A Goodstein Sequence

After watching a YouTube video titled The sequence that grows remarkably large, then drops to zero!, I became aware of Goodstein sequences. These sequences make use of what is called hereditary base-\(n\) notation that Wikipedia explains as follows:

Goodstein sequences are defined in terms of a concept called "hereditary base-''n'' notation". This notation is very similar to usual base-\(n\) positional notation, but the usual notation does not suffice for the purposes of Goodstein's theorem.

In ordinary base-\(n\) notation, where \(n\) is a natural number greater than 1, an arbitrary natural number \(m\) is written as a sum of multiples of powers of \(n\): $$ m = a_k n^k + a_{k-1} n^{k-1} + \cdots + a_0 $$where each coefficient \(a_i\) satisfies \( 0 ≤ a_i<n \), and \(a_k<≠ 0\). For example, in base 2:$$35 = 32 + 2 + 1 = 2^5 + 2^1 + 2^0 $$Thus the base-2 representation of 35 is 100011, which means \(2^5 + 2 + 1\).  Similarly, 100 represented in base-3 is 10201:$$100 = 81 + 18 + 1 = 3^4 + 2 \cdot 3^2 + 3^0 $$Note that the exponents themselves are not written in base-\(n\) notation. For example, the expressions above include \( 2^5\) and \(3^4\), and \(5>2,\) and \(4>3\).

To convert a base-\(n\) representation to hereditary base-\(n\) notation, first rewrite all of the exponents in base-\(n\) notation. Then rewrite any exponents inside the exponents, and continue in this way until every number appearing in the expression (except the bases themselves) has been converted to base-\(n\) notation.

For example, while 35 in ordinary base-2 notation is \(2^5 + 2 + 1\), it is written in hereditary base-2 notation as:$$35 = 2^{2^{2^1}+1}+2^1+1 $$using the fact that \(5 = 2^{2^1} + 1\). Similarly, 100 in hereditary base-3 notation is $$100 = 3^{3^1+1} + 2 \cdot 3^2 + 1$$

The article goes on to explain:

The Goodstein sequence G(\(m\)) of a number \(m\) is a sequence of natural numbers. The first element in the sequence G(\(m\)) is \(m\) itself. To get the second, G(\(m\))(2), write \(m\) in hereditary base-2 notation, change all the 2s to 3s, and then subtract 1 from the result.

This process continues until zero is reached. Let's use \(m=4\) as an example of the process.

\(a(0) = 4 = 2^2\)

\(a(1) = 3^3 - 1 = 26 = 2 \times 3^2 + 2 \times 3 + 2 \)

\(a(2) = 2 \times 4^2 + 2 \times 4 + 2 - 1 = 41 = 2 \times 4^2 + 2 \times 4 + 1 \)

\( a(3) = 2 \times 5^2 + 2 \times 5 + 1 - 1 = 60 = 2 \times 5^2 + 2 \times 5 \)

\(a(4) = 2 \times 6^2 + 2 \times 6 - 1 = 83 = 2 \times 6^2 + 6 + 5\)

\(a(5) = 2 \times 7^2 + 7 + 5 - 1 = 109 \text{ etc.} \)

The above example is taken from the comments to OEIS  A056193: 

A056193

Goodstein sequence starting with 4: to calculate a(\(n\)+1), write a(\(n\)) in the hereditary representation in base \(n\)+2, then bump the base to \(n\)+3, then subtract 1.

The initial members of the sequence (up to 40000) are: 

4, 26, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, 401, 458, 519, 584, 653, 726, 803, 884, 969, 1058, 1151, 1222, 1295, 1370, 1447, 1526, 1607, 1690, 1775, 1862, 1951, 2042, 2135, 2230, 2327, 2426, 2527, 2630, 2735, 2842, 2951, 3062, 3175, 3290, 3407, 3525, 3645, 3767, 3891, 4017, 4145, 4275, 4407, 4541, 4677, 4815, 4955, 5097, 5241, 5387, 5535, 5685, 5837, 5991, 6147, 6305, 6465, 6627, 6791, 6957, 7125, 7295, 7467, 7641, 7817, 7995, 8175, 8357, 8541, 8727, 8915, 9105, 9297, 9491, 9687, 9885, 10085, 10287, 10491, 10697, 10905, 11115, 11327, 11540, 11755, 11972, 12191, 12412, 12635, 12860, 13087, 13316, 13547, 13780, 14015, 14252, 14491, 14732, 14975, 15220, 15467, 15716, 15967, 16220, 16475, 16732, 16991, 17252, 17515, 17780, 18047, 18316, 18587, 18860, 19135, 19412, 19691, 19972, 20255, 20540, 20827, 21116, 21407, 21700, 21995, 22292, 22591, 22892, 23195, 23500, 23807, 24116, 24427, 24740, 25055, 25372, 25691, 26012, 26335, 26660, 26987, 27316, 27647, 27980, 28315, 28652, 28991, 29332, 29675, 30020, 30367, 30716, 31067, 31420, 31775, 32132, 32491, 32852, 33215, 33580, 33947, 34316, 34687, 35060, 35435, 35812, 36191, 36572, 36955, 37340, 37727, 38116, 38507, 38900, 39295, 39692

I've chosen G(4) because it has this large numbers of members in the range covering my diurnal age. 27316 is the next number that will come up for me in about 200 days time. Most other Goodstein sequences increase very quickly. This is just the briefest of introductions to Goodstein sequences and I don't pretend to understand the following quote from Wikipedia.

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Kirby and Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem gave another example.

Laurence Kirby and Jeff Paris introduced a graph-theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" (named for the mythological multi-headed Hydra of Lerna) is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. Kirby and Paris proved that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very long time. Just like for Goodstein sequences, Kirby and Paris showed that it cannot be proven in Peano arithmetic alone.