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 "natural" example of a true statement that is unprovable in Peano arithmetic (after Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic and the Paris–Harrington theorem). Earlier statements of this type had either been, except for Gentzen, extremely complicated, ad-hoc constructions (such as the statements generated by the construction given in Gödel's incompleteness theorem) or concerned metamathematics or combinatorial results.
Laurie Kirby and Jeff Paris gave an interpretation of the Goodstein's theorem as a hydra game: the "Hydra" 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. The Kirby–Paris interpretation of the theorem says 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, very long time.
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.
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:
where each coefficient satisfies , and . For example, in base 2,
Thus the base 2 representation of 35 is . (This expression could be written in binary notation as 100011.) Similarly, one can write 100 in base 3:
Note that the exponents themselves are not written in base-n notation. For example, the expressions above include and .
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 digit appearing in the expression is n or less.
For example, while 35 in ordinary base-2 notation is , it is written in hereditary base-2 notation as
using the fact that Similarly, 100 in hereditary base 3 notation is
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 next element A056004, write m in hereditary base 2 notation, change all the 2s to 3s, and then subtract 1 from the result; this is the second element of G(m). To get the third element of G(m) A057650, write the second element in hereditary base 3 notation, change all 3s to 4s, and subtract 1 again. Continue until the result is zero, at which point the sequence terminates.
Early Goodstein sequences terminate quickly. For example, G(3) A215409 terminates at the sixth step:
|2||3||Write 3 in base 2 notation|
|3||3||Switch the 2 to a 3, then subtract 1|
|4||3||Switch the 3 to a 4, then subtract 1. Now there are no more 4s left|
|5||2||No 4s left to switch to 5s. Just subtract 1|
Later Goodstein sequences increase for a very large number of steps. For example, G(4) A056193 starts as follows:
Elements of G(4) continue to increase for a while, but at base , they reach the maximum of , stay there for the next steps, and then begin their first and final descent.
However, even 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 spite of this rapid growth, Goodstein's theorem states that every Goodstein sequence eventually terminates at 0, no matter what the starting value is.
Proof of Goodstein's theorem
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.
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 4s in 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, decreases to 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.
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. What Kirby showed is that Goodstein's theorem leads to Gentzen's theorem, i.e. it can substitute for induction up to ε0.
Sequence length as a function of the starting value
The Goodstein function, , is defined such that is the length of the Goodstein sequence that starts with n. (This is a total function since every Goodstein sequence terminates.) The extreme growth-rate of can be calibrated by relating it to various standard ordinal-indexed hierarchies of functions, such as the functions in the Hardy hierarchy, and the functions in the fast-growing hierarchy of Löb and Wainer:
- Kirby and Paris (1982) proved that
- has approximately the same growth-rate as (which is the same as that of ); more precisely, dominates for every , and dominates
- (For any two functions , is said to dominate if for all sufficiently large .)
- Cichon (1983) showed that
- where is the result of putting n in hereditary base-2 notation and then replacing all 2s with ω (as was done in the proof of Goodstein's theorem).
- Caicedo (2007) showed that if with then
|4||3·2402653211 − 2|
|8||> A3(3,3) = A(A(61, 61), A(61, 61))|
|12||> fω+1(64) > Graham's number|
(For Ackermann function and Graham's number bounds see fast-growing hierarchy#Functions in fast-growing hierarchies.)
Application to computable functions
Goodstein's theorem can be used to construct a total computable function that Peano arithmetic cannot prove to be total. The Goodstein sequence of a number can be effectively enumerated by a Turing machine; thus the function which maps n to the number of steps required for the Goodstein sequence of n to terminate is computable by a particular Turing machine. This machine merely enumerates the Goodstein sequence of n and, when the sequence reaches 0, returns the length of the sequence. Because every Goodstein sequence eventually terminates, this function is total. But because Peano arithmetic does not prove that every Goodstein sequence terminates, Peano arithmetic does not prove that this Turing machine computes a total function.
- Non-standard model of arithmetic
- Fast-growing hierarchy
- Paris–Harrington theorem
- Kanamori–McAloon theorem
- Kruskal's tree theorem
- Kirby, L.; Paris, J. (1982). "Accessible Independence Results for Peano Arithmetic". Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112/blms/14.4.285.
- Goodstein, R. (1944), "On the restricted ordinal theorem", Journal of Symbolic Logic 9: 33–41, JSTOR 2268019.
- Cichon, E. (1983), "A Short Proof of Two Recently Discovered Independence Results Using Recursive Theoretic Methods", Proceedings of the American Mathematical Society 87: 704–706, JSTOR 2043364.
- Caicedo, A. (2007), "Goodstein's function", Revista Colombiana de Matemáticas 41 (2): 381–391.
- Weisstein, Eric W., "Goodstein Sequence", MathWorld.
- Some elements of a proof that Goodstein's theorem is not a theorem of PA, from an undergraduate thesis by Justin T Miller
- A Classification of non standard models of Peano Arithmetic by Goodstein's theorem - Thesis by Dan Kaplan, Franklan and Marshall College Library
- Definitions of Goodstein sequences in the programming languages Ruby and Haskell, as well as a large-scale plot
- The Hydra game implemented as a Java applet
- Goodstein Sequences: The Power of a Detour via Infinity - good exposition with illustrations of Goodstein Sequences and the hydra game.