Goodstein's theorem
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^{[1]} showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as secondorder arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε_{0}induction in Peano arithmetic. The Paris–Harrington theorem was a later example.
Laurence Kirby and Jeff Paris introduced a graphtheoretic hydra game with behavior similar to that of Goodstein sequences: 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. 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.^{[1]}
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Hereditary basen notation[edit]
Goodstein sequences are defined in terms of a concept called "hereditary basen notation". This notation is very similar to usual basen positional notation, but the usual notation does not suffice for the purposes of Goodstein's theorem.
In ordinary basen 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 a_{i} satisfies 0 ≤ a_{i} < n, and a_{k} ≠ 0. For example, in base 2,
Thus the base2 representation of 35 is 100011, which means 2^{5} + 2 + 1. Similarly, 100 represented in base 3 is 10201:
Note that the exponents themselves are not written in basen notation. For example, the expressions above include 2^{5} and 3^{4}.
To convert a basen representation to hereditary basen notation, first rewrite all of the exponents in basen notation. Then rewrite any exponents inside the exponents, and continue in this way until every number appearing in the expression has been converted to basen notation.
For example, while 35 in ordinary base2 notation is 2^{5} + 2 + 1, it is written in hereditary base2 notation as
using the fact that 5 = 2^{2} + 1. Similarly, 100 in hereditary base3 notation is
Goodstein sequences[edit]
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 base2 notation, change all the 2s to 3s, and then subtract 1 from the result. In general, the (n + 1)st term G(m)(n + 1) of the Goodstein sequence of m is as follows:
 Take the hereditary basen + 1 representation of G(m)(n).
 Replace each occurrence of the basen + 1 with n + 2.
 Subtract one. (Note that the next term depends both on the previous term and on the index n.)
 Continue until the result is zero, at which point the sequence terminates.
Early Goodstein sequences terminate quickly. For example, G(3) terminates at the 6th step:
Base  Hereditary notation  Value  Notes 

2  3  Write 3 in base2 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  
6  1  No 5s left to switch to 6s. Just subtract 1  
7  0  No 6s left to switch to 7s. Just subtract 1 
Later Goodstein sequences increase for a very large number of steps. For example, G(4) OEIS: A056193 starts as follows:
Hereditary notation  Value 

4  
26  
41  
60  
83  
109  
253  
299  
1151 
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.
The value 0 is reached at base . (Curiously, this is a Woodall number: . This is also the case with all other final bases for starting values greater than 4.^{[citation needed]})
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:
Hereditary notation  Value 

19  
625597484990 7  




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[edit]
Goodstein's theorem can be proved (using techniques outside Peano arithmetic, see below) as follows: Given a Goodstein sequence G(m), we construct a parallel sequence P(m) of ordinal numbers which is strictly decreasing and terminates. Then G(m) must terminate too, and it can terminate only when it goes to 0. A common misunderstanding of this proof is to believe that G(m) goes to 0 because it is dominated by P(m). Actually, the fact that P(m) dominates G(m) plays no role at all. The important point is: G(m)(k) exists if and only if P(m)(k) exists (parallelism). Then if P(m) terminates, so does G(m). And G(m) can terminate only when it comes to 0.
We define a function f which takes the hereditary base k representation of G(m)(n) and replaces each occurrence of the base k with the first infinite ordinal number ω. Each term P(m)(n) of the sequence P(m) is then defined as f(G(m)(n),n+1). For example, G(3)(1) = 3 = 2^{1} + 2^{0} and P(3)(1) = f(2^{1} + 2^{0},2) = ω^{1} + ω^{0} = ω + 1. Addition, multiplication and exponentiation of ordinal numbers are well defined.
We claim that . Let be G(m)(n) after applying the basechanging operation of the Goodstein sequence. Then clearly, . Now we apply the minus 1 operation, and obviously , as .
For example, and , so and , which is strictly smaller. Note that in order to calculate f(G(m)(n),n+1), we first need to write G(m)(n) in hereditary base n+1 notation, as for instance the expression either makes no sense or is equal to .
Thus the sequence P(m) is strictly decreasing. As the standard order < on ordinals is wellfounded, an infinite strictly decreasing sequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals terminates (and cannot be infinite). But P(m)(n) is calculated directly from G(m)(n). Hence the sequence G(m) must terminate as well, meaning that it must reach 0.
While this proof of Goodstein's theorem is fairly easy, the Kirby–Paris theorem,^{[1]} which shows 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.
Extended Goodstein's theorem[edit]
Suppose the definition Goodstein sequence is changed so that instead of replacing each occurrence of the base b with b+1 it replaces it with b+2. Would the sequence still terminate? More generally, let b_{1}, b_{2}, b_{3}, … be any sequences of integers. Then let the n+1st term G(m)(n+1) of the extended Goodstein sequence of m be as follows: take the hereditary base b_{n} representation of G(m)(n), and replace each occurrence of the base b_{n} with b_{n+1} and then subtract one. The claim is that this sequence still terminates. The extended proof defines P(m)(n) = f(G(m)(n), n) as follows: take the hereditary base b_{n} representation of G(m)(n), and replace each occurrence of the base b_{n} with the first infinite ordinal number ω. The basechanging operation of the Goodstein sequence when going from G(m)(n) to G(m)(n+1) still does not change the value of f. For example, if b_{n} = 4 and if b_{n+1} = 9, then , hence the ordinal is strictly greater than the ordinal
Sequence length as a function of the starting value[edit]
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 growthrate of can be calibrated by relating it to various standard ordinalindexed hierarchies of functions, such as the functions in the Hardy hierarchy, and the functions in the fastgrowing hierarchy of Löb and Wainer:
 Kirby and Paris (1982) proved that
 has approximately the same growthrate 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 base2 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
 .
Some examples:
n  

1  2  
2  4  
3  6  
4  3·2^{402653211} − 2 ≈ 6.895080803×10^{121210694 }  
5  > A(4,4)>  
6  > A(6,6)  
7  > A(8,8)  
8  > A^{3}(3,3) = A(A(61, 61), A(61, 61))  
12  > f_{ω+1}(64) > Graham's number  
19 
(For Ackermann function and Graham's number bounds see fastgrowing hierarchy#Functions in fastgrowing hierarchies.)
Application to computable functions[edit]
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.
See also[edit]
 Nonstandard model of arithmetic
 Fastgrowing hierarchy
 Paris–Harrington theorem
 Kanamori–McAloon theorem
 Kruskal's tree theorem
References[edit]
 ^ ^{a} ^{b} ^{c} Kirby, L.; Paris, J. (1982). "Accessible Independence Results for Peano Arithmetic" (PDF). Bulletin of the London Mathematical Society. 14 (4): 285. doi:10.1112/blms/14.4.285.
Bibliography[edit]
 Goodstein, R. (1944), "On the restricted ordinal theorem", Journal of Symbolic Logic, 9: 33–41, doi:10.2307/2268019, 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, doi:10.2307/2043364, JSTOR 2043364.
 Caicedo, A. (2007), "Goodstein's function" (PDF), Revista Colombiana de Matemáticas, 41 (2): 381–391.
External links[edit]
 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 largescale plot
 The Hydra game implemented as a Java applet
 Javascript implementation of a variant of the Hydra game
 Goodstein Sequences: The Power of a Detour via Infinity  good exposition with illustrations of Goodstein Sequences and the hydra game.
 Goodstein Calculator