**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:

[1]

where the first term denotes the unique hereditary representation ofthe natural number in the natural number base :

e. g.

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:

[2]

where:

(a) over

(b)

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:

[3]

**C: The hereditary representation of **

Now we note that:

(a) if then the hereditary representation of is:

[4]

(b) whilst if for all , then the hereditary representation of is:

[5]

where:

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:

Now:

(a) if we have:

[6]

(b) whilst if for all we have:

[7]

Further:

(c) if in equation (6) we replace the base by the base in the term:

[8]

and the base also by the base in the term:

[9]

then we have:

[10]

since ;

(d) whilst if in equation (7}) we replace the bases similarly, then we have:

[11]

where ,

and .

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!

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