Goodstein Sequence: Power of Infinity

A postive integer number can be expressed in a recursive expotenial form form. For example, in the base of 2, the number 299 can be written as the following binary tree form.

{Plus, {Power, 2, {Plus, {Power, 2, {Plus, {Power, 2, 1}, 1}}, 0}}, {Plus, {Power, 2, {Plus, {Power, 2, {Plus, {Power, 2, 1}, 0}}, 1}}, {Plus, {Power, 2, {Plus, {Power, 2, 1}, 1}}, {Plus, {Power, 2, 1}, 1}}}}

The Goodstein sequence is generated in the following steps:
1. Start with any positive integer number and write it in the above recursive expotenial form with the base 2;
2. Replace the base 2 in the above form with 3 and then substract 1 from 1.
3. Recursively repeat the above steps to generate the sequence.

Apparently, the sequence will increase very rapidly. For example, if we start at 5, the first 61 numbers in the Goodstein sequence are:

{5, 27, 255, 467, 775, 1197, 1751, 2454, 3325, 4382, 5643, 7126, 8849, 10830, 13087, 15637, 18499, 21691, 25231, 29137, 33427, 38119, 43231, 48781, 54787, 61267, 68239, 75721, 83731, 92287, 101407, 111108, 121409, 132328, 143883,156092, 168973, 182544, 196823, 211828, 227577, 244088, 261379,279468, 298373, 318112, 338703, 360164, 382513, 405768, 429947, 455068,481149, 508208, 536263, 565332, 595433, 626584, 658803, 692108, 726517}.

We can graph the sequence as follows.



It seems that the sequence increases in a power law. However, after a finite number of steps, though very large, the sequence will become 0 at last. More amazingly, this cannot be proved in the Peano arithmetic. Baez showed how the infinity of ordinals can be used to prove this result, http://math.ucr.edu/home/baez/week236.html.