Let be the Goodstein sequence starting with n and ending at 0.
Let be the base of the hereditary notation of the last term of (Alternatively, it is the length of the Goodstein sequence + 1). We shall call these the Goodstein numbers
Now, in fact, if , then . I show that this function has a meaning.
Growth of Goodstein Numbers
Let us define a family of functions:
Notice the pattern? If appears in then (where B is the base and k<B). Likewise, if appears, then .
In fact, let's rename our functions (here is a label, not a variable — in fact, it is actually an ordinal number, or generalized index, who's properties I hope to take advantage of):
And add a new one:
Thus, if we have a value of the form at base B in , then .
Derivation of Growth Function
- Note: It turns out that the function I define here is a variant of the fast-growing hierarchy much like the Hardy hierarchy.
Goodstein talks about the hereditary form of a number and the unique ordinal number associated with each hereditary form. For example:
- is in form
Let us identify the hereditary form with that ordinal number.
If N has hereditary form with base B, then let be the base at which the the Goodstein sequence starting at N in base B will end.
For some values of :
Note: . So Graham's number .
Now where we remember that . So,
Now let's look back at the table: