all .
Clearly if for all non-zero terms of the Goodstein sequence, then in each of the cases—equation (10) and equation (11).
(Since we then have:
in equation (11).)
The sequence is thus a descending sequence of natural numbers, and must terminate finitely in if .
Since if , Goodstein’s sequence too must terminate finitely in if .
Obviously, since we can always find a for all non-zero terms of the Goodstein sequence if it terminates finitely in , the condition that we can always find some for all non-zero terms of any Goodstein sequence is equivalent to the assumption that any Goodstein sequence terminates finitely in .
E: Goodstein’s argument over the ordinal numbers
We shall see next how Goodstein mirrors the above argument over the ordinals and curiously concludes the Theorem that, since we now have for any finite ordinal , therefore any Goodstein sequence over the natural numbers must terminate finitely since a corresponding Goodstein sequence of transfinite ordinals cannot be an infinitely descending sequence of ordinals!