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.

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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.


An interesting sequence

May 16, 2008

Mandar has an interesting set of posts on convergence and divergence of infinite series. It made me recall a fascinating result I had come across once. It is not about an infinite series, though. But it is a very interesting sequence.

The sequence needs some explanation of a notation called hereditary base-n notation. Let me explain this with an example. Let us write the number 26 in its hereditary base-2 notation. First we start by writing 26 as the sum of powers of 2.
26 = 24 + 23 + 21
Next, the powers are written as sums of powers of 2 as well.
So, 26 = 222 + 221 + 1 + 21

Similarly, the hereditary base-3 notation of 1000 is the following.

1000 = 36 + 34 + 33 + 1 = 33 + 3 + 33 + 1 + 33 + 1

Note that the bases and the powers cannot be bigger than n. Also, we can write the terms as a product of a base power n and a number smaller than n. For example – 26 can be written in hereditary base-3 notation as, 2.32 + 2.3 + 2.

Now, let us take a number, say 26, and do the following:

  1. Take the number. Start with base, n = 2.
  2. Express number in hereditary-n notation
  3. The next number is formed by changing all the n’s to n+1’s. That is, “increase” the base of the sequence by 1. [n = n + 1]
  4. Subtract 1 from the above number and goto 2. [number = number – 1]

Let’s take an example. Let’s start with 4.

4 = 22 (step 2)

33 (step 3)

33 – 1 = 26 (step 4) — Iteration 1

26 = 2.32 + 2.3 + 2 (step 2)

2.42 + 2.4 + 2 (step 3)

2.42 + 2.4 + 2 – 1 = 41 (step 4) — Iteration 2

41 = 2.42 + 2.4 + 1 (step 2)

2.52 + 2.5 + 1 (step 3)

2.52 + 2.5 + 1 – 1 = 60 (step 4) — Iteration 3

60 = 2.52 + 2.5 (step 2)

2.62 + 2.6 (step 3)

2.62 + 2.6 – 1 = 83 (step 4) — Iteration 4

83 = 2.62 + 6 + 5 (step 2)

2.72 + 7 + 5 (step 3)

2.72 + 7 + 5 – 1 = 109 (step 4) — Iteration 5

And so on. Now the question is, does this sequence converge (or terminate)? If it converges, what does it converge to? Or if you think it does not converge, explain why.

As always, people who already know this result may please defer commenting.


Seminar on Complex Networks

May 14, 2008

A series of seminars on Complex Systems has been scheduled through the summer over here. I had been asked to give the first seminar. I gave a talk yesterday. I gave an overview of the philosophy, design and analyses of  complex networks. Among other things, I talked about how several graph theoretic measures can be used in both design and analyses of complex networks. There was a lot of lively discussion. The total seminar spanned about 2.5 hours, which probably means the seminar went off quite well. In case you are interested, here are the slides I had used.