aa r X i v : . [ m a t h . L O ] A p r A walk with Goodstein
David Fern´andez-Duque ∗ and Andreas Weiermann † Department of Mathematics: Analysis, Logic and DiscreteMathematics, Ghent University
Abstract
Goodstein’s principle is arguably the first purely number-theoreticstatement known to be independent of Peano arithmetic. It involves se-quences of natural numbers which at first appear to grow very quickly,but eventually decrease to zero. These sequences are defined relative toa notation system based on exponentiation for the natural numbers. Inthis article, we explore notions of optimality for such notation systemsand apply them to the classical Goodstein process, to a weaker variantbased on multiplication rather than exponentiation, and to a strongervariant based on the Ackermann function. In particular, we introduce thenotion of base-change maximality, and show how it leads to far-reachingextensions of Goodstein’s result.
Ever since G¨odel’s first incompleteness theorem [4], we know that Peano arith-metic ( PA ) cannot prove every true arithmetical statement. However, G¨odel’sproof is based on a specifically constructed statement that can be argued to beartificial from the perspective of mainstream mathematics. Since then, severalfacts of a purely combinatorial nature have been shown to be independent from PA [3, 9, 12], but the oldest example is a theorem of Goodstein [5], which willbe the focus of this work.Informally, one writes a natural number m in hereditary base 2, meaningthat m is written in base 2 in the usual way, then so is each exponent thatappears, and so on. A precise definition will be given later but for example, m = 22 would be written as 2 + 2 + 2. The Goodstein sequence based on m is a sequence ( G i m ) αi =0 with α ≤ ∞ , such that G m = m and, if G i m is definedand positive, G i +1 m is obtained by first writing G i m in hereditary base i + 2,then replacing every instance of i + 2 by i + 3, and finally subtracting 1. The ∗ [email protected] † [email protected] G
22 = 3 + 3 + 3 − + 3 + 2 . This number is already large enough to be rather cumbersome to write out and,in fact, the sequence will grow quite rapidly for some time. This should makeGoodstein’s principle quite surprising: for any m that we start with, there willbe a value of i such that G i m = 0. The proof uses transfinite induction, andKirby and Paris showed that this was, in a precise sense, unavoidable, leadingto unprovability in PA [8]. However, in this paper we do not assume familiaritywith transfinite induction, and its use will be kept to a minimum.A natural question to ask is if this particular way of writing natural numbersis ‘canonical’ in some way. For example, we could just as well have written22 = 2 + 2 + 2 + 2. This would lead to a different candidate for G
22, namely,3 + 3 + 3 + 2. Is there some sense in which the standard representation of 22is preferable? Will the Goodstein process still terminate if we choose a differentrepresentation of each natural number?Remarkably, the answer to both of these questions is ‘yes’. In fact, thetwo questions are intimately connected, as we will see throughout the paper.Regarding the first question, we identify two criteria for a canonical system ofnotations: first, it should be norm minimizing, meaning that we use the leastpossible number of symbols to write a number. Second, it should be base-changemaximal, which roughly states that G i +1 m will be as large as possible given G i m . This latter property is surprisingly useful. In particular, termination fora Goodstein process based on a base-change maximal notation system impliesthat any other notation system (based on the same primitive functions) will alsoyield a terminating Goodstein process.As we will see, the hereditary exponential normal form for natural numbersenjoys both norm minimization and base-change maximality. This tells us thatevery Goodstein walk is finite, by which we mean every sequence of numbers( m i ) αi =0 , where m i +1 is obtained by writing m i in an arbitrary fashion usingaddition, mutiplication and base-( i + 2) exponentiation, then replacing everyinstance of i + 2 by i + 3 and subtracting one. This is a powerful extension ofGoodstein’s original theorem, which itself did not involve multiplication (aside,possibly, for multiplication by single digits).There are variants of Goodstein’s result based on other primitive functions.One can base systems of notation on addition and multiplication, or add newfunctions such as the Ackermann function. In each of these cases we will explorethe above-mentioned optimality conditions, and use them to obtain extendedGoodstein-style principles. In the case of the multiplicative Goodstein principle,we give a full proof of termination and an Ackermannian lower bound, followinga structure similar to a modern presentation of Goodstein’s proof. Layout
In Section 2 we review Goodstein’s classical result and set up an ab-stract framework which allows for generalizations. Section 3 then introducesthe notions of norm minimality and base-change maximality, which will be a2ocus of the paper. With these notions in mind, the following sections studyvarious Goodstein processes: Section 4 considers a weakened Goodstein prin-ciple, for which an Ackermannian lower bound is given in Section 5. Section6 studies the optimality of hereditary exponential normal forms, and Section 7shows that base-change maximality holds even if we extend the notation systemto include multiplication. Section 8 then considers a Goodstein process basedon the Ackermann function. The optimality results obtained are used in Sec-tion 9 to provide generalizations of Goodstein’s theorem and its Ackermannianvariant. Section 10 provides some concluding remarks.
Let us discuss the original Goodstein principle from an abstract perspective,which will be useful in the rest of the text. A notation system is a family offunction symbols F so that each f ∈ F is equipped with an arity n f > | f | : N n f → N . For a function symbol f ( x , . . . , x n ) of arity n + 1, theparameter x will be regarded as a ‘base’ and usually donoted k or ℓ .Given fixed k ≥
2, the set of (closed) base k terms, T F k , is defined inductivelyso that if τ , . . . , τ n are terms and f is a function symbol with arity n + 1, then f ( k, τ , . . . , τ n ) is a term. We write T F for S ∞ k =2 T F k . The value of a term τ = f ( k, σ , . . . , σ n ) is defined recusrively by | τ | = | f | ( k, | σ | , . . . , | σ n | ). The norm of τ is defined inductively by k τ k = 1 + P ni =1 k σ i k ; note that constantsymbols (i.e., function symbols that depend only on k ) have norm one.It is important to make a conceptual distinction between function symbolsand the functions they represent, as for example we can do induction on thecomplexity of a term, independently of its numerical value. However, we willusually not make a notational distinction, and whether an expression shouldbe treated as a function or a symbol will be made clear from context. For theclassical Goodstein process, we will work with the functions/function symbols0 , x + y and k x ; we denote this notation by E for ‘exponential’, and write E k instead of T E k .It is not required that each natural number have a single notation, buta canonical one may be chosen nonetheless. A normal form assignment for anotation system F is a function nf · ( · ) : [2 , ∞ ) × N → T F such that nf k ( n ) ∈ T F k and | nf k ( n ) | = n for all k ≥ n . A notation system equipped with a normalform assignment is a normalized notation system. In the case of E k , the normalform for n ∈ N is defined as follows. Set nf k (0) = 0. For n >
0, assume that nf k ( m ) is defined for all m < n . Let r be the unique natural number such that k r ≤ n < k r +1 , and b = n − k r . Then set nf k ( n ) = k nf k ( r ) + nf k ( b ).Finally, we need to define a base change operation to define the Goosteinprocess. Given k ≤ ℓ and τ ∈ T F k , we define h ℓ i τ recursively by h ℓ i f ( k, σ , . . . , σ n ) = f ( ℓ, h ℓ i σ , . . . , h ℓ i σ n ) . If a normal form assignment is given, we can extend operations on terms tonatural numbers by first computing their normal form. In particular, we define3 n k k = k nf k ( n ) k and h ℓ/k i n as h ℓ i nf k ( n ). To ease notation, we will sometimeswrite h ℓ i n instead of h ℓ/k i n , where it is assumed that k = ℓ − k is explicitly specified. We may also write nf k ( τ ) instead of nf k ( | τ | ). Definition 2.1.
Let F be a normalized notation system and m ∈ N . We definethe F -Goodstein sequence beginning on m to be the unique sequence (G F i m ) αi =0 ,where α ∈ N ∪ {∞} , so that1. G F m = m G F i +1 m = h i + 3 i G F i m − if G F i m > α = i if G F i m = 0 ; if there is no such i , then α = ∞ . With this, we can state Goodstein’s principle within our general framework.
Theorem 2.2 (Goodstein) . For every m ∈ N , there is i ∈ N such that G E i m =0 . The proof proceeds by transfinite induction, but we will not go into detail.However, in Section 4, we will consider a weaker version of the Goodstein prin-ciple which highlights the key features of the proof but is easier to present in aself-contained manner.
For a given notation system F , there may be many ways to assign normal formsto natural numbers. The question thus arises: is there an ‘optimal’ way to definenormal forms? The following two criteria could help answer this question. Wesay that a normal form assignment nf is: • norm minimizing if whenever k ≥ τ ∈ T F k , it follows that k τ k ≥k nf k ( τ ) k ; • base-change maximal if whenever k ≥ τ ∈ T F k , it follows that |h ℓ i τ | ≤|h ℓ i nf k ( τ ) | for all ℓ ≥ k .The motivation for norm minimizing normal forms should be clear, as theseprovide the most succinct way to represent natural numbers. Base-change max-imality is perhaps a less obvious criterion, although the intuition is that weare using the fastest-growing functions available in order to represent numbers;from this perspective, one may expect that the two notions will often coincide(although not always). Moreover, as we will see, base change maximal normalforms are rather useful. For one thing, under some mild assumptions, theysatisfy a natural monotonicity property. Proposition 3.1.
Let F be a normalized notation system which includes ad-dition and a term which does not depend on k . Suppose that F is base-change maximal. Then, whenever ≤ k < ℓ and m < n , it follows that h ℓ/k i m < h ℓ/k i n . roof. Working inductively, we may assume that n = m + 1. Then, we havethat n = | nf k ( m ) + 1 | , and by base-change maximality, h ℓ/k i m < h ℓ i nf k ( m ) + 1 = h ℓ i ( nf k ( m ) + 1) ≤ h ℓ i nf k ( n ) = h ℓ/k i n. In fact, this monotonicity property is crucial for proving that Goodsteinprocesses terminate, and Proposition 3.1 tells us that we have this property forfree, given base-change maximality.
Remark 3.2.
Note that Proposition 3.1 can also be applied ‘locally’: if we knowthat F is base-change maximal whenever | τ | < N for some fixed value of N ,then from m < n < N we can deduce that h ℓ/k i m < h ℓ/k i n . This restrictedversion will be useful in inductive arguments. In the sequel we will evaluate various Goodstein-like processes according tothese criteria. We begin by considering a weak variant of Goodstein’s originalresult.
In this section we consider a Goodstein principle for which it is feasible topresent a full proof of termination within the present work. It is based on the‘multiplicative’ normalized notation system M , whose functions are 0 , , + and kx (which we may also denote k · x ). We will write M k instead of T M k . For easeof notation, we will omit parentheses around addition and treat terms ( σ + τ )+ ρ and τ + ( σ + ρ ) as identical; this will not be an issue, as all of the propertieswe consider are invariant under associativity. For q ∈ N , we define a term ¯ q bysetting ¯0 = 0 and, for q >
0, ¯ q = 1 + 1 + · · · + 1 ( q times); note that k ¯ q k = 2 q − m = k k · p + q if p, q are the unique positive integers suchthat q < k and m = k · p + q . If p = 0, set nf k ( m ) = ¯ q , and if p >
0, defineinductively nf k ( m ) = k · nf k ( p )+¯ q . Note that k m k k = k p k k +2 q +1. Throughoutthis section, all notation (e.g. nf k ( m ), k τ k , etc.) will refer exclusively to thisrepresentation of natural numbers. Lemma 4.1. If m ∈ N and ℓ > k ≥ , then nf ℓ ( h ℓ/k i m ) = h ℓ i nf k ( m ) . Proof.
This is clear since q < k yields q < ℓ .This normalized notation system satisfies both optimality properties, as wesee next.
Let us begin by showing that our multiplicative notation system satisfies thenorm minimality property.
Theorem 4.2. If τ ∈ M k , then k nf k ( τ ) k ≤ k τ k . roof. Write m = | τ | and proceed by induction on m , considering several cases. Case 1 ( τ = 0). Then, m = 0 and k m k k = k τ k = 1. Case 2 ( τ = kσ ). If | σ | = 0 then nf k ( τ ) = 0 and k k < k τ k . Otherwise, bythe induction hypothesis, k τ k ≥ k k · nf k ( σ ) k , and clearly k · nf k ( σ ) = nf k ( τ ). Case 3 ( τ = k · σ + ¯ n ). We may assume that n >
0, otherwise m = | k · σ | withsmaller norm, and we can apply the previous case.Write n = k · p + q with q < k , so that m = k k · | σ + ¯ p | + q . By the inductionhypothesis, k| σ + ¯ p |k k ≤ k σ + ¯ p k = k σ k + 2 p . Hence, k m k k ≤ k| σ + p |k k + 2 q + 1 ≤ k σ k + 2( p + q ) + 1 ≤ k σ k + 2 n + 1 = k k · σ + ¯ n k . Case 4 ( τ = ¯ m ). If m < k , then τ is already in normal form. Otherwise,we note that m = | k · m − k | (omitting m − k if m = k ) and, moreover, k k · m − k k = 3 + 2( m − k ) ≤ m −
2) = 2 m −
1. Thus we obtain k τ k > k k · m − k k , and can apply one of the previous cases. Case 5 ( τ = τ + τ , but not one of the above). By the induction hypothesis,we have that k τ k ≤ k ( nf k ( τ ) + nf k ( τ )) k , and thus we may assume that τ and τ are in normal form. Write τ = kσ + ¯ n and τ = kσ + ¯ n , whereany of the displayed terms are omitted if their value is zero. Then, | τ | = | k ( σ + σ ) + ( n + n ) | (also omitting null terms), and it is easy to check that k τ k ≥ k k ( σ + σ ) + ( n + n ) k . We can then apply the appropriate one of theprevious cases. Recall that our second optimality criterion was optimality under base change.We will show that multiplicative normal forms also enjoy this property. Thiswill follow from the next lemma.
Lemma 4.3. If m = k · r + s and ℓ ≥ k , then h ℓ/k i m ≥ ℓ · h ℓ/k i r + s, and equality holds if and only if m = k k · r + s .Proof. In this proof, we write h ℓ i x instead of h ℓ/k i x . Proceed by induction on m . If m = 0, then r = s = 0 and h ℓ/k i ℓ · h ℓ/k i m >
0. Write s = k k · p + q (with p possibly zero), so that m = k k · ( r + p ) + q .Write r = ku + v in normal form. Then, the induction hypothesis yields h ℓ i ( r + p ) = h ℓ i (cid:0) ku + ( v + p ) (cid:1) ≥ ih ℓ · h ℓ i u + v + p = h ℓ i r + p. h ℓ i m = ℓ · h ℓ i ( r + p ) + q ≥ ℓ · h ℓ i r + ℓp + q ≥ ℓ · h ℓ i r + s, and the last inequality is strict unless p = 0, so that m = k kr + s . Theorem 4.4. If τ ∈ M k , ℓ > k ≥ , and m = | τ | , then h ℓ/k i m ≥ |h ℓ i τ | . Proof.
Proof by induction on k τ k . Once again, write h ℓ i x for h ℓ/k i x . Write m = k k · p + q , so that h ℓ i m = k · h ℓ i p + q . Consider several cases. Case 1 ( τ = k · σ + ¯ q ′ ). By the induction hypothesis, |h ℓ i σ | ≤ h ℓ i| σ | . Therefore,by Lemma 4.3, h ℓ i m = ℓ · h ℓ i p + q ≥ ℓ · h ℓ i| σ | + q ′ ≥ ℓ · |h ℓ i σ | + q ′ = h ℓ i τ. Case 2 ( τ = k · σ or τ = ¯ q ′ ). In either case, we replace τ by τ ′ = k · σ + ¯ q ′ ,where either σ or q ′ may be zero; note that |h ℓ i τ ′ | = |h ℓ i τ | . Then, we may applythe previous case. Case 3 ( τ = τ + τ , but not of the above forms). Writing nf k ( τ i ) = k · σ i + ¯ q i (omitting parameters that are zero), the induction hypothesis yields |h ℓ i τ | ≤ |h ℓ i ( k · σ + ¯ q ) + h ℓ i ( k · σ + ¯ q ) | = |h ℓ i ( k ( σ + σ ) + q + q ) | . We can then apply one of the previous cases to k ( σ + σ ) + q + q . Our goal is to prove the following variant of Goodstein’s theorem.
Theorem 4.5.
For every m ∈ N , there is i ∈ N such that G M i m = 0 . We prove this using transfinite induction. Let ω be a variable. For a term τ ∈ M k , define h ω i τ ∈ N [ ω ] by replacing every occurrence of k by ω . Moreprecisely, we set h ω i h ω i h i ( σ + τ ) = h ω i σ + h ω i τ , and h ω i k · σ = ω · h ω i σ . If m ∈ N , we set h ω/k i m = h ω i nf k ( m ).Given f, g ∈ N [ ω ], we set f < g if f ( n ) < g ( n ) whenever n is large enough.The following is easy to check using basic calculus. Below, we treat polynomialsas infinite sums P ∞ i =0 a n ω n , with the understanding that a n = 0 for large enough n . Lemma 4.6. If f = P ∞ i =0 a n ω n and g = P ∞ i =0 b n ω n , then f < g if and only if a n < b n for the largest n such that a n = b n . We may use this order on N [ ω ] to perform transfinite induction, in view ofthe following. 7 roposition 4.7. Any non-empty set X ⊂ N [ ω ] has a least element.Proof. We prove by induction on n ∈ {−∞} ∪ N that if X contains a polynomial f of degree deg( f ) ≤ n , then X has a least element. For n = −∞ this is clear,since this means that 0 ∈ X and 0 is clearly the least element of N [ ω ]. So wemay assume n ≥ m < n and suppose that there is ax n + g ∈ X such that deg( g ) < n . Choose a ∗ minimal such that there is some such g with a ∗ x n + g ∈ X . By the induction hypothesis, there is a least g ∗ withdeg( g ∗ ) < n such that f ∗ := a ∗ x n + g ∗ ∈ X . Then, it is easy to check that f ∗ isthe least element of X . Remark 4.8.
Readers familiar with ordinal notation systems will recognizeelements of N [ ω ] as representing ordinals below ω ω . However, for purposes ofthe present article we wish to avoid direct reference to ordinals whenever possible,so we will treat ω as a variable and work within the semi-ring of polynomials. Inparticular, addition refers to polynomial, rather than ordinal, addition. On theother hand, the ordering we use can be checked to be identical to the standardordinal ordering. Below, we note that h ω i τ is defined according to the base of τ ; if τ ∈ M k ,then h ω i τ is obtained by replacing every occurrence of k by ω , and if τ ∈ M ℓ ,then h ω i τ is obtained by replacing every occurrence of ℓ by ω . Lemma 4.9.
Fix k ≥ .1. If m < n then h ω/k i m < h ω/k i n .2. If τ ∈ T M k and ℓ > k , then h ω ih ℓ i τ = h ω i τ. Proof.
1. The claim is clear when m = 0, so we may assume otherwise. Write m = k k · σ + q and n = k k · σ ′ + q ′ . Case 1 ( | σ | < | σ ′ | ). The induction hypothesis yields h ω i σ < h ω i σ ′ . Hence, ω · h ω i σ + q < ω · h ω i σ + ω ≤ ω · h ω i σ ′ ≤ ω · h ω i σ ′ + q ′ = h ω/k i n. Case 2 ( | σ | = | σ ′ | and q < q ′ ). Then, h ω/k i m = ω · h ω i σ + q < ω · h ω i σ ′ + q ′ = h ω/k i n.
2. By induction on k τ k . The claim is clear for τ ∈ { , } . If τ = σ + ρ , then h ω ih ℓ i τ = h ω ih ℓ i ( σ + ρ ) = h ω ih ℓ i σ + h ω ih ℓ i ρ = ih h ω i σ + h ω i ρ = h ω i ( σ + ρ ) = h ω i τ. τ = k · σ for some σ , and then h ω ih ℓ i τ = h ω ih ℓ i ( k · σ ) = h ω i ( ℓ · h ℓ i σ )= ω · h ω ih ℓ i σ = ih ω · h ω i σ = h ω i τ. Corollary 4.10. If ≤ k < ℓ , then h ω/ℓ ih ℓ/k i m = h ω/k i m. Proof.
Write τ := nf k ( m ). By Lemma 4.1, nf ℓ ( m ) = h ℓ i τ . Hence, by Lemma4.9.2, h ω/k i m = h ω i τ = h ω ih ℓ i τ = h ω/ℓ ih ℓ/k i m. Proof of Theorem 4.5.
Let ( m i ) i<α be the Goodstein sequence beginning on m (so that m i = G M i m ) and o ( i ) := h ω/i + 2 i m i . If m i >
0, then o ( i + 1) = h ω/i + 3 i m i +1 = h ω/i + 3 i ( h i + 3 i m i − < h ω/i + 3 ih i + 3 i m i = h ω/i + 2 i m i = o ( i ) . Hence o ( i ) is decreasing as long as G M i m >
0, and since there are no infinitedecreasing sequences on N [ ω ] by Proposition 4.7, it follows that G M i m = 0 forsome i . Despite the multiplicative Goodstein process always being finite, terminationcan be quite slow. In fact, the termination time grows about as fast as theAckermann function, which is well-known to grow very quickly, certainly muchfaster than any elementary function. The precise definition of the Ackermannfunction can vary in the literature, so we will work with a general presentationwhich allows for defining many different versions.
Definition 5.1.
Let f : N → N , and a, b, k ∈ N . We define A a b = A a ( f, k, b ) ∈ N recursively as follows. First, as an auxiliary value, define A a ( −
1) = 1 . Then,set:1. A b = f ( b ) ;2. A a +1 b = A ka A a +1 ( b − . It can be checked that A a ( f, k, b ) is well-defined by induction on a witha secondary induction on b . In the rest of this section, we write A a b insteadof A a ( S, , b ), where S is the successor function x x + 1. However, in latersections we will consider other variants of this function. Note that A a b is strictlyincreasing on both a and b , as can be checked by a simple induction.9e will use the Ackermann function to give a lower bound on terminationfor the multiplicative Goodstein process. It is well-known that this function in-creases very quickly, with a A a primitive recursive function [10].To this end, to each natural number m and each k ≥
2, we assign a function b A km as follows. Let b A k be the identity. For m >
0, write m = a + k r , where r is maximal so that m is a multiple of k r ; note that a, r are unique. Assumeinductively that we have defined b A ka . Then, set b A km = b A ka ◦ A r . Thus, if m = a r k r + · · · + a k in base k , we obtain b A km = A a r r ◦ · · · ◦ A a . Note that for ℓ > k ,we have that h ℓ/k i m = a r ℓ r + · · · + a ℓ in base ℓ , and so b A km = b A ℓ h ℓ/k i m . Theorem 5.2.
Let m ∈ N and let i ∗ be least with G M i ∗ m = 0 . Then, if ≤ k ≤ i ∗ , we have that b A kG M k m ( k ) ≤ i ∗ .Proof. For x ∈ N and k ≥
2, let us write G k m instead of G M k m and F k insteadof b A kG M k m . Let ℓ = k + 1. Proceed by induction on i ∗ − k and consider severalcases. Case 1 ( G k m = 0). In this case, F k is the identity, hence F k ( k ) = k = i ∗ . Case 2 ( G k m > m = a + k r with r maximal so that m is a multipleof k r . We then have that F k = b A ka A r , and the induction hypothesis yields F ℓ ( ℓ ) ≤ i ∗ . Consider two sub-cases. Case 2 .1 ( r = 0). Then, h ℓ i m − h ℓ i a , so that F k ( k ) = b A ka A k = b A ℓ h ℓ i a ( k + 1) = F ℓ ( ℓ ) ≤ i ∗ . Case 2 .2 ( r = 1). Then, h ℓ i m − h ℓ i a + kℓ , so that F ℓ = b A ℓ h ℓ i a A k . Then, F k ( k ) = b A ka A k = b A ka A k A b A ℓ h ℓ i a A k < b A ℓ h ℓ i a A k ( k + 1) = F ℓ ( ℓ ) ≤ i ∗ . Case 2 .3 ( r > h ℓ i m − h ℓ i a + kℓ r − + · · · + kℓ , so that F ℓ = b A ℓ h ℓ i a ◦ A kr − ◦ · · · ◦ A k . Hence, F k ( k ) = b A ka A r k = b A ka A kr − A r b A ka A kr − A r − b A ka A kr − A r − A r − b A ℓ h ℓ i a A kr − A r − A r − < b A ℓ h ℓ i a A kr − A kr − · · · A k ( k + 1) = F ℓ ( ℓ ) ≤ i ∗ . Corollary 5.3.
The multiplicative Goodstein process starting on a terminatesin time at least A a , hence the termination time is not primitive recursive on a . Optimality of Exponential Normal Forms
Now we turn our attention to the original Goodstein process. In this setting, it isalready known that the process terminates [6], and that this fact is independentof Peano arithmetic [8]. We will not provide a proof here, and instead take theseresults as given. Our focus will be on whether the notation system used satisfiesour optimality criteria, and what consequences we can extract from this. Webegin by establishing some useful basic properties.
Recall that E has as primitive functions 0, x + y , and k x , and is equipped withnormal forms as defined in Section 2. We will treat terms modulo associativity ofaddition and hence omit parentheses. However, we will not treat term additionas commutative. With this in mind, it is easy to check that k ρ + . . . k ρ n − isin normal form if and only if each ρ i is in normal form, ρ i ≥ ρ i +1 whenever i + 1 < n , and ρ i > ρ i + k whenever i + k < n . We will extend the notation = k towrite m = k τ ( k, a , . . . , a n ), where a i ∈ N , if m = k τ ( k, nf k ( a ) , . . . , nf k ( a n ));for example, we may write 15 = + 2 + 3 or 12 = but not, say,15 = . Sums should be read from right to left, i.e. P ni =0 τ i = τ n + . . . + τ .Multiplication is used as a shorthand: p · τ = τ + . . . + τ ( p times).With this notation at hand, the following is easily checked. Lemma 6.1.
Fix k ≥ , m ∈ N and σ, τ ∈ E k .1. If σ + τ is in normal form, then σ and τ are each in normal form.2. If m = k k a and b < a , then m − k b = k P a − i = b ( k − k i .3. If m = a + k b and n = k c + d are in normal form with b > c , then m + n is in normal form. In this subsection, we will show that the hereditary exponential notation satisfiesnorm minimality. We begin with some useful inequalities.
Lemma 6.2. If k ≥ and m ∈ N , then k m + 1 k k ≤ k m k k + 3 .Proof. Write m + 1 = k a + k b . If b = 0, then m = a and k m + k k k = k m k k + 3(one for the term 0, one for + and one for k · ). Otherwise, using Lemma 6.1, wesee that m = k a + P b − i =0 ( k − k i , and k m k k + 3 ≥ k a + k b − k k + 3 = k a k k + k b − k k + 5 ≥ ih k a k + k b k + 2 = k m + 1 k k . Lemma 6.3. If m = k a + b , then k m k k ≤ k a k k + k b k k + 2 .Proof. Induction on m . Write m = k k p + q . Consider the following cases.11 ase 1 ( a = p ). Then also q = b , so that k a k k + k b k k + 2 = k p k k + k q k k + 2 = k m k k . Case 2 ( b ≥ k p ). Then, b = k k p + c for c = q − k a , and the induction hypothesisyields k a k k + k b k k + 2 = k a k k + k p k k + k c k k + 4 ≥ ih k p k k + k k a + c k k + 2 = k p k k + k q k k + 2 = k m k k . Case 3 ( a < p and b < k p ). From k p > b = k p + q − k a we obtain q < k a . Thusby Lemma 6.1, b = ( k p − k a ) + q = k p − X i = a ( k − k i + q. Hence, k b k k ≥ k k p − + q k k = k p − k k + k q k k + 2 ≥ k p k k + k q k k − , where the last inequality is by Lemma 6.2, so that k a k k + k b k k + 2 ≥ k a k k + k p k k + k q k k + 1 ≥ k m k k . From this, it readily follows that E is norm minimal. Theorem 6.4. If τ ∈ E k and m = | τ | , then k m k k ≤ k τ k .Proof. Simple induction on term complexity using Lemma 6.3.
We have seen that hereditary exponential notation satisfies norm minimality.Let us now show that it is base-change maximal as well. This will follow fromthe next lemma. If F is a normalized notation system, say that F is base-change maximal below m ∈ N if, whenever τ ∈ T F k and | τ | < m , it follows that |h ℓ i τ | ≤ |h ℓ i nf k ( τ ) | . Recall from Remark 3.2 that, if F is base-change maximalbelow m , then whenever x < y < m , we may conclude that h ℓ/k i x < h ℓ/k i y .As we wish to appeal to this property in the proof of the following lemma,we will assume inductively that hereditary exponential notation is base-changemaximal below m . Lemma 6.5.
Fix ℓ > k ≥ and suppose that the normalized notation system E is base-change maximal below m . If m = k a + b , then h ℓ/k i m ≥ ℓ h ℓ/k i a + h ℓ/k i b. Proof.
Induction on m . To simplify notation, we write h ℓ i x instead of h ℓ/k i x throughout this proof. Write m = k k p + q and consider the following cases.12 ase 1 ( a = p ). Then also b = q and h ℓ i m = ℓ h ℓ i a + h ℓ i b. Case 2 ( a < p and b ≥ k p ). Then, b = k k p + c for some c , and the inductionhypothesis yields ℓ h ℓ i a + h ℓ i b ≤ ℓ h ℓ i a + ℓ h ℓ i p + h ℓ i c = ℓ h ℓ i p + ℓ h ℓ i a + h ℓ i c ≤ ℓ h ℓ i p + h ℓ i ( k a + c ) . Then we can apply
Case 1 . Case 3 ( a < p and b < k p ). As in the proof of Lemma 6.3, q < k a , and thus b = ( k p − k a ) + q = k p − X i = a ( k − k i + q, hence h ℓ i b = ( k − p − X i = a ℓ h ℓ i i + h ℓ i q = ( k − p − a X i =1 ℓ h ℓ i ( p − i ) + h ℓ i q. Since p < m , we may use the assumption that E is base-change maximal below m to obtain h ℓ i ( p − i ) ≤ h ℓ i p − i , and hence p − a X i =1 ℓ h ℓ i ( p − i ) ≤ p − a X i =1 ℓ h ℓ i p − i = ℓ h ℓ i p ( ℓ p − a − − ℓ p − a − ( ℓ − < ℓ h ℓ i p k . By monotonicity below m , we have that ℓ h ℓ i a < ℓ h ℓ i p . Therefore, ℓ h ℓ i a + h ℓ i b < ℓ h ℓ i p + ( k − ℓ h ℓ i p k + h ℓ i q = h ℓ i m. Theorem 6.6. If τ ∈ E k and m = | τ | , then h k + 1 i m ≥ |h k + 1 i τ | .Proof. Induction on term complexity using Lemma 6.5.In view of Proposition 3.1, we immediately obtain monotonicity of the base-change operation.
Corollary 6.7. If m < n and ≤ k < ℓ , then h ℓ/k i m < h ℓ/k i n . In this section we consider an extension of E with product and study whetherhereditary exponential normal forms are still optimal in this context. Define L = { , x + y, x · y, k x } . Then for example(5 + 5 + 5 ) · (5 + 5 ) = 5 + 2 · + 2 · + 5 = 5 + 5 + 5 + 5 + 5 + 5 , k . However, as we will see, we still obtain maximality underbase change. Theorem 7.1.
Let ℓ ≥ k ≥ , m ∈ N , and τ ∈ L k . Then, |h ℓ i τ | ≤ |h ℓ i nf k ( τ ) | .Proof. By induction on | τ | with a secondary induction on k τ k . Case 1 ( τ = 0). Trivial. Case 2 ( τ = σ + ρ ). Let nf k ( σ ) = σ ′ and nf k ( ρ ) = ρ ′ . Then, σ ′ + ρ ′ ∈ E k , | σ ′ | , | ρ ′ | ≤ | τ | , and k σ ′ k , k ρ ′ k < k τ k . Thus we may use the induction hypothesisand Theorem 6.6 to see that h ℓ i τ ≤ h ℓ i ( σ ′ + ρ ′ ) ≤ h ℓ i nf k ( σ ′ + ρ ′ ) = h ℓ i nf k ( τ ) . Case 3 ( τ = k σ ). Then, h ℓ i τ = ℓ h ℓ i σ ≤ ih ℓ h ℓ i nf k ( σ ) = h ℓ i nf k ( τ ) . Case 4 ( τ = σ · ρ ). If | σ | = 0, then from k σ k < k τ k and the secondary inductionhypothesis we have that h ℓ i τ = h ℓ i σ · h ℓ i ρ ≤ h ℓ/k i · h ℓ i ρ = 0 = h ℓ i nf k ( τ ) , so we may assume that | σ | > | ρ | >
0. Write nf k ( σ ) = k α + β and nf k ( ρ ) = k γ + δ . Define τ ′ = k α · δ + k γ · β + β · δ . Then, h ℓ i τ = h ℓ i σ · h ℓ i ρ ≤ ih h ℓ i ( k α + β ) · h ℓ i ( k γ + δ )= ℓ h ℓ i α + h ℓ i β + h ℓ i τ ′ ≤ ih ℓ h ℓ i α + h ℓ i β + h ℓ i nf ( τ ′ ) = h ℓ i ( k α + β + nf k ( τ ′ )) . Note that k α + β + nf ( τ ′ ) ∈ E k , and moreover | τ | = | k α + β + τ ′ | = | k α + β + nf k ( τ ′ ) | . Thus by Theorem 6.6, |h ℓ i ( k α + β + nf k ( τ ′ )) | ≤ |h ℓ i nf k ( k α + β + nf k ( τ ′ )) | = |h ℓ i nf k ( τ ) | . We conclude that |h ℓ i τ | ≤ |h ℓ i nf k ( τ ) | , as required.Theorem 7.1 might seem surprising, as hereditary exponential normal formsdo not involve multiplication, yet they remain base-change maximal even com-pared to arbitrary elementary terms. Later, we will see that this result leads toa wide generalization of Goodstein’s principle. But first, we consider notationsystems based on a different set of functions.14 Ackermannian Goodstein Sequences
Exponential notation falls short when attempting to represent large numbersthat sometimes arise in combinatorics, such as Graham’s number [7]. Thesenumbers may instead be written in terms of fast-growing functions such as theAckermann function, as given by Definition 5.1. Throughout this section, wewrite A a ( k, b ) for A a ( x k x , k, b ), so that the Ackermann functions used herewill be based on the exponential, rather than the successor, function. Theparameter k is regarded as the base of our notation system. There will typicallybe several ways to write a number in the form A a ( k, b ) + c , so a suitable normalform is chosen in Definition 8.1. Our normal forms are based on iterativelyapproximating m via a ‘sandwiching’ procedure. With this, we can define the Ackermannian Goodstein process.
We define normal forms based on the Ackermann function. Fix a ‘base’ k . When k is clear from context, we will write A a b instead of A a ( k, b ). The general ideais to represent m > A a b + c . It is tempting to choose a maximal so that there is b with A a b ≤ m < A a ( b + 1). However, c may stillbe quite large; so large, in fact, that there is a ′ < a with A a ′ A a b ≤ m . In thiscase, A a ′ A a b is a better approximation to m than A a b and, moreover, we maychoose b ′ maximal so that A a ′ b ′ ≤ m . We then have that A a b < A a ′ b ′ ≤ m < A a ′ ( b ′ + 1) ≤ A a ( b + 1) , ‘sandwiching’ m between better and better approximations. Continuing in thisfashion, we can find the ‘best’ approximation to m ; this will be the basis for ournormal forms. Definition 8.1.
Fix k ≥ and let A x y = A x ( k, y ) . Given m, a, b, c ∈ N with m > , we define A a b + c to be the k -normal form of m , in symbols m = k A a b + c ,if m = A a b + c and there exist sequences a , . . . , a n of sandwiching indices, b , . . . , b n of sandwiching arguments, and m , . . . m n of sandwiching values suchthat for i < n ,1. m = 0 ;2. A a i +1 m i ≤ m < A a i +1 +1 m i ;3. A a i +1 b i +1 ≤ m < A a i +1 ( b i +1 + 1) ;4. m i +1 = A a i +1 b i +1 ;5. A m n > m , and6. a = a n and b = b n . e denote the sequence of pairs ( a i , b i ) by ( A a i b i ) ni =1 and call it the k -sandwichingsequence of m .We set nf k (0) = 0 and inductively define nf k ( m ) = A nf k ( a ) nf k ( b ) + nf k ( c ) . We write simply sandwiching sequence when k is clear from context. Everypositive integer has a unique k -sandwiching sequence and hence a unique normalform. The intuition is that we obtain the normal form of m by ‘sandwiching’ itin smaller and smaller intervals, so that (cid:2) A a ( b ) , A a ( b + 1) (cid:1) ) . . . ) (cid:2) A a n ( b n ) , A a n ( b n + 1) (cid:1) ∋ m. Example 8.2.
Let us write A a b for A a (2 , b ) , and let us compute the Acker-mannian -normal form of . We note that A A , while A A > A A A A > , so that a = 1 by item 2 and (from A > ) b = 0 by item 3. It follows that m = A . We then seethat A m = 2 < , while A A A > > , so that a = 0 , while A > , yielding b = 4 and m = A . Since A
16 = 2 > ,the sequence terminates, and thus the sandwiching sequence for is ( A , A .We thus have that
20 = k A . A similar analysis shows that k A and k A , so nf (20) = A A A A A . These normal forms can then be used to define a Goodstein process usingDefinition 2.1. These were first introduced in [11], and the Goodstein processwas shown to be terminiating in [1].
Theorem 8.3.
Given any m ∈ N , there is i ∈ N such that G A i m = 0 . Nevertheless, the termination is much slower than that of the standard Good-stein process, let alone the Ackermannian bounds we have provided for themultiplicative version. The following example shows that the elements of thesequence can grow rather quickly.
Example 8.4.
Let us write A a b for A a (2 , b ) and B a b for A a (3 , b ) . Recall fromExample 8.2 that nf (20) = A A A A A . Let us compute the next ele-ment of the Goodstein sequence starting on . We have that h / i A A B B B B B , hence G A
20 = B B B B B − > B B . But B B , hence G A > B . We will need to review some basic properties of the Ackermannian normal forms.These properties will be needed later to prove that Ackermannian normal formsare base-change maximal. Here, we present them without proof and refer thereader to [1] for details. If m has sandwiching sequence ( A a i b i ) ni =1 , we defineˇ m = A a n − b n − . Lemma 8.5. If m = k A a b with a > , then A a − ˇ m is in normal form. m = k A a b + c , we may still have that c ≥ A a b . For suchcases we define the extended k -normal form of m to be A a b · p + q , in symbols m ≃ k A a b · p + q , if m = k A a b + c for some c , m = A a b · p + q , and 0 ≤ q < A a b ;note that p and q are uniquely defined. If m ≃ k A a b · p + q and d = A a b wewrite m ≈ k d · p + q and call it the simplified k -normal form of m . In fact, thebase change operation commutes with the parameter p . Lemma 8.6. If m ≃ k A a ( k, b ) · p + q and < k ≤ ℓ ≤ ω , then h ℓ/k i m = A h ℓ/k i a ( ℓ, h ℓ/k i b ) · p + h ℓ/k i q. Given an expression A a b in normal form, expanding the function A a on the right gives rise to expressions of the form A ska − A a ( b − s ). It is important torecognize when such expressions are in normal form; here, the following lemma,also found in [1], is useful. Lemma 8.7.
Let m = A a b with a, b > , and let s ∈ [1 , b + 1] . Then:1. A a b = A ska − A a ( b − s ) .
2. Let ℓ ∈ [1 , k ] and c = A ℓa − A a ( b − s ) . If m = k A a b , b = ˇ m , and A a − b ≤ c < A a b, then it follows that c is in normal form as written. To clarify, the claim in item 2 is merely that c = k A a − d , where d = A ℓ − a − A a ( b − s ). It is not necessarily the case that d or its sub-expressions arein normal form. A similar remark applies to subsequent lemmas. We may alsoexpand the function A a on the left, giving rise to the following sequences. Definition 8.8.
Let A a b be in normal form with a > and define a sequence c , . . . , c a by recursion as follows:1. c = A a ( b − ;2. c i = A a − i ( A k − a − i c i − − if i > .We call the sequence ( c i ) i ≤ a the left expansion sequence for A a b . Lemma 8.9.
Let m = k A a b with a > .1. If < ℓ < k and < i ≤ a then A ℓa − i c i − is in normal form.2. If i ≤ a and either i > or b > ˇ m , then c i has normal form as written inDefinition 8.8. We remark that some care must be taken when computing the normal formof A a ( b −
1) assuming that m = k A a b , as we must apply Lemma 8.7 when b = ˇ m ,and Lemma 8.9 when b > ˇ m . With this, we can describe the normal form of m − m = k A a b . The operations A x y and h z/y i x are always assumed to be performed before multiplication. emma 8.10. If m = k A a b with a > and left expansion sequence ( c i ) i ≤ a ,then m − ≈ k ( k − · c a + ( c a − . Ackermannian normal forms do not produce minimal norms. Let m = A A −
1. Then m ≃ A · p + q with a large p of about k A − log k A . However, m = P A − i =0 A i , which has norm of about ( A .There are other notions of normal form that we may consider aside fromthose of Definition 8.1. For example, we may skip the sandwiching procedureand choose a , b so that A a b is maximal with the property that A a b ≤ m ,then choose a maximal such that there exists b ≥ A a b = A a b .However, the alternative normal forms are also not norm minimizing. Let n = A ( A k − A ( k −
1) + 1). Then n is in alternative normal form with normabout 4 k , since k − A · ( k − n = A k A A k .Currently, we do not know if there is a primitive recursive procedure which,given a term τ , yields the norm-minimal τ ∗ with | τ ∗ | = | τ | . Despite the failure of norm minimality, our Ackermannian notation does havethe base-change maximality property. As we have done previously, we willassume that base-change maximality holds below m in order to use monotonicitywhen needed, as per Remark 3.2. We begin with a preparatory lemma.
Lemma 8.11.
Assume that the normalized notation system A is base-changemaximal below m . Let ℓ > k ≥ and write A x y for A x ( k, y ) and B x y for A x ( ℓ, y ) . Suppose that m = k A a b with a > . Then, h ℓ/k i A a ( b − ≤ B h ℓ/k i a ( h ℓ/k i b − . Proof.
Let c = A a ( b −
1) and d = B h ℓ/k i a ( h ℓ/k i b − h ℓ i x for h ℓ/k i x .We remark that c may or may not be in normal form as written, so we mustdivide the proof into several cases. Case 1 ( b = 0). We have that h ℓ/k i c = h ℓ/k i A a ( −
1) = h ℓ/k i ≤ d . Case 2 ( b > ˇ m ). In this case, c = k A a ( b − h ℓ i c = B h ℓ i a h ℓ i ( b − ≤ d, where we use Proposition 3.1 and Remark 3.2 to see that h ℓ i ( b − ≤ h ℓ i b − In fact, monotonicity is alreay proven in [1]. However, we will not use it here, so that ourwork may serve as an alternative proof. ase 3 ( b = ˇ m > s ∈ [1 , b ], A a ( b −
1) = A k ( s − a − A a ( b − s ). Since A a ( −
1) = 1 ≤ b ,we have that there is a least t ∈ [2 , b +1] such that A a ( b − t ) ≤ b . Similarly, thereis a greatest r < k such that u := A ra − A a ( b − t ) ≤ b . Note that A a − u > b ,and hence A a − u > A a − b = A a − ˇ m . It follows by Lemma 8.7.2 that A va − u isin normal form whenever 1 < v ≤ k ( t −
2) + k − r. Similarly, by Lemma 8.5, A a − b is in normal form, since by assumption, b = ˇ m . Then, h ℓ i c = h ℓ i A k ( t − k − r − a − A a − u = B k ( t − k − r − h ℓ i ( a − h ℓ i A a − u ≤ B k ( t − k − h ℓ i ( a − h ℓ i A a − b = B k ( t − k − h ℓ i ( a − B h ℓ i ( a − h ℓ i b = B k ( t − h ℓ i ( a − h ℓ i b< B k ( t − h ℓ i ( a − B t − h ℓ i ( a − B h ℓ i a ( h ℓ i b − t ) ≤ B ℓ ( t − h ℓ i a − B h ℓ i a ( h ℓ i b − t )= B h ℓ i a ( h ℓ i b −
1) = d, where the first and third inequalities use base-change maximality below m andRemark 3.2. Lemma 8.12.
Suppose that A is base-change maximal below m . Fix ℓ > k ≥ and let ℓ = k + 1 . Then, if A d e ≤ m = k A a b + c , it follows that A h ℓ/k i d h ℓ/k i e ≤ A h ℓ/k i a h ℓ/k i b. Proof.
Let ( A a i b i ) ni =1 be the k -sandwiching sequence for m and let j be maximalso that a j ≥ d ; such a j exists since A a +1 > m ≥ A d e ≥ A d
0, so a ≥ d .If d = a j , the claim is immediate using monotonicity (Remark 3.2), so assumethat d < a j .First note that e < m j . If j < n , this follows from A a j +1 +1 e ≤ A d e ≤ m < A a j +1 +1 m j , and if j = n , this follows from A e ≤ A d e ≤ m < A m j . Let ( c i ) i ≤ a j be the left expansion sequence for m j and write h ℓ i x for h ℓ/k i x .Then, Lemma 8.10 and monotonicity below m yield h ℓ i e ≤ h ℓ i ( m j −
1) = ( k − h ℓ i c a j + h ℓ i ( c a j − < k h ℓ i c a j . (1)We claim that ℓ h ℓ i c a j ≤ B k h ℓ i a j − B h ℓ i a j ( h ℓ i b j − . (2)The lemma would then follow, as B h ℓ i d h ℓ i e ≤ B h ℓ i d k h ℓ i c a j < B ℓ h ℓ i a j − B h ℓ i a j ( h ℓ i b j −
1) = B h ℓ i a h ℓ i b, where the first inequality follows from (1) and the second from h ℓ i d ≤ h ℓ i a j − m ) and (2). 19t remains to prove (2). In order to show this, we first claim that ℓ h ℓ i c a j ≤ B k h ℓ i c a j − (3)and that, for 0 ≤ u < i < a j , B k h ℓ i a j − i − h ℓ i c i ≤ B k h ℓ i a j − u − h ℓ i c u . (4)From this we obtain (2), since ℓ h ℓ i c a j ≤ B k h ℓ i c a j − ≤ B k h ℓ i a j − ( a j − − h ℓ i c a j − ≤ B k h ℓ i a j − h ℓ i c ≤ B k h ℓ i a j − B h ℓ i a j ( h ℓ i b j − , where the first inequality follows from (3), the third from (4), and the lastinequality follows from h ℓ i c ≤ B h ℓ i a j ( h ℓ i b j −
1) by Lemma 8.11.To see that (3) holds, ℓ h ℓ i c a j = ℓ h ℓ i A ( A k − c a j − − ≤ ℓB ( h ℓ i A k − c a j − − ℓB ( B k − h ℓ i c a j − −
1) = B k h ℓ i c a j − , where in the inequality we use monotonicity below m to obatin h ℓ i ( A k − c a j − − ≤ h ℓ i A k − c a j − − , and the last equality comes from ℓB x = ℓ · ℓ x = ℓ x +1 = B ( x + 1).Finally, we prove (4). We may assume that u = i −
1, since the general claimfollows by induction. But then, B k h ℓ i a j − i − h ℓ i c i = B k h ℓ i a j − i − h ℓ i A a j − i ( A k − a j − i c i − − ≤ B ℓ h ℓ i a j − i − B h ℓ i a j − i ( B k − h ℓ i a j − i h ℓ i c i − −
1) = B k h ℓ i a j − i h ℓ i c i − , where the inequality again uses monotonicity below m . This yields the desiredinequalities, and concludes the proof. Proposition 8.13.
Assume that A is base-change maximal below m and let ℓ > k ≥ . If A d e + s ≤ m , then A h ℓ i d h ℓ i e + h ℓ i s ≤ h ℓ i m .Proof. By induction on m . Write m = k A a b + c , h ℓ i x = h ℓ/k i x , and considerthree cases. Case 1 ( s = 0). This case is an immediate consequence of Lemma 8.12. Case 2 ( s > Case 2 .1 ( A d e ≥ A a b ). Then, s ≤ c , so that h ℓ i s ≤ h ℓ i c . Moreover, Lemma8.12 yields B h ℓ i d h ℓ i e ≤ B h ℓ i a h ℓ i b . Thus, B h ℓ i b h ℓ i e + h ℓ i s ≤ B h ℓ i a h ℓ i b + h ℓ i c. ase 2 .2 ( A d e < A a b ). Let ( c i ) i ≤ a be the left expansion sequence for m . Since A d e is in the range of A and A a b = A k c a − , we obtain A d e ≤ A ( A k − c a − −
1) = c a . Thus the induction hypothesis yields B h ℓ i d h ℓ i e ≤ h ℓ i c a . Similarly, s ≤ m −
1, and monotonicity below m yields h ℓ i s ≤ h ℓ i ( m −
1) = ( k − h ℓ i c a + h ℓ i ( c a − < k h ℓ i c a . But then, B h ℓ i b h ℓ i e + h ℓ i s < ℓ h ℓ i c a ≤ B k h ℓ i a − B h ℓ i a ( h ℓ i b − < B h ℓ i a h ℓ i b ≤ h ℓ i m, where the second inequality is an instance of (2).From this, we obtain base-change maximality by an easy induction on termcomplexity. Theorem 8.14. If τ ∈ E k and ℓ > k ≥ , then h ℓ i τ ≤ h ℓ i nf k ( τ ) . In view of Proposition 3.1, we immediately obtain an alternative proof ofmonotonicity of the base-change operator, also proven directly in [1].
Corollary 8.15. If n < m and ℓ > k ≥ , then h ℓ/k i n < h ℓ/k i m . In this section we introduce and study Goodstein walks. These are Goodstein-like properties which are defined independently of a normal form representation;natural numbers may be written in an arbitrary way using the functions from F . Aside from this, the definition is analogous to that of standard Goodsteinprocesses. Definition 9.1.
Fix a notation system F . A Goodstein walk (for F ) is asequence ( m i ) αi =0 , where α ≤ ∞ , such that for every i < α , there is a term τ i with | τ i | = m i and m i +1 = h i + 1 i τ − . Theorem 9.2.
Let F be a normalized notation system with + and . Supposethat F is base-change maximal, and that for every m ∈ N there is i ∈ N suchthat G F i m = 0 . Then, every Goodstein walk for F is finite.Proof. Let F satisfy the assumptions of the theorem and ( m i ) αi =0 be a Goodsteinwalk for F . Let m = m . By induction on i , we check that m i ≤ G F i m . For thebase case this is clear. Otherwise, m i +1 = |h i + 3 i τ i |− τ i ∈ T F i +2 ,and thus m i +1 = |h i + 3 i τ i | − ≤ h i + 3 i m i − ih ≤ h i + 3 i G F i m − F i +1 m, i such thatG F i m = 0, we must have α ≤ i .As a corollary, we obtain the following extension of Goodstein’s theorem, aswell as its Ackermannian variant from [1]. Theorem 9.3.
Any Goodstein walk for L or A is finite. Example 9.4.
Consider alternative normal forms based on L as follows. Let k ≥ and m ≥ . First, set nf k (0) = 0 . For m > , let p · · · p n be thedecomposition of m into prime factors. If n ≤ , we write m = k r + b with m Consider alternative Ackermannian normal forms obtained bywriting m = A a b + c , where a is maximal so that A a ≤ m and b is maximal sothat A a b ≤ m . Then, set nf k ( m ) = A nf k ( a ) nf k ( b ) + nf k ( c ) . Such normal formsgive alternative Goodstein sequences which we have shown to be terminating [2].However, this result would be an immediate corollary of Theorem 9.3.On the other hand, the lower bounds given in [2] do not immediately followfrom Theorem 9.3, and such bounds must still be computed on a case-by-casebasis. 10 Concluding remarks We have explored the notion of optimality in Goodstein-style notation systems,arriving at the notions of norm minimality and base-change maximality. Thebenefits of the former are evident, as for practical purposes one wishes to usenotations that are compact. Meanwhile, we have shown the second to providenew insights into the Goodstein principle. Most notably, the termination of abase-change maximal Goodstein process implies the termination of any Good-stein walk, in particular yielding Theorem 9.3, a far-reaching extension both ofGoodstein’s original theorem and our recent results regarding the AckermannianGoodstein process.One intersting and challenging question that is left open is whether a nota-tion system which allows for expressions τ σ with arbitrary terms τ, σ will enjoysimilar termination properties. Similarly, one can include multiplication in our22ckermannian notation and ask the same question. It is currently unclear if thetechniques presented here are sufficient to settle these challenging problems. References [1] T. Arai, D. Fern´andez-Duque, S. Wainer, and A. Weiermann. Predicativelyunprovable termination of the Ackermannian Goodstein principle. Proceed-ings of the American Mathematical Society , 2019. Accepted for publication.[2] D. Fern´andez-Duque and A. Weiermann. Ackermannian goodstein se-quences of intermediate growth. In Computability in Europe , 2020.[3] H. Friedman and F. Pelupessy. Independence of Ramsey theorem variantsusing ε . Proceedings of the American Mathematical Society , 144(2):853–860, 2016. English summary.[4] K. G¨odel. ¨Uber Formal Unentscheidbare S¨atze der Principia Mathemat-ica und Verwandter Systeme, I. Monatshefte f¨ur Mathematik und Physik ,38:173–198, 1931.[5] R.L. Goodstein. On the restricted ordinal theorem. Journal of SymbolicLogic , 9(2):3341, 1944.[6] R.L. Goodstein. Transfinite ordinals in recursive number theory. Journalof Symbolic Logic , 12(4):123–129, 12 1947.[7] R.L. Graham and B.L. Rothschild. Ramsey’s theorem for n -dimensionalarrays. Bulletin of the American Mathematical Society , 75(2):418–422, 031969.[8] L. Kirby and J. Paris. Accessible independence results for Peano arithmetic. Bulletin of the London Mathematical Society , 14(4):285–293, 1982.[9] J. Paris and L. Harrington. A mathematical incompletenss in Peano arith-metic. In J. Barwise, editor, Handbook of Mathematical Logic , pages 1133–1142. North-Holland Publishing Company, 1977.[10] P. R´ozsa. Recursive functions . Academic Press, New York-London; Akad-miai Kiado, Budapest, third revised edition edition, 1967. translated fromthe German by Istv´an F¨oldes.[11] A. Weiermann. Ackermannian Goodstein principles for first order Peanoarithmetic. In Sets and computations , volume 33 of Lecture Notes Se-ries, Institute for Mathematical Sciences, National University of Singapore ,pages 157–181, Hackensack, NJ, 2018. World Scientific. English summary.[12] A. Weiermann and W. Van Hoof. Sharp phase transition thresholds for theParis Harrington Ramsey numbers for a fixed dimension.