Goodstein's theorem

Goodstein's theorem is a straightforward statement about the natural numbers that is undecidable in Peano arithmetic but can be proven to be true using the stronger axiom system of set theory. It stands as an example that not all undecidable theorems are as peculiar or contrived as those constructed by Gödel's incompleteness theorem.

The theorem states that every Goodstein sequence (see below) eventually terminates at 0.

Table of contents

1 Definition of a Goodstein sequence
2 Examples of Goodstein sequences
3 Proof

Definition of a Goodstein sequence

In order to define what a Goodstein sequence is, we must first define hereditary base-n notation. To write a natural number in hereditary base-n notation, we first write it in the form , where each is an integer between 0 and n-1; we then recursively write all the exponents in hereditary base n notation. For example, 35 in ordinary base-2 notation is , and in hereditary base-2 notation is .

The Goodstein sequence on a number m, notated G(m), is defined as follows: the first element of the sequence is m. To get the next element, write m in hereditary base 2 notation, change all the 2's to 3's, and then subtract 1 from the result; this is the second element of G(m). To get the third element of G(m), write the previous number in hereditary base 3 notation, change all 3's to 4's, and subtract 1 again. Continue until the result is zero, at which point the sequence terminates.

Examples of Goodstein sequences

Elements of a Goodstein sequence appear to increase rapidly. For example, G(4) starts as follows:

Hereditary notation Value
2² 4
2·3² + 2·3 + 2 26
2·4² + 2·4 + 1 41
2·5² + 2·5 60
2·6² + 6 + 5 83
2·7² + 7 + 4 109
...
2·11² + 11 253
2·12² + 11 299
...

Hereditary notation	Value
2²	4
2·3² + 2·3 + 2	26
2·4² + 2·4 + 1	41
2·5² + 2·5	60
2·6² + 6 + 5	83
2·7² + 7 + 4	109
...
2·11² + 11	253
2·12² + 11	299
...

Elements of G(4) continue to increase for a while, but after approximately 2.6 × 10^60605351 steps, the elements begin to decrease again, and eventually get to 0. However, the example of G(4) doesn't give a good idea of just how quickly the elements of a Goodstein sequence can increase. G(19) increases much more rapidly, and starts as follows:

Hereditary notation Value
2^2²+2+1 19
3^3³+3 7625597484990
4^4⁴+3 approximately 1.3 × 10¹⁵⁴
5^5⁵+2 approximately 1.8 × 10²¹⁸⁴
6^6⁶+1 approximately 2.6 × 10³⁶³⁰⁵
7^7⁷ approximately 3.8 × 10⁶⁹⁵⁹⁷⁴
7·8^{7·8⁷ +
7·8⁶ +
7·8⁵ +
7·8⁴ +
7·8³ +
7·8² +
7·8 + 7}
+
7·8^{7·8⁷ +
7·8⁶ +
7·8⁵ +
7·8⁴ +
7·8³ +
7·8² +
7·8 + 6}
+
...
+ 7·8² + 7·8 + 7

approximately 6 × 10^15151335
7·9^{7·9⁷ +
7·9⁶ +
7·9⁵ +
7·9⁴ +
7·9³ +
7·9² +
7·9 + 7}
+
7·9^{7·9⁷ +
7·9⁶ +
7·9⁵ +
7·9⁴ +
7·9³ +
7·9² +
7·9 + 6}
+
...
+ 7·9² + 7·9 + 6

approximately 4.3 × 10^369693099
...

Hereditary notation	Value
2^2²+2+1	19
3^3³+3	7625597484990
4^4⁴+3	approximately 1.3 × 10¹⁵⁴
5^5⁵+2	approximately 1.8 × 10²¹⁸⁴
6^6⁶+1	approximately 2.6 × 10³⁶³⁰⁵
7^7⁷	approximately 3.8 × 10⁶⁹⁵⁹⁷⁴
7·8^{7·8⁷ + 7·8⁶ + 7·8⁵ + 7·8⁴ + 7·8³ + 7·8² + 7·8 + 7} + 7·8^{7·8⁷ + 7·8⁶ + 7·8⁵ + 7·8⁴ + 7·8³ + 7·8² + 7·8 + 6} + ... + 7·8² + 7·8 + 7	approximately 6 × 10^15151335
7·9^{7·9⁷ + 7·9⁶ + 7·9⁵ + 7·9⁴ + 7·9³ + 7·9² + 7·9 + 7} + 7·9^{7·9⁷ + 7·9⁶ + 7·9⁵ + 7·9⁴ + 7·9³ + 7·9² + 7·9 + 6} + ... + 7·9² + 7·9 + 6	approximately 4.3 × 10^369693099
...

In spite of this rapid growth, Goodstein's theorem states that every Goodstein sequence eventually terminates at 0, no matter what the start value m is.

Proof

Goodstein's theorem can be proved (using techniques outside Peano arithmetic, see below) as follows: Given a Goodstein sequence G(m), we will contruct a parallel sequence of ordinal numbers whose elements are no smaller than those in the given sequence. If the elements of the parallel sequence go to 0, the elements of the Goodstein sequence must also go to 0.

To construct the parallel sequence, take the hereditary base n representation of the (n-1)-th element of the Goodstein sequence, and replace every instance of n with the first infinite ordinal number ω. Addition, multiplication and exponentiation of ordinal numbers is well defined, and the resulting ordinal number clearly cannot be smaller than the original element.

The 'base-changing' operation of the Goodstein sequence does not change the element of the parallel sequence: replacing all the 4's in 4^(4^4)+4 with ω is the same as replacing all the 4's with 5's and then replacing all the 5's with ω. The 'subtracting 1' operation, however, corresponds to decreasing the infinite ordinal number in the parallel sequence; for example, ω^(ω^ω) + ω decreases to ω^(ω^ω) + 4 if the step above is performed. Because the ordinals are well-ordered, there are no infinite strictly decreasing sequences of ordinals. Thus the parallel sequence must terminate at 0 after a finite number of steps. The Goodstein sequence, which is bounded above by the parallel sequence, must terminate at 0 also.

Need a proof sketch that the theorem cannot be proved in PA