Abstract
The original Goodstein process proceeds by writing natural numbers in nested exponential k-normal form, then successively raising the base to 
and subtracting one from the end result. Such sequences always reach zero, but this fact is unprovable in Peano arithmetic. In this paper we instead consider notations for natural numbers based on the Ackermann function. We define two new Goodstein processes, obtaining new independence results for 
and 
, theories of second order arithmetic related to the existence of Turing jumps.
Keywords: Goodstein sequences, Independence proofs, Ordinal notation systems
Introduction
Goodstein’s principle [6] is arguably the oldest example of a purely number-theoretic statement known to be independent of 
, as it does not require the coding of metamathematical notions such as Gödel’s provability predicate [4]. The proof proceeds by transfinite induction up to the ordinal 
[5]. 
does not prove such transfinite induction, and indeed Kirby and Paris later showed that Goodstein’s principle is unprovable in 
[8].
Goodstein’s original principle involves the termination of certain sequences of numbers. Say that m is in nested (exponential) base-k normal form if it is written in standard exponential base k, with each exponent written in turn in base k. Thus for example, 20 would become 
in nested base-2 normal form. Then, define a sequence 
by setting 
and defining 
recursively by writing 
in nested base-
normal form, replacing every occurrence of 
by 
, then subtracting one (unless 
, in which case 
).
In the case that 
, we obtain
 |
and so forth. At first glance, these numbers seem to grow superexponentially. It should thus be a surprise that, as Goodstein showed, for every m there is 
for which 
.
By coding finite Goodstein sequences as natural numbers in a standard way, Goodstein’s principle can be formalized in the language of arithmetic, but this formalized statement is unprovable in 
. Independence can be shown by proving that the Goodstein process takes at least as long as stepping down the fundamental sequences below 
; these are canonical sequences 
such that 
for all 
and for limit 
, 
as 
. For standard fundamental sequences below 
, 
does not prove that the sequence 
is finite.
Exponential notation is not suitable for writing very big numbers (e.g. Graham’s number [7]), in which case it may be convenient to use systems of notation which employ faster-growing functions. In [2], T. Arai, S. Wainer and the authors have shown that the Ackermann function may be used to write natural numbers, giving rise to a new Goodstein process which is independent of the theory 
of arithmetical transfinite recursion; this is a theory in the language of second order arithmetic which is much more powerful than 
. The main axiom of 
states that for any set X and ordinal 
, the 
-Turing jump of X exists; we refer the reader to [13] for details.
The idea is, for each 
, to define a notion of Ackermannian normal form for each 
. Having done this, we can define Ackermannian Goodstein sequences analogously to Goodstein’s original version. The normal forms used in [2] are defined using an elaborate ‘sandwiching’ procedure first introduced in [14], approximating a number m by successive branches of the Ackermann function. In this paper, we consider simpler, and arguably more intuitive, normal forms, also based on the Ackermann function. We show that these give rise to two different Goodstein-like processes, independent of 
and 
, respectively. As was the case for 
, these are theories of second order arithmetic which state that certain Turing jumps exist. 
asserts that, for all 
and 
, the n-Turing jump of X exists, while 
asserts that its 
-jump exists; see [13] for details. The proof-theoretic ordinal of 
is 
[1], and that of 
is 
[9]; we will briefly review these ordinals later in the text, but refer the reader to standard texts such as [10, 12] for a more detailed treatment of proof-theoretic ordinals.
Basic Definitions
Let us fix 
and agree on the following version of the Ackermann function.
Definition 1
For 
we define 
by the following recursion.
,
,
.
Here, the notation 
refers to the k-fold composition of the function 
. It is well known that for every fixed a, the function 
is primitive recursive and the function 
is not primitive recursive. We use the Ackermann function to define k normal forms for natural numbers. These normal forms emerged from discussions with Toshiyasu Arai and Stan Wainer, which finally led to the definition of a more powerful normal form defined in [14] and used to prove termination in [2].
Lemma 1
Let 
. For all 
, there exist unique 
such that 
We write 
in this case. This means that we have in mind an underlying context fixed by k and that for the number c we have uniquely associated the numbers a, b, m, n. Note that it could be possible that 
, so that we have to choose the right representation for the context; in this case, item 2 guarantees that a is chosen to take the maximal possible value.
By rewriting iteratively b and n in such a normal form, we arrive at the Ackermann k-normal form of c. If we also rewrite a iteratively, we arrive at the nested Ackermann k-normal form of c. The following properties of normal forms are not hard to prove from the definitions.
Lemma 2
is in k-normal form for every 
such that 
.if
is in k-normal form, then for every 
, the number 
is also in k-normal form.
In the sequel we work with standard notations for ordinals. We use the function 
to enumerate the fixed points of 
. With 
we denote the binary Veblen function, where 
enumerates the common fixed points of all 
with 
. We often omit parentheses and simply write 
. Then 
, 
, 
is the first fixed point of the function 
, 
is the first common fixed point of the function 
, and 
is the first ordinal closed under 
. In fact, not much ordinal theory is presumed in this article; we almost exclusively work with ordinals less than 
, which can be written in terms of addition and the functions 
, 
. For more details, we refer the reader to standard texts such as [10, 12].
Goodstein Sequences for 
In this section we define a Goodstein process that is independent of 
. We do so by working with unnested Ackermannian normal forms. Such normal forms give rise to the following notion of base change.
Definition 2
Given 
and 
, define 
by:
.
if 
.
With this, we may define a new Goodstein process, based on unnested Ackermannian normal forms.
Definition 3
Let 
. Put 
Assume recursively that 
is defined and 
. Then 
. If 
, then 
.
We will show that for every 
there is i with 
. In order to prove this, we first establish some natural properties of the base-change operation.
Lemma 3
Fix 
and let 
. Then:
.If
, then 
.
Proof
The first assertion is proved by induction on c. It clearly holds for 
. If 
then the induction hypothesis yields 
The second assertion is harder to prove. The proof is by induction on d with a subsidiary induction on c. The assertion is clear if 
. Let 
and 
. We distinguish cases according to the position of a relative to 
, the position of b relative to 
, etc.
-
Case 1 (
). We sub-divide into two cases.- Case 1.1 (
). Then, the induction hypothesis applied to 
yields 
. - Case 1.2 (
). In this case, 
, 
, 
, and 
. We have 
. For 
we have that 
is in k-normal form by Lemma 2. Thus the induction hypothesis yields 
. The number 
is in k-normal form and so the induction hypothesis applied to 
yields 
. Moreover we have that 
. This yields
where the second inequality follows from
and the last from
1
Case 2 (
). This case does not occur since then 
.-
Case 3 (
and 
). The induction hypothesis yields 
and 
. Now, consider two sub-cases.- Case 3.1 (
). Since d is in k-normal form and 
we see that 
is in k-normal form by Lemma 2. Then, the induction hypothesis yields 
. -
Case 3.2 (
). We know that 
. Consider two further sub-cases.- Case 3.2.1 (
). This means that 
, 
, and 
, where d has k-normal form 
. The induction hypothesis yields 
and 
. We then have that 
. - Case 3.2.2 (
). Then,
where the second inequality uses
Case 4 (
and 
). This case does not appear since otherwise 
.- Case 5 (
and 
and 
). Then the induction hypothesis yields 
Case 6 (
and 
and 
). This case is not possible given the assumptions.- Case 7 (
and 
and 
). Then 
and the induction hypothesis yields 
Thus, the base-change operation is monotone. Next we see that it also preserves normal forms.
Lemma 4
If 
is in k-normal form, then 
is in 
normal form.
Proof
Assume that 
. Then, 
, 
, and 
. Clearly, 
. By Lemma 2, 
is in k-normal form, so that by Lemma 3, 
yields 
. Since 
is in k-normal form, Lemma 3 yields 
. It remains to check that we also have 
.
If 
, then 
means that 
with 
and 
. Then, 
and 
. Thus 
and thus 
In the remaining case, we have for 
that
 |
So 
is in 
-normal form.
These Ackermannian normal forms give rise to a new Goodstein process. In order to prove that this process is terminating, we must assign ordinals to natural numbers, in such a way that the process gives rise to a decreasing (hence finite) sequence. For each k, we define a function 
, where 
is a suitable ordinal, in such a way that 
is computed from the k-normal form of m. Unnested Ackermannian normal forms correspond to ordinals below 
, as the following map shows.
Definition 4
For 
, define 
as follows:
.
if 
.
Lemma 5
If 
then 
.
Proof
Proof by induction on d with subsidiary induction on c. The assertion is clear if 
. Let 
and 
. We distinguish cases according to the position of a relative to 
, the position of b relative to 
, etc.
Case 1 (
). We have 
and, since 
, the induction hypothesis yields 
We have 
and the induction hypothesis yields 
It follows that 
, hence 
Case 2 (
). This case is not possible since this would imply that 
-
Case 3 (
). We consider several sub-cases.- Case 3.1 (
). The induction hypothesis yields 
. Hence 
. We have 
, and the subsidiary induction hypothesis yields 
. Putting things together we see 
- Case 3.2 (
). This case is not possible since this would imply 
. -
Case 3.3 (
). This case is divided into further sub-cases.- Case 3.3.1 (
). We have 
and the subsidiary induction hypothesis yields 
Hence 
- Case 3.3.2 (
). This case is not possible since this would imply 
. - Case 3.3.3 (
). The inequality 
yields 
and the induction hypothesis yields 
. Hence 
.
Our ordinal assignment is invariant under base change, in the following sense.
Lemma 6
.
Proof
Proof by induction on c. The assertion is clear for 
Let 
. Then, 
, and the induction hypothesis yields
 |
It is well-known that the so-called slow-growing hierarchy at level 
matches up with the Ackermann function, so one might expect that the corresponding Goodstein process can be proved terminating in 
. This is true but, somewhat surprisingly, much less is needed here. We can lower 
to 
.
Theorem 1
For all 
, there exists a 
such that 
This is provable in 
.
Proof
Define 
If 
, then, by the previous lemmata,
 |
Since 
cannot be an infinite decreasing sequence of ordinals, there must be some k with 
, yielding 
.
Now we are going to show that for every 
, 
This will require some work with fundamental sequences.
Definition 1
Let 
be an ordinal. A system of fundamental sequences on
is a function 
such that 
with equality holding if and only if 
, and 
whenever 
. The system of fundamental sequences is convergent if 
whenever 
is a limit, and has the Bachmann property if whenever 
, it follows that 
.
It is clear that if 
is an ordinal then for every 
there is n such that 
, but this fact is not always provable in weak theories. The Bachmann property that will be useful due to the following.
Proposition 1
Let 
be an ordinal with a system of fundamental sequences satisfying the Bachmann property, and let 
be a sequence of elements of 
such that, for all n, 
. Then, for all n, 
.
Proof
Let 
be the reflexive transitive closure of 
. We need a few properties of these orderings. Clearly, if 
, then 
. It can be checked by a simple induction and the Bachmann property that, if 
, then 
. Moreover, 
is monotone in the sense that if 
, then 
, and if 
, then 
(see, e.g., [11] for details).
We claim that for all n, 
, from which the desired inequality immediately follows. For the base case, we use the fact that 
is transitive by definition. For the successor, note that the induction hypothesis yields 
, hence 
. Then, consider three cases.
Case 1 (
). By transitivity and monotonicity, 
yields 
.Case 2 (
). Then, 
.Case 3 (
). The Bachmann property yields 
, and since 
, monotinicity and transitivity yield 
.
Let 
and 
. Let us define the standard fundamental sequences for ordinals less than 
as follows.
If
with 
, then 
.If
, then we set 
if 
, 
if 
, and 
if 
.If
, then 
if 
, 
if 
, and 
if 
.
This system of fundamental sequences enjoys the Bachmann property [11].
In view of Proposition 1, the following technical lemma will be crucial for proving our main independence result for 
.
Lemma 7
Given 
with 
, 
.
Proof
We prove the claim by induction on c. Let 
-
Case 3 (
and 
). We consider several sub-cases.- Case 3.3 (
and 
). Then the induction hypothesis yields similarly as in Case 3.1:
since
is in 
normal form. - Case 3.4 (
and 
). The assertion follows trivially since then 
.
Theorem 2
Let 
. Then 
Hence 
Proof
Assume for a contradiction that 
Then 
Recall that 
. We have 
. Lemma 7 and Lemma 5 yield 
, hence Proposition 1 yields 
So the least k such that 
is at least as big as the least k such that 
. But by standard results in proof theory [3], 
does not prove that this k is always defined as a function of 
. This contradicts 
Goodstein Sequences for 
In this section, we indicate how to extend our approach to a situation where the base change operation can also be applied to the first argument of the Ackermann function. The resulting Goodstein principle will then be independent of 
. The key difference is that the base-change operation is now performed recursively on the first argument, as well as the second.
Definition 5
For 
and 
, define 
by:
if 
.
Note that in this section, 
will always indicate the operation of Definition 5. We can then define a Goodstein process based on this new base change operator.
Definition 6
Let 
. Put 
Assume recursively that 
is defined and 
. Then, 
. If 
, then 
.
Termination and independence results can then be obtained following the same general strategy as before. We begin with the following lemmas, whose proofs are similar to those for their analogues in Sect. 3.
Lemma 8
If 
and 
, then 
.
Lemma 9
If 
is in k-normal form, then 
is in 
normal form.
It is well-known that the so-called slow-growing hierarchy at level 
matches up with the functions which are elementary in the Ackermann function, so one might expect that the corresponding Goodstein process can be proved terminating in 
. This is true but, somewhat surprisingly, much less is needed here. Indeed, nested Ackermannian normal forms are related to the much smaller ordinal 
by the following mapping.
Definition 7
Given 
, define a function 
given by:
.
if 
.
As was the case for the mappings 
, the maps 
are strictly increasing and invariant under base change, as can be checked using analogous proofs to those in Sect. 3.
Lemma 10
Let 
with 
.
If
, then 
.
.
Theorem 3
For all 
, there exists a 
such that 
This is provable in 
.
Next, we show that for every 
, 
For this, we need the following analogue of Lemma 7.
Lemma 11
.
Proof
We proceed by induction on c. Let 
. Let us concentrate on the critical case 
and 
, where 
and 
.
The induction hypothesis yields
 |
since 
is in 
normal form.
The remaining details of the proof of the theorem can be carried out similarly as before.
Theorem 4
For every 
Hence 
Contributor Information
Marcella Anselmo, Email: manselmo@unisa.it.
Gianluca Della Vedova, Email: gianluca.dellavedova@unimib.it.
Florin Manea, Email: flmanea@gmail.com.
Arno Pauly, Email: arno.m.pauly@gmail.com.
David Fernández-Duque, Email: David.FernandezDuque@UGent.be.
Andreas Weiermann, Email: Andreas.Weiermann@UGent.be.
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