In mathematical logic, Goodstein's theorem is a 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, in particular using the axiom of infinity. The theorem states that every Goodstein sequence eventually terminates at 0. It stands as an example that not all undecidable theorems are peculiar or contrived, as those constructed by Gödel's incompleteness theorem are sometimes considered. Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...

Definition of a Goodstein sequence

In order to define a Goodstein sequence, first define hereditary base-n notation. To write a natural number in hereditary base-n notation, first write it in the form , where each a_i is an integer between 0 and n − 1; then break up each term into individual powers of n:a_kn^k becomes . Then write all the exponents k in hereditary base n notation, and continue recursively until every digit appearing in the expression is n or 0 - every non-exponent is n and every exponent in a tower of exponents is n or 0 (note that n⁰ = 1). In elementary mathematics, a term is either a single number or variable, or the product of several numbers and/or variables. ...

For example, 35 in ordinary base-2 notation is 2⁵ + 2 + 1, 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

Early Goodstein sequences terminate quickly; for example G(3):

Many later Goodstein sequences increase for a very large number of steps. For example, G(4) starts as follows (in the second line, 2·3² is an abbreviation for and so in other cases) :

Elements of G(4) continue to increase for a while, but at base 3 · 2^402653209, they reach the maximum of 3 · 2^402653210 − 1, stay there for the next 3 · 2^402653209 steps, and then begin their first and final descent.

The value 0 is reached at base 3 · 2^402653211 − 1, which, curiously, is a Woodall number, just as all other final bases for starting values greater than 4. 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: In mathematics, a Woodall number is a natural number of the form n · 2n − 1 (written Wn). ...

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 construct 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. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

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. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

The 'base-changing' operation of the Goodstein sequence does not change the element of the parallel sequence: replacing all the 4s in 4^(4^4) + 4 with ω is the same as replacing all the 4s with 5s and then replacing all the 5s 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. Sometimes the phrase well-ordering principle (or the axiom of choice) is taken to be synonymous with well-ordering theorem. On other occasions the phrase is taken to mean the proposition that the set of natural numbers {1, 2, 3, ....} is well-ordered, i. ...

While this proof of Goodstein's theorem is fairly easy, the Kirby-Paris theorem which says that Goodstein's theorem is not a theorem of Peano arithmetic, is technical and considerably more difficult. It makes use of countable nonstandard models of Peano arithmetic.

References

Goodstein, R., On the restricted ordinal theorem, Journal of Symbolic Logic, 9 (1944), 33-41.

Kirby, L. and Paris, J., Accessible independence results for Peano arithemtic, Bull. London. Math. Soc., 14 (1982), 285-93.

External links

Some elements of a proof that Goodstein's theorem is not a theorem of PA can be found here: http://www.u.arizona.edu/~miller/thesis/node11.html

Definitions of Goodstein sequences in the programming languages Ruby and Haskell, as well as a large-scale plot, may be found at http://www.cwi.nl/~tromp/pearls.html#goodstein