Goodstein’s Theorem

May 20, 2008

In the previous post, I had talked about a special kind of sequence. Such sequences are called Goodstein’s sequences. And Goodsten’s theorem states that all Goodstein sequences converge to 0. Now, this is a rather counter-intuitive result given the way the series expands. Remember, in every iteration, you are increasing the base and the exponent of the terms in a sum by 1, whereas you are subtracting the whole sum by just 1. Surely, the number is increasing very rapidly notwithstanding the subtractions? For example, even a small first number, say 4, increases in this manner: 4, 26, 41, 60, 83, 109, 139, 173,…, 3319048, …, 93850207,…

How can this sum start reducing and converge to 0? One has to look more carefully to see what is happening. Let us start with a smaller number, say 3.

Iteration 1 (hereditary base-2):
(1) 3 = 2 + 1
(2) 3 + 1
(3) 3 + 1 – 1 = 3
Iteration 2 (hereditary base-3):
(1) 3 = 3
(2) 4
(3) 4 + 1 – 1 = 3
Iteration 3 (hereditary base-4)
(1) 3 = 3
(2) 3
(3) 3 – 1 = 2

See what’s happening? Firstly, note that in step(2) of iteration 1, the last term, which is 1, does not get increased, because the number is less than the current base. Say the base is n. Then increasing the base of numbers 1 through (n – 1), does not change them. For example, 2 is base 4 is 40 + 40. And 2 in base 5 is, 50 + 50. Correct? So, in this case, the last term is not increased and there is a subtraction following that.

What happens is, even when you are increasing the bases of the terms, the subtraction by 1 operation is slowly “eating” into the last term of the sum. This process may be arbitrarily slow, but eventually, it so happens that, the power of the last term in the sum “falls off”, and the last term falls into a lower base than the ongoing one, hence ceasing to increase. So then, the subtraction can now eat away the last term easily, before it starts attacking the next term (which will now be the last term).

So, you can now see that, that the sum does in fact start reducing, albeit after numerous steps, even for small numbers. In fact, you can only check this for the numbers 1, 2 and 3 by hand. You can probably write a program to check this for bigger numbers, though it might take extremely long.

***

Now, another important thing about Goodstein’s theorem is that it is one of Godel type theorems. The truth of Goodstein’s theorem cannot be proved using the axioms of first order arithmetic (Peano arithmetic). There is a proof for this, as well as a proof for Goodsteins’s theorem using techniques that are outside of Peano arithmetic.