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 showed that it is unprovable in Peano 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 ε0-induction in Peano arithmetic. The Paris–Harrington theorem was a later example.
Laurence Kirby and Jeff Paris introduced a graph-theoretic 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", 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.
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 ai satisfies, and. For example, in base 2,
Thus the base-2 representation of 35 is 100011, which means. Similarly, 100 represented in base 3 is 10201:
Note that the exponents themselves are not written in base-n notation. For example, the expressions above include 25 and 34, and 5>2, 4>3.
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 number appearing in the expression has been converted to base-n notation.
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
Goodstein sequences
The Goodstein sequence G of a number m is a sequence of natural numbers. The first element in the sequence G is m itself. To get the second, G, write m in hereditary base-2 notation, change all the 2s to 3s, and then subtract 1 from the result. In general, the term of the Goodstein sequence of m is as follows:- Take the hereditary base- representation of G.
- Replace each occurrence of the base- with.
- Subtract one.
- Continue until the result is zero, at which point the sequence terminates.
| 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 | 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 starts as follows:
| Hereditary notation | Value |
| 4 | |
| 26 | |
| 41 | |
| 60 | |
| 83 | |
| 109 | |
| 253 | |
| 299 | |
| 1151 |
Elements of G 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.
However, even G doesn't give a good idea of just how quickly the elements of a Goodstein sequence can increase.
G increases much more rapidly and starts as follows:
| Hereditary notation | Value |
| 19 | |
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 as follows: Given a Goodstein sequence G, we construct a parallel sequence P of ordinal numbers which is strictly decreasing and terminates. Then G must terminate too, and it can terminate only when it goes to 0. A common misunderstanding of this proof is to believe that G goes to 0 because it is dominated by P. Actually, the fact that P dominates G plays no role at all. The important point is: G exists if and only if P exists. Then if P terminates, so does G. And G can terminate only when it comes to 0.We define a function which computes the hereditary base representation of and then replaces each occurrence of the base with the first infinite ordinal number ω. For example,.
Each term P of the sequence P is then defined as f. For example, and. Addition, multiplication and exponentiation of ordinal numbers are well defined.
We claim that :
Let be G after applying the first,
base-changing operation in generating the next element of the Goodstein sequence,
but before the second minus 1 operation in this generation.
Observe that.
Then clearly,. Now we apply the minus 1 operation, and, as.
For example, and, so and, which is strictly smaller. Note that in order to calculate f, we first need to write G in hereditary base notation, as for instance the expression either makes no sense or is equal to.
Thus the sequence P is strictly decreasing. As the standard order < on ordinals is well-founded, an infinite strictly decreasing sequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals terminates. But P is calculated directly from G. Hence the sequence G must terminate as well, meaning that it must reach 0.
While this proof of Goodstein's theorem is fairly easy, the Kirby–Paris theorem, 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
Suppose the definition Goodstein sequence is changed so that instead ofreplacing each occurrence of the base b with
it replaces it with. Would the sequence still terminate?
More generally, let b1, b2, b3, … be any sequences of integers.
Then let the
term of the extended Goodstein sequence of m be as
follows: take the hereditary base bn representation of
G, and replace each occurrence of the base bn
with and then subtract one.
The claim is that this sequence still terminates.
The extended proof defines as
follows: take the hereditary base bn representation of
G, and replace each occurrence of the base
bn with the first infinite ordinal number ω.
The base-changing operation of the Goodstein sequence when going
from G to still does not change the value of f.
For example, if and if,
then
, hence the ordinal is strictly greater than the ordinal
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. 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 proved that
- Cichon showed that
- Caicedo showed that if with then