# Shadows of infinity cast over the finite realm

### We shall fight the fearsome Hydra and in defeating it we shall see how the existence of transfinite numbers can have surprising consequences within the finite realm.

### Goodstein sequences

Let us undertake a certain fantastical number process. Begin with any positive number, such as the number 41, and write it in *hereditary* base 2, which means to write it in base 2 as a sum of powers of 2, but also to write the exponents and iterated exponents in this way. We denote this initial number by *a*_{2}, in part to indicate this initial choice of base.

We form the next number *a*_{3} simply by replacing all the 2s in this representation with 3s and subtracting 1, writing the result in hereditary base 3.

This number *a*_{3} is 22876792455042 in decimal, or about 2 · 10^{13}, which is much larger than our original number 41.

The succeeding number *a*_{4} is formed similarly by replacing all 3s with 4s and subtracting 1.

In decimal notation, *a*_{4} is the number:

which is about 5· 10^{155}, a big step up from *a*_{3}.

We form *a*_{5} similarly by replacing 4s with 5s in the hereditary base 4 representation and subtracting 1, and so on. Starting from any given initial value *a*_{2}, we generate in this way a sequence

This is known as a *Goodstein* sequence in light of the remarkable theorem about them proved by Reuben Goodstein in 1944. The amazing fact is that for any natural number starting value *a*_{2}, if we form the corresponding Goodstein sequence according to the process we have described, then at some point *n* the value will become zero *a _{n}* = 0.

What?! It had seemed that the sequence would always grow larger, with each next value *a _{n}*

_{+1}much bigger than the previous value

*a*. How can we reach

_{n}*a*= 0? Won't

_{n}*a*

_{n}_{+1}always be larger than

*a*? Actually, no. According to Goodstein's theorem, the initial impression that the numbers grow ever larger is simply mistaken. Eventually, the numbers start getting smaller and they will eventually reach 0.

_{n}Let me explain.

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