Editing Goodstein's theorem (section)

== Hereditary base-''n'' notation ==
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.

To achieve the 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'':
:<math>m = a_k n^k + a_{k-1} n^{k-1} + \cdots + a_0,</math>
where each coefficient ''a<sub>i</sub>'' satisfies {{nowrap|0 ≤ ''a<sub>i</sub>'' &lt; ''n''}}, and {{nowrap|''a<sub>k</sub>'' ≠ 0}}. For example, to achieve the [[Binary number|base 2 notation]], one writes
:<math>35 = 32 + 2 + 1 = 2^5 + 2^1 + 2^0.</math>
Thus the base-2 representation of 35 is 100011, which means {{nowrap|2<sup>5</sup> + 2 + 1}}.  Similarly, 100 represented in [[Ternary numeral system|base-3]] is 10201:
:<math>100 = 81 + 18 + 1 = 3^4 + 2 \cdot 3^2 + 3^0.</math>
Note that the exponents themselves are not written in base-''n'' notation. For example, the expressions above include 2<sup>5</sup> and 3<sup>4</sup>, and 5&nbsp;>&nbsp;2, 4&nbsp;>&nbsp;3.

To convert a base-n notation (which is a step in achieving base-''n'' representation) to a hereditary base-''n'' notation, first rewrite all of the exponents as a sum of powers of ''n'' (with the limitation on the coefficients {{nowrap|0 ≤ ''a<sub>i</sub>'' &lt; ''n''}}). Then rewrite any exponent inside the exponents in base-''n'' notation (with the same limitation on the coefficients), and continue in this way until every number appearing in the expression (except the bases themselves) is written in base-''n'' notation.

For example, while 35 in ordinary base-2 notation is {{nowrap|2<sup>5</sup> + 2 + 1}}, it is written in hereditary base-2 notation as
:<math>35 = 2^{2^{2^1}+1}+2^1+1, </math>
using the fact that {{nowrap|1=5 = 2<sup>2<sup>1</sup></sup> + 1.}} Similarly, 100 in hereditary base-3 notation is
:<math>100 = 3^{3^1+1} + 2 \cdot 3^2 + 1.</math>