Goodstein's theorem
Encyclopedia
In mathematical logic
, Goodstein's theorem is a statement about the natural number
s, made by Reuben Goodstein
, which states that every Goodstein sequence eventually terminates at 0. 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.
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
Early Goodstein sequences terminate quickly. For example, G(3) terminates at the sixth step:
Later Goodstein sequences increase for a very large number of steps. For example, G(4) 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.
The value 0 is reached at base
(curiously, this is a generalized Woodall number
:
. This is also the case with all other final bases for starting values greater than 4).
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.
s 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.
, 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:
Some examples:
(For Ackermann function
and Graham's number
bounds see fast-growing hierarchy#Functions in fast-growing hierarchies.)
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.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, Goodstein's theorem is a statement about the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, made by Reuben Goodstein
Reuben Goodstein
Reuben Louis Goodstein was an English mathematician with a strong interest in the philosophy and teaching of mathematics....
, which states that every Goodstein sequence eventually terminates at 0. showed that it is unprovable
Independence (mathematical logic)
In mathematical logic, independence refers to the unprovability of a sentence from other sentences.A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that...
in Peano arithmetic
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...
(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
Gerhard Gentzen
Gerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus...
's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic and the Paris–Harrington theorem
Paris–Harrington theorem
In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory is true, but not provable in Peano arithmetic...
). 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
Jeff Paris
Jeffrey Bruce Paris is a British mathematician known for his work on mathematical logic, in particular provability in arithmetic, uncertain reasoning and inductive logic with an emphasis on rationality and common sense principles....
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 asusing the fact that
Similarly, 100 in hereditary base 3 notation isGoodstein sequences
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, 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), 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) terminates at the sixth step:
| Base | Hereditary notation | Value | Notes |
|---|---|---|---|
| 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 |
| 6 |  |
1 | |
| 7 |  |
0 |
Later Goodstein sequences increase for a very large number of steps. For example, G(4) starts as follows:
| Hereditary notation | Value |
|---|---|
 |
4 |
 |
26 |
 |
41 |
 |
60 |
 |
83 |
 |
109 |
 |
 |
 |
253 |
 |
299 |
 |
 |
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 generalized Woodall numberWoodall number
In number theory, a Woodall number is any natural number of the formfor some natural number n. The first few Woodall numbers are:Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly-defined Cullen numbers...
:
. This is also the case with all other final bases for starting values greater than 4).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 |
 |
7,625,597,484,990 |
 |
 |
 |
 |
 |
 |
 |
 |
     |
 |
     |
 |
 |
 |
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 numberOrdinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s 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
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
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-orderedWell-ordering principle
The "well-ordering principle" is a property of ordered sets. A set X is well ordered if every subset of X has a least element. An example of a well ordered set is the set of natural numbers. An example of set that is not well ordered is the set of integers...
, 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
Gentzen's consistency proof
Gentzen's consistency proof is a result of proof theory in mathematical logic. It "reduces" the consistency of a simplified part of mathematics, not to something that could be proved , but to clarified logical principles.-Gentzen's theorem:In 1936 Gerhard Gentzen proved the consistency of...
, 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 hierarchyHardy hierarchy
In computability theory, computational complexity theory and proof theory, the Hardy hierarchy, named after G. H. Hardy, is an ordinal-indexed family of functions hα: N → N . It is related to the fast-growing hierarchy and slow-growing hierarchy...
, and the functions
in the fast-growing hierarchyFast-growing hierarchy
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy is an ordinal-indexed family of rapidly increasing functions fα: N → N...
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 
> g(n)\,\! 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
.
Some examples:
| n |  |
||||
|---|---|---|---|---|---|
| 1 |  |
 |
 |
 |
2 |
| 2 |  |
 |
 |
 |
4 |
| 3 |  |
 |
 |
 |
6 |
| 4 |  |
 |
 |
 |
3·2402653211 − 2 |
| 5 |  |
 |
 |
 |
> A Ackermann function In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive... (4,4) |
| 6 |  |
 |
 |
 |
> A(6,6) |
| 7 |  |
 |
 |
 |
> A(8,8) |
| 8 |  |
 |
 |
 |
> A3(3,3) = A(A(61, 61), A(61, 61)) |
 |
|||||
| 12 |  |
 |
 |
 |
> fω+1(64) > Graham's number Graham's number Graham's number, named after Ronald Graham, is a large number that is an upper bound on the solution to a certain problem in Ramsey theory.The number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of Scientific American in November 1977,... |
 |
|||||
| 19 |  |
 |
 |
 |
|
(For Ackermann function
Ackermann function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive...
and Graham's number
Graham's number
Graham's number, named after Ronald Graham, is a large number that is an upper bound on the solution to a certain problem in Ramsey theory.The number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of Scientific American in November 1977,...
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 functionComputable function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any concrete model of computation such as Turing machines or register...
that Peano arithmetic cannot prove to be total. The Goodstein sequence of a number can be effectively enumerated by a Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
; 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.