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.
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 a2, in part to indicate this initial choice of base.
We form the next number a3 simply by replacing all the 2s in this representation with 3s and subtracting 1, writing the result in hereditary base 3.
This number a3 is 22876792455042 in decimal, or about 2 · 1013, which is much larger than our original number 41.
The succeeding number a4 is formed similarly by replacing all 3s with 4s and subtracting 1.
In decimal notation, a4 is the number:
which is about 5· 10155, a big step up from a3.
We form a5 similarly by replacing 4s with 5s in the hereditary base 4 representation and subtracting 1, and so on. Starting from any given initial value a2, 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 a2, if we form the corresponding Goodstein sequence according to the process we have described, then at some point n the value will become zero an = 0.
What?! It had seemed that the sequence would always grow larger, with each next value an+1 much bigger than the previous value an. How can we reach an = 0? Won't an+1 always be larger than an? 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.
Let me explain.
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