Deducibility and Independence in Beklemishev's Autonomous Provability Calculus
aa r X i v : . [ m a t h . L O ] A ug Deducibility and Independence in Beklemishev’sAutonomous Provability Calculus
David Fern´andez-Duque ∗ Eduardo Hermo Reyes † September 1, 2020
Abstract
Beklemishev introduced an ordinal notation system for the Feferman-Sch¨utte ordinal Γ based on the autonomous expansion of provability al-gebras. In this paper we present the logic BC (for Bracket Calculus ). Thelanguage of BC extends said ordinal notation system to a strictly positivemodal language. Thus, unlike other provability logics, BC is based on aself-contained signature that gives rise to an ordinal notation system in-stead of modalities indexed by some ordinal given a priori. The presentedlogic is proven to be equivalent to RC Γ , that is, to the strictly positivefragment of GLP Γ . We then define a combinatorial statement based on BC and show it to be independent of the theory ATR of Arithmeti-cal Transfinite Recursion, a theory of second order arithmetic far morepowerful than Peano Arithmetic. In view of G¨odel’s second incompleteness theorem, we know that the consistencyof any sufficiently powerful formal theory cannot be established using purely‘finitary’ means. Since then, the field of proof theory, and more specifically ofordinal analysis, has been successful in measuring the non-finitary assumptionsrequired to prove consistency assertions via computable ordinals. Among thebenefits of this work is the ability to linearly order natural theories of arith-metic with respect to notions such as their ‘consistency strength’ (e.g., theirΠ ordinal) or their ‘computational strength’ (their Π ordinal). Nevertheless,the assignment of these proof-theoretic ordinals to formal theories depends ona choice of a ‘natural’ presentation for such ordinals, with well-known patho-logical examples having been presented by Kreisel [26] and Beklemishev [7]. ∗ [email protected] † [email protected] The Π ordinal of a theory is another measure of its strength and does not have suchsensitivity to a choice of notation system. However, there are some advantages to consideringΠ ordinals, among others that they give a finer-grained classification of theories. PA ) and related theories via their provability al-gebras. Consider the Lindenbaum algebra of the language of arithmetic moduloprovability in a finitary theory U such as primitive recursive arithmetic ( PRA )or the weaker elementary arithmetic ( EA ). For each natural number n and eachformula ϕ , the n -consistency of ϕ is the statement that all Σ n consequences of U + ϕ are true, formalizable by some arithmetical formula h n i ϕ (where ϕ isidentified with its G¨odel number). In particular, h i ϕ states that ϕ is consis-tent with U . An iterated consistency assertion, also called worm, is then anexpression of the form h n i . . . h n k i⊤ , where ⊤ is some fixed tautology.The operators h n i and their duals [ n ] satisfy Japaridze’s provability logic GLP [24], a multi-modal extension of the G¨odel-L¨ob provability logic GL [11].As Beklemishev showed, the set of worms is well-ordered by their consistencystrength < , where A < B if A → h i B is derivable in GLP . Moreover, thiswell-order is of order-type ε , which characterizes the proof-theoretical strengthof PA . This tells us that proof-theoretic ordinals already appear naturallywithin Lindenbaum algebras of arithmetical theories. Using these ideas, Bek-lemishev has shown how the logic GLP gives rise to the
Worm principle, arelatively simple combinatorial principle which is independent of PA [1].Beklemishev also observed that this process can be extended by consideringworms with ordinal entries. Extensions of GLP , denoted
GLP Λ , have beenconsidered in cases where Λ is an ordinal [2, 15, 19] or even an arbitrary lin-ear order [5]. Proof-theoretic interpretations for GLP Λ have been developedby Fern´andez-Duque and Joosten [18] for the case where Λ is a computablewell-order. Nevertheless, we now find ourselves in a situation where an expres-sion h λ i ϕ requires a system of notation for the ordinal λ . Fortunately we may‘borrow’ this notation from finitary worms and represent λ itself as a worm.Iterating this process we obtain the autonomous worms, whose order types areexactly the ordinals below the Feferman-Sch¨utte ordinal Γ . By iterating thisprocess we obtain a notation system for worms which uses only parentheses,as ordinals (including natural numbers) can be iteratively represented in thisfashion. Thus the worm h i⊤ becomes () , h i⊤ becomes (()) , h ω i⊤ becomes ((())) , etc.These are Beklemishev’s brackets, which provide a notation system for Γ without any reference to an externally given ordinal [2]. However, it has thedrawback that the actual computation of the ordering between different wormsis achieved via a translation into a traditional ordinal notation system. Wewill remove the need for such an intermediate step by providing an autonomouscalculus for determining the ordering relation (and, more generally, the logicalconsequence relation) between bracket notations. To this end we present the Bracket Calculus ( BC ). We show that our calculus is sound and complete withrespect to the intended embedding into GLP Γ . We then show that the Wormprinciple can be naturally extended to BC to yield independence for theories ofstrength Γ , particularly the theory Arithmetical Transfinite Recursion ATR ,2ne of the ‘Big Five’ of reverse mathematics [29]. Japaridze’s logic
GLP gained much interest due to Beklemishev’s proof-theoreticapplications [1]; however, from a modal logic point of view, it is not an easysystem to work with. To this end, in [3, 4, 12] Beklemishev and Dashkov intro-duced the system called
Reflection Calculus , RC , that axiomatizes the fragmentof GLP ω consisting of implications of strictly positive formulas. This systemis much simpler than GLP ω but yet expressive enough to maintain its mainproof-theoretic applications. In this paper we will focus exclusively on reflec-tion calculi, but the interested reader may find more information on the full GLP in the references provided.Similar to
GLP Λ , the signature of RC Λ contains modalities of the form h α i for α ∈ Λ. However, since this system only considers strictly positive formulas,the signature does not contain negation, disjunction or modalities [ α ]. Thus,consider a modal language L with a constant ⊤ , a set of propositional variables p, q, . . . , a binary connective ∧ and unary connectives h α i , for each α ∈ Λ. Theset of formulas in this signature is defined as follows:
Definition 1.
Fix an ordinal Λ . By F Λ we denote the set of formulas built upby the following grammar: ϕ := ⊤ | p | ( ϕ ∧ ψ ) | h α i ϕ for α ∈ Λ . Next we define a consequence relation over F Λ . For the purposes of thispaper, a deductive calculus is a pair X = ( F X , ⊢ X ) such that F X is some set, the language of X , and ⊢ X ⊆ F X × F X . We write ϕ ≡ X ψ for ϕ ⊢ X ψ and ψ ⊢ X ϕ .We will omit the subscript X when this does not lead to confusion, including inthe definition below, where ⊢ denotes ⊢ RC Λ . Definition 2.
Given an ordinal Λ , the calculus RC Λ over F Λ is given by thefollowing set of axioms and rules:Axioms:1. ϕ ⊢ ϕ, ϕ ⊢ ⊤ ;2. ϕ ∧ ψ ⊢ ϕ, ϕ ∧ ψ ⊢ ψ ;3. h α ih α i ϕ ⊢ h α i ϕ ; 4. h α i ϕ ⊢ h β i ϕ for α > β ;5. h α i ϕ ∧ h β i ψ ⊢ h α i (cid:0) ϕ ∧ h β i ψ (cid:1) for α > β .Rules:1. If ϕ ⊢ ψ and ϕ ⊢ χ , then ϕ ⊢ ψ ∧ χ ; 2. If ϕ ⊢ ψ and ψ ⊢ χ , then ϕ ⊢ χ ;3. If ϕ ⊢ ψ , then h α i ϕ ⊢ h α i ψ ; RC Λ -formula ϕ , we can define the signature of ϕ as the set ofordinals occurring in any of its modalities. Definition 3.
For any ϕ ∈ F Λ , we define the signature of ϕ , S ( ϕ ) , as follows:1. S ( ⊤ ) = S ( p ) = ∅ ;2. S ( ϕ ∧ ψ ) = S ( ϕ ) ∪ S ( ψ ) ;3. S ( h α i ϕ ) = { α } ∪ S ( ϕ ) . With the help of this last definition we can make the following observation:
Lemma 4.
For any ϕ, ψ ∈ F Λ :1. If S ( ψ ) = ∅ and ϕ ⊢ ψ , then max S ( ϕ ) ≥ max S ( ψ ) ;2. If S ( ϕ ) = ∅ and ϕ ⊢ ψ , then S ( ψ ) = ∅ .Proof. By an easy induction on the length of the derivation of ϕ ⊢ ψ .The reflection calculus has natural arithmetical [18], Kripke [12, 4], alge-braic [10] and topological [6, 15, 22, 23] interpretations for which it is soundand complete, but in this paper we will work exclusively with reflection calculifrom a syntactical perspective. Other variants of the reflection calculus havebeen proposed, for example working exclusively with worms [13], admitting thetransfinite iteration of modalities [21], or allowing additional conservativity op-erators [8, 9]. In this section we review the consistency ordering between worms, along withsome of their basic properties.
Definition 5.
Fix an ordinal Λ . The set of worms in F Λ , W Λ , is recursivelydefined as follows: 1. ⊤ ∈ W Λ ; 2. If A ∈ W Λ and α < Λ , then h α i A ∈ W Λ .Similarly, we inductively define for each α ∈ Λ the set of worms W ≥ α Λ whereall ordinals are at least α : 1. ⊤ ∈ W ≥ α Λ ; 2. If A ∈ W ≥ α Λ and β ≥ α , then h β i A ∈ W ≥ α Λ . From now on, we shall make use of the usual arithmetical operations onordinal numbers such as addition ( α + β ) or exponentiation ( α β ); see e.g. [16]for definitions. Definition 6.
Let A = h ξ i . . . h ξ n i⊤ and B = h ζ i . . . h ζ m i⊤ be worms. Then,define AB = h ξ i . . . h ξ n ih ζ i . . . h ζ m i⊤ . Given an ordinal λ , define λ ↑ A to be h λ + ξ i . . . h λ + ξ n i⊤ . Often we will want to put an extra ordinal between two worms, and we write B h λ i A for B ( h λ i A ). Next, we define the consistency ordering between worms. Definition 7.
Given an ordinal Λ , we define a relation < on W Λ by B < A if and only if A ⊢ h i B. We also define B ≤ A if B < A or B ≡ A . ≤ has some nice properties. Recall that if A is a set (or class),a preorder on A is a transitive, reflexive relation ⊆ A × A . The preorder is total if, given a, b ∈ A , we always have that a b or b a , and antisymmetric if whenever a b and b a , it follows that a = b . A total, antisymmetricpreorder is a linear order. We say that h A, i is a pre-well-order if is a totalpreorder and every non-empty B ⊆ A has a minimal element (i.e., there is m ∈ B such that m b for all b ∈ B ). A well-order is a pre-well-order that isalso linear. Note that pre-well-orders are not the same as well-quasiorders (thelatter need not be total). Pre-well-orders will be convenient to us because, aswe will see, worms are pre-well-ordered but not linearly ordered. The followingwas first proven by Beklemishev [2], and a variant closer to our presentationmay be found in [16]. Theorem 8.
For any ordinal Λ , the relation ≤ is a pre-well-order on W Λ . This yields as a corollary a nice characterization of ≤ . Corollary 9.
Given an ordinal Λ and A, B ∈ W Λ , A ≥ B if and only if A ⊢ RC Γ0 B or A ⊢ RC Γ0 h i B .Proof. By definition, A ≥ B if and only if A ≡ RC Γ0 B or A ⊢ RC Γ0 h i B , soit remains to prove that A ⊢ RC Γ0 B implies that A ≥ B . Otherwise, since ≤ is a pre-well-order, A < B , so that B ⊢ RC Γ0 h i A ⊢ RC Γ0 h i B , yielding B < B , and violating the well-foundedness of < .Note that ≤ fails to be a linear order merely because it is not antisymmetric.To get around this, one may instead consider worms modulo provable equiva-lence. Alternatively, as Beklemishev has done [2], one can choose a canonicalrepresentative for each equivalence class. Definition 10 (Beklemishev Normal Form) . A worm A ∈ W Λ is defined recur-sively to be in BNF if either1. A = ⊤ , or2. A := A k h α i A k − h α i . . . h α i A with • α = min S ( A ) ; • k ≥ ; • A i ∈ W ≥ α +1Λ , for i ≤ k ;such that A i ∈ BNF and A i ⊢ RC Γ0 h α + 1 i A i +1 for each i < k . This definition essentially mirrors that of Cantor normal forms for ordinals.The following was proven in [2].
Theorem 11.
Given any worm A there is a unique A ′ ∈ BNF such that A ≡ RC Γ0 A ′ . It follows immediately that for any ordinal Λ, ( W Λ ∩ BNF , ≤ ) is a well-order.5 Hyperexponential notation for Γ A = h A, i is any pre-well-order, for a ∈ A we may define a function o : A → Ord given recursively by o ( a ) = sup b ≺ a ( o ( b ) + 1), where by conventionsup ∅ = 0, representing the order-type of a ; this definition is sound since A ispre-well-ordered. The rank of A is then defined as sup a ∈ A ( o ( a ) + 1).The following lemma is useful in characterizing the rank function [16]. Lemma 12.
Let h A, i be a well-order. Then o : A → Ord is the unique functionsuch that1. x ≺ y implies that o ( x ) < o ( y ) ,2. if ξ < o ( x ) then ξ = o ( y ) for some y ∈ A . In order to compute the ordinals o ( A ), let us recall a notation system for Γ using hyperexponentials [17]. The class of all ordinals will be denoted Ord , and ω denotes the first infinite ordinal. Recall that many number-theoretic operationssuch as addition, multiplication and exponentiation can be defined on the classof ordinals by transfinite recursion. The ordinal exponential function ξ ω ξ isof particular importance for representing ordinal numbers. When working withorder types derived from reflection calculi, it is convenient to work with a slightvariation of this exponential. Definition 13 (Exponential function) . The exponential function is the function e : Ord → Ord given by ξ
7→ − ω ξ . Observe that for ξ = 0, we have that e ξ = − ω = − e is an example of a normal function, i.e. f : Ord → Ord which isstrictly increasing and continuous, in the sense that if λ is a limit then f ( λ ) =sup ξ<λ f ( ξ ). Giving a mapping f : X → X , it is natural and often useful toask whether f has fixed points, i.e., solutions to the equation x = f ( x ). Inparticular, normal functions have many fixed points. Proposition 14.
Every normal function f : Ord → Ord has arbitrarily largefixed points. The least fixed point of f greater or equal than α is given by lim n →∞ f n α . The first ordinal α such that α = ω α is the limit of the ω -sequence( ω, ω ω , ω ω ω , . . . ), and is usually denoted ε . Every ξ < ε can be written interms of 0 using only addition and the function ω ω ξ via its Cantor normalform. The hyperexponential function is then a natural transfinite iteration ofthe ordinal exponential which remains normal after each iteration. Definition 15 (Hyperexponential functions) . The hyperexponential functions( e ζ ) ζ ∈ Ord are the unique family of normal functions that satisfy . e = id ,2. e = e ,3. e α + β = e α ◦ e β for all α and β , and4. if ( f ξ ) ξ ∈ Ord is a family of functions satisfying 2 and 3, then for all α, β ∈ Ord , e α β ≤ f α β . Fern´andez-Duque and Joosten proved that the hyperexponentials are well-defined [17]. If α > e α β is always additively indecomposable in thesense that ξ, ζ < e α β implies that ξ + ζ < e α β ; note that zero is additivelyindecomposable according to our definition. In [16] it is also shown that thefunction ξ e ξ . Just like ordinalsbelow ε may be written using 0, addition, and ω -exponentiation, every ordinalbelow Γ may be written in terms of 0, 1, addition and the function ( ξ, ζ ) e ξ ζ .The following was first proven in [2] with different notation, and in the currentform in [17]. Theorem 16.
Let
A, B be worms and α be an ordinal. Then,1. o ( ⊤ ) = 0 ,2. o ( B h i A ) = o ( A ) + 1 + o ( B ) , and3. o ( α ↑ A ) = e α o ( A ) . While hyperexponential notation is more convenient for computing ordertypes of worms, it can easily be translated back and forth into notation basedon Veblen functions. Given an ordinal α , recall that φ α is defined recursively sothat φ β := ω β and for α > φ α β is the β -th member of { η : ( ∀ ξ < α )[ φ ξ η = η ] } . Then, Γ is the first non-zero ordinal closed under ( α, β ) φ α β . We thenhave the following equivalences [16, Proposition 5.15]: Proposition 17.
Given ordinals α, β :1. e α (0) = 0 ;2. e (1 + β ) = φ (1 + β ) ;3. e ω α (1 + β ) = φ α ( β ) . Finally we mention a useful property of o proven in [16]. Lemma 18.
Let A = ⊤ be a worm and µ an ordinal. Moreover, let α be thegreatest ordinal appearing in A . Then,1. if µ ≤ α , then o ( h µ i⊤ ) ≤ o ( A ) , and2. if α < µ , then o ( A ) < o ( h µ i⊤ ) . Beklemishev’s bracket notation system for Γ Definition 19. By W () we denote the smallest set such that: 1. ⊤ ∈ W () ; 2. if a, b ∈ W () , then ( a ) b ∈ W () . By convention we shall write () a , for a ∈ W () , to the denote ( ⊤ ) a ∈ W () .We shall also omit the use of ⊤ at the end of any worm a = ⊤ , e.g., we shalluse ( a ) . . . ( a k ) to denote ( a ) . . . ( a k ) ⊤ .We can define a translation ∗ : W () → W Λ in such a way that an element a ∈ W () will denote the ordinal o ( a ∗ ):1. ⊤ ∗ = ⊤ (cid:0) ( a ) b (cid:1) ∗ = h o ( a ∗ ) i b ∗ .Therefore, we can also define o ∗ : W () → Ord as o ∗ ( a ) = o ( a ∗ ). Example 20.
We have that o ∗ ( ((())) ) = ε . To see this, first note that () ∗ = h i⊤ , and o ( h i⊤ ) = 1 , where the calculation is performed using Theorem 16 inthe ‘degenerate’ case where A = B = ⊤ . It follows that (()) ∗ = h o ( h i⊤ ) i⊤ = h i⊤ , so that o ∗ ( (()) ) = o ( h i⊤ ) = e o ( h i⊤ ) = e ω . By similar reasoning, o ∗ ( ((())) ) = o ( h ω i⊤ ) = e ω φ ε , where the second-to-last equality usesProposition 17 and the last is the definition of ε . In fact, o ∗ : W () → Γ , and this map is surjective. In order to prove this, wemake some observations about how the ordinals represented by worms in W () can be bounded in terms of the maximum number of nested brackets occurringin them. For this purpose, we introduce the following two definitions. Definition 21.
For a ∈ W () , we define the nesting of a , N ( a ) , as the maximumnumber of nested brackets. That is:1. N ( ⊤ ) = 0 ;2. N ( ( a ) b ) = max (cid:0) N ( a ) + 1 , N ( b ) (cid:1) . Definition 22.
We recursively define the function h : N → Γ as follows:1. h (0) = 0 ;2. h ( n + 1) = e h ( n ) . Note that h is a strictly monotone function. Using Proposition 14, we seethat lim n →∞ h ( n ) = Γ . In the following proposition we can find upper andlower bounds for any ordinal o ∗ ( a ), with a ∈ W () , in terms of the nesting of a . Proposition 23.
For a ∈ W () , if N ( a ) = n , then h ( n ) ≤ o ∗ ( a ) < h ( n + 1) . roof. By induction on n . If n = 0 then we must have a = ⊤ , hence h (0) =0 = o ∗ ( a ) < h (1).For n = n ′ + 1 , we have that a = ( a ) . . . ( a m ) for some m ∈ ω . Moreover,1. N ( a i ) ≤ n ′ for i, ≤ i ≤ m ;2. there is j ∈ { , . . . , m } such that N ( a J ) = n ′ .Thus by the I.H. we get that a ∗ = h α i . . . h α m i⊤ such that:1. For each i , α i < h ( n ′ + 1);2. there is j ∈ { , . . . , m } such that α j ≥ h ( n ′ ).By Lemma 18, o ( h h ( n ′ ) i⊤ ) ≤ o ( a ∗ ) < o ( h h ( n ′ + 1) i⊤ );but by Theorem 16 o ( h h ( n ′ ) i⊤ ) = e h ( n ′ ) h ( n ), while o ( h h ( n ′ + 1) i⊤ ) = e h ( n ) h ( n + 1), as needed.As a consequence of this last proposition, we get the following corollaries. Corollary 24.
For a ∈ W () , if N ( a ) = n , then a ∗ ∈ W h ( n ) .Proof. For N ( a ) = 0, clearly we have that a ∗ ∈ W h ( n ) . For N ( a ) = n >
0, let a := ( a ) . . . ( a k ) . Thus, a ∗ := h o ∗ ( a ) i . . . h o ∗ ( a k ) i⊤ where by Proposition23 each o ∗ ( a i ) < h ( n ). Corollary 25.
For a, b ∈ W () , o ∗ ( a ) ≥ o ∗ ( b ) ⇒ N ( a ) ≥ N ( b ) . Proof.
We reason by contrapositive applying Proposition 23.
In this section we introduce the
Bracket Calculus , denoted BC . This system isanalogous to RC Γ and, as we will see later, both systems can be shown to beequivalent under a natural translation of BC -formulas into RC Γ -formulas.The main feature of BC is that it is based on a signature that uses au-tonomous notations instead of modalities indexed by ordinals, whose orderingmust be computed using a separate calculus. Moreover, since the order betweenthese notations can be established in terms of derivability within the calculus,the inferences in this system can be carried out without using any externalproperty of ordinals. In this sense, we say that BC provides an autonomousprovability calculus.The set of BC -formulas, F () , is defined by extending W () to a strictly pos-itive signature containing a constant ⊤ , a binary connective ∧ and a set ofpropositional variables. Definition 26. By F () we denote the set of formulas built up by the followinggrammar: ϕ := ⊤ | p | ϕ ∧ ψ | ( a ) ϕ for a ∈ W () . RC , BC is based on sequents , i.e. expressions of the form ϕ ⊢ ψ ,where ϕ, ψ ∈ F () . In addition to this, we will also use b E a , for a, b ∈ W () , todenote that either a ⊢ () b or a ⊢ b is derivable. Analogously, we will use b ⊳ a to denote that the sequent a ⊢ () b is derivable. Definition 27. BC is given by the following set of axioms and rules:Axioms: 1. ϕ ⊢ ϕ, ϕ ⊢ ⊤ ; 2. ϕ ∧ ψ ⊢ ϕ, ϕ ∧ ψ ⊢ ψ ;Rules:1. If ϕ ⊢ ψ and ϕ ⊢ χ , then ϕ ⊢ ψ ∧ χ ;2. If ϕ ⊢ ψ and ψ ⊢ χ , then ϕ ⊢ χ ;3. If ϕ ⊢ ψ and b E a , then ( a ) ϕ ⊢ ( b ) ψ and ( a ) ( b ) ϕ ⊢ ( b ) ψ ;4. If b ⊳ a , then ( a ) ϕ ∧ ( b ) ψ ⊢ ( a ) (cid:0) ϕ ∧ ( b ) ψ (cid:1) . In this section we introduce a way of interpreting BC -formulas as RC Γ -formulas, and prove that under this translation, both systems can derive exactlythe same sequents. Definition 28.
We define a translation τ between F () and F Γ , τ : F () → F Γ ,as follows:1. ⊤ τ = ⊤ ;2. p τ = p ; 3. ( ϕ ∧ ψ ) τ = ( ϕ τ ∧ ψ τ ) ;4. ( ( a ) ϕ ) τ = h o ∗ ( a ) i ϕ τ . Note that for a ∈ W () , a τ = a ∗ . Using this and routine induction, thefollowing can readily be verified. Lemma 29.
Given ϕ ∈ F () and α ∈ S ( ϕ τ ) , there is a subformula a ∈ W () of ϕ such that α = o ∗ ( a ) . The following lemma establishes the preservability of BC with respect to RC Γ , under τ . Lemma 30.
For any ϕ, ψ ∈ F () : ϕ ⊢ BC ψ = ⇒ ϕ τ ⊢ RC Γ0 ψ τ . Proof.
By induction on the length of the derivation. We can easily check thatthe set of axioms of BC is preserved under τ . Likewise, the cases for a derivationending on Rules 1 or 2 are straightforward. Thus, we only check Rules 3 and 4.Regarding Rule 3, we need to prove that if a D b then both sequents h o ∗ ( a ) i ϕ τ ⊢ h o ∗ ( b ) i ψ τ and h o ∗ ( a ) ih o ∗ ( b ) i ϕ τ ⊢ h o ∗ ( b ) i ψ τ are derivable in RC Γ .We can make the following observations by applying the I.H.:10. Since a D b , we have that either a τ ⊢ h i b τ or a τ ⊢ b τ are derivable in RC Γ . Therefore, by Corollary 9, o ( a τ ) ≥ o ( b τ ). Since o ∗ ( a ) = o ( a ∗ ) = o ( a τ ) and the same equality holds for b , we have that o ∗ ( a ) ≥ o ∗ ( b ).2. We also have that ϕ τ ⊢ RC Γ0 ψ τ and thus, by Rule 3 of RC Γ we obtainthat h o ∗ ( a ) i ϕ τ ⊢ h o ∗ ( a ) i ψ τ and h o ∗ ( a ) ih o ∗ ( b ) i ϕ τ ⊢ h o ∗ ( a ) ih o ∗ ( b ) i ψ τ arederivable in RC Γ .On the one hand, by these two facts together with Axiom 4 we obtain that h o ∗ ( a ) i ϕ τ ⊢ RC Γ0 h o ∗ ( b ) i ψ τ . On the other hand, we can combine Axioms 4 and3 to get that h o ∗ ( a ) ih o ∗ ( b ) i ϕ τ ⊢ RC Γ0 h o ∗ ( b ) i ψ τ .We follow an analogous reasoning in the case of Rule 4. By the I.H. wehave that a τ ⊢ RC Γ0 h i b τ . Therefore o ∗ ( a ) > o ∗ ( b ) and by Axiom 5, h o ∗ ( a ) i ϕ ∧h o ∗ ( b ) i ψ ⊢ RC Γ0 h o ∗ ( a ) i (cid:0) ϕ ∧ h o ∗ ( b ) i ψ (cid:1) .With the following definition we fix a way of translating F Γ -formulas intoformulas in F () . However, since different words in W () might denote the sameordinal, we need a normal form theorem for W () . Definition 31.
We define NF ⊂ W () to be the smallest set of W () -words suchthat ⊤ ∈ NF and for any ( a ) b ∈ W () , if a, b ∈ NF and (cid:0) ( a ) b (cid:1) ∗ ∈ BNF , then ( a ) b ∈ NF . Every element of W () has a unique normal form, as shown by L. Beklemishevin [2]. Theorem 32 (Beklemishev) . For each α ∈ Γ we can associate a unique a α ∈ NF such that o ∗ ( a α ) = α . Now we are ready to translate F Γ -formulas into F () -formulas. Definition 33.
We define a translation ι between F Γ and F () , ι : F Γ → F () ,as follows:1. ⊤ ι = ⊤ ;2. p ι = p ; 3. ( ϕ ∧ ψ ) ι = ( ϕ ι ∧ ψ ι ) ;4. ( h α i ϕ ) ι = ( a α ) ϕ ι . The following remark follows immediately from the definitions of τ and ι . Remark 34.
For any ϕ ∈ F Γ , ( ϕ ι ) τ = ϕ . In particular, if A ∈ W Γ is a wormthen A ι ∈ W () and o ∗ ( A ι ) = o (( A ι ) ∗ ) = o (( A ι ) τ ) = o ( A ) . With the next definition, we extend the nesting N ( a ) of a ∈ W () to F () -formulas. Definition 35.
For ϕ ∈ F () , we define the nesting of ϕ , Nt ( ϕ ) , as the maximumnumber of nested brackets. That is:1. Nt ( ⊤ ) = Nt ( p ) = N ( ⊤ ) ; . Nt ( ϕ ∧ ψ ) = max (cid:0) Nt ( ϕ ) , Nt ( ψ ) (cid:1) ;3. Nt ( ( a ) ϕ ) = max (cid:0) N ( ( a ) ) , Nt ( ϕ ) (cid:1) = max (cid:0) N ( a ) + 1 , Nt ( ϕ ) (cid:1) . The upcoming remark collects a useful observation concerning the nesting Nt ( ϕ ) of a formula ϕ and its subformulas. This fact can be verified by an easyinduction. Remark 36.
For any ϕ ∈ F () which is either ⊤ or Nt ( ϕ ) ≥ , there is asubformula a ∈ W () of ϕ such that Nt ( ϕ ) = Nt ( a ) . Moreover, if Nt ( ϕ ) ≥ ,there is a subformula a ∈ W () of ϕ such that Nt ( ϕ ) = Nt ( a ) + 1 . The following lemma relates the derivability in RC Γ under τ , and thenesting of formulas in F () . Lemma 37.
For any ϕ, ψ ∈ F () : ϕ τ ⊢ RC Γ0 ψ τ = ⇒ Nt ( ϕ ) ≥ Nt ( ψ ) . Proof.
Suppose that ϕ τ ⊢ RC Γ0 ψ τ . If S ( ψ τ ) = ∅ then it is easy to check that Nt ( ψ ) = 0 and there is nothing to prove, so assume otherwise. Then, by Lemma4.1, max S ( ϕ τ ) ≥ max S ( ψ τ ). Using Lemma 29, let a ∈ W () be a subformula of ϕ such that o ∗ ( a ) = max S ( ϕ τ ). Moreover, since S ( ψ τ ) = ∅ , then Nt ( ψ ) ≥ b ∈ W () , a subformulaof ψ such that Nt ( ψ ) = N ( b ) + 1. If we had N ( a ) < N ( b ) then it would followfrom Corollary 25 that o ∗ ( a ) < o ∗ ( b ), contradicting max S ( ϕ τ ) ≥ max S ( ϕ τ ).Thus N ( a ) ≥ N ( b ) and Nt ( ϕ ) ≥ N ( a ) + 1 ≥ Nt ( ψ ), as needed.With the following theorem we conclude the proof of the preservability be-tween BC and RC Γ . Theorem 38.
For any ϕ, ψ ∈ F () : ϕ τ ⊢ RC Γ0 ψ τ ⇐⇒ ϕ ⊢ BC ψ. Proof.
The right-to-left direction is given by Lemma 30, so we focus on theother. Proceed by induction on Nt ( ϕ ). For the base case, assume Nt ( ϕ ) = 0and ϕ τ ⊢ RC Γ0 ψ τ . By a subsidiary induction on the length of the derivationof ϕ τ ⊢ RC Γ0 ψ τ , we set to prove ϕ ⊢ BC ψ . If the derivation has length one itsuffices to check RC Γ -Axioms 1 and 2, which is immediate. If it has lengthgreater than one it must end in a rule. The case for RC Γ -Rule 1 follows by theI.H.. For RC Γ -Rule 2, we have that there is χ ∈ F Γ such that ϕ τ ⊢ RC Γ0 χ and χ ⊢ RC Γ0 ψ τ . By Remark 34 and Lemma 37, we get that ϕ τ ⊢ RC Γ0 ( χ ι ) τ and ( χ ι ) τ ⊢ RC Γ0 ψ τ with Nt ( χ ι ) = 0. Thus, by the subsidiary I.H., ϕ ⊢ BC χ ι and χ ι ⊢ BC ψ and by BC -Rule 2, ϕ ⊢ BC ψ .For the inductive step, let Nt ( ϕ ) = n + 1. We proceed by a subsidiaryinduction on the length of the derivation. If ϕ τ ⊢ RC Γ0 ψ τ is obtained by meansof RC Γ -Axioms 1 and 2, then clearly ϕ ⊢ BC ψ . If ϕ τ ⊢ RC Γ0 ψ τ is an instanceof RC Γ -Axiom 3, then we have that ϕ := ( a )( b ) χ and ψ := ( c ) χ for some12 ∈ F () and a, b, c ∈ W () such that o ∗ ( a ) = o ∗ ( b ) = o ∗ ( c ). Hence, a ∗ ⊢ RC Γ0 b ∗ and b ∗ ⊢ RC Γ0 c ∗ . Since Nt ( w ) < n + 1 for w ∈ { a, b, c } , by the main I.H. wehave that a ⊢ BC b and b ⊢ BC c . Thus, we have the following BC -derivation: χ ⊢ χ b ⊢ c (Rule 3) ( b ) χ ⊢ ( c ) χ a ⊢ b (Rule 3) ( a )( b ) χ ⊢ ( b )( c ) χ χ ⊢ χ b ⊢ c (Rule 3) ( b )( c ) χ ⊢ ( c ) χ (Rule 2) ( a )( b ) χ ⊢ ( c ) χ If ϕ τ ⊢ RC Γ0 ψ τ is obtained by using RC Γ -Axiom 4, then ϕ := ( a ) χ and ψ := ( b ) χ for some χ ∈ F () and a, b, ∈ W () with o ∗ ( a ) > o ∗ ( b ). Therefore a ∗ ⊢ RC Γ0 h i b ∗ and since ϕ := ( a ) χ , we have that Nt ( a ) < Nt ( ϕ ). Thus, bythe main I.H. a ⊢ BC () b and by BC -Rule 3, ( a ) χ ⊢ BC ( b ) χ . If ϕ τ ⊢ RC Γ0 ψ τ is an instance of RC Γ -Axiom 5, then we have that ϕ := ( a ) χ ∧ ( b ) χ and ψ := ( a ) (cid:0) χ ∧ ( b ) χ (cid:1) , for some χ , χ ∈ F () and a, b ∈ W () with o ∗ ( a ) >o ∗ ( b ). Since Nt ( a ) < Nt ( ϕ ) and a ∗ ⊢ RC Γ0 h i b ∗ , by the main I.H. we obtainthat a ⊢ BC () b and by applying BC -Rule 4, ( a ) χ ∧ ( b ) χ ⊢ ( a ) (cid:0) χ ∧ ( b ) χ (cid:1) . Regarding rules, RC Γ -Rule 1 is immediate and RC Γ -Rule 3 followsan analogous reasoning to that of Axiom 4. This way, we only check RC Γ -Rule 2. Assume ϕ τ ⊢ RC Γ0 ψ τ is obtained by an application of RC Γ -Rule 2.Then, there is χ ∈ F Γ such that ϕ τ ⊢ RC Γ0 χ and χ ⊢ RC Γ0 ψ τ . By Remark 34together with Lemma 37 we obtain that ϕ τ ⊢ RC Γ0 ( χ ι ) τ and ( χ ι ) τ ⊢ RC Γ0 ψ τ with Nt ( χ ) ≤ n + 1. By the subsidiary I.H. ϕ ⊢ BC χ ι and χ ι ⊢ BC ψ and hence,by BC -Rule 2, ϕ ⊢ BC ψ .With this we obtain our main result: an autonomous calculus for represent-ing ordinals below Γ . Theorem 39. ( NF , E ) is a well-order of order-type Γ .Proof. By Theorem 38, a ⊳ b if and only if b τ ⊢ RC Γ0 h i a τ if and only if o ∗ ( a ) < o ∗ ( b ). Moreover if ξ < o ∗ ( a ) then since o ∗ ( a ) = o ( a τ ) and o is theorder-type function on BNF , by item 2 of Lemma 12 there is some
B < a τ suchthat ξ = o ( B ), hence in view of Remark 34, ξ = o ∗ ( B ι ). Thus by Lemma 12, o ∗ is the order-type function on NF . That the range of o ∗ is Γ follows fromProposition 23 which tells us that o ∗ ( a ) < h ( N ( a ) + 1) < Γ for all a ∈ W () ,while if we define recursively a = ⊤ and a n +1 = ( a n ) , Theorem 16 and an easyinduction readily yield Γ = lim n →∞ h ( n ) = lim n →∞ o ∗ ( a n ).It remains to check that E is antisymmetric. If a, b ∈ NF and a ≡ BC b , then a τ ≡ RC Γ0 b τ and are in BNF . By Theorem 11, a τ = b τ . Writing a = ( a ) . . . ( a n ) and b = ( b ) . . . ( b m ) , it follows that n = m and that for0 < i ≤ n , o ∗ ( a i ) = o ∗ ( b i ). It follows that a τi ≡ RC Γ0 b τi , so that a i ≡ BC b i .Induction on nesting depth yields a i = b i , hence a = b .13 The Bracket Principle
In this section, we adapt Beklemishev’s system of fundamental sequences forworms [1] to elements of W () . These are sequences ( a { n } ) n ∈ N which convergeto a with respect to the ⊳ ordering whenever a is a limit worm. Beklemishev usesthese fundamental sequences to present a combinatorial statement independentof PA . As we will see, these fundamental sequences generalize smoothly to W () and provide an independent statement for the theory ATR of ArithmeticalTransfinite Recursion, which we briefly recall in the next section.Let a ∈ W () and write a = ( a ) . . . ( a m ) , with m ≥
0. Then, consider thefollowing cases for arbitrary n .1. ⊤{ n } = ⊤ .2. a { n } = ( a ) . . . ( a k ) if a = ⊤ .3. If min( a ) = a , let ℓ be least such that a ℓ ⊳ a , otherwise let ℓ = k + 1.Let b = ( a { n } ) . . . ( a ℓ − ) and c = ( a ℓ ) . . . ( a k ) (with c possibly empty).Then, a { n } = b n +1 c , where b n is inductively defined as ⊤ for n = 0 and b b n ′ for n = n ′ + 1. Remark 40.
Beklemishev’s definition of fundamental sequences is almost iden-tical, except that each a i is a natural number and a { n } is replaced by a − when a > . Note, however, that ( () k +1 ) { n } = () k , so the two definitionscoincide if we represent natural numbers as elements of W () . Proposition 41. If a = ⊤ , then a { n } ⊳ a .Proof. We proceed by induction on the nesting of a , N ( a ). For N ( a ) = 1, wehave that a := () k +1 ⊤ for some k < ω . Moreover, for any n , we have that a { n } = () k ⊤ . Thus, we have that a ⊢ a and a = () a { n } , that is, a { n } ⊳ a .For N ( a ) = m + 1, if a is of the form () a ′ , we reason as in the previous case.Otherwise, let a be of the form ( a ) . . . ( a k ) with a = ⊤ , then by the I.H. wehave that a ⊢ () a { n } . Thus, ( a ) . . . ( a k ) ⊢ ( a ) . . . ( a k ) a ⊢ () a { n } (Rule 3) ( a ) . . . ( a k ) ⊢ ( a { n } ) . . . ( a k ) a ⊢ ( a { n } ) . . . ( a k ) It follows that a ⊢ ( a { n } ) . . . ( a ℓ − )( a ℓ ) . . . ( a k ) . where ℓ is the least such that a ℓ ⊳ a (recall that if a = min( a ), then ℓ = k + 1,and the part ( a ℓ ) . . . ( a k ) is empty). Hence, we get that: • a { n } ⊳ a and • a E a j for j, ≤ j < ℓ ,and so, a { n } ⊳ a j . We can reason as follows:14 ⊢ ( a { n } ) . . . ( a ℓ − )( a ℓ ) . . . ( a k ) a ⊢ ( a ) . . . ( a ℓ − ) (Rule 1) a ⊢ ( a { n } ) . . . ( a ℓ − )( a ℓ ) . . . ( a k ) ∧ ( a ) . . . ( a ℓ − ) Let a ′ := ( a { n } ) . . . ( a ℓ − )( a ℓ ) . . . ( a k ) . Combining the fact that a j ⊢ () a { n } for any j, ≤ j < ℓ , we have that we can iteratively apply Rule 4 andRule 3 to place the modalities in ( a ) . . . ( a ℓ − ) outside the conjunction. Moreprecisely, we start by applying Rule 3, obtaining a ⊢ ( a ) . . . ( a ℓ − ) ∧ a ′ (Rule 3) a ⊢ ( a ) (cid:16) ( a ) . . . ( a ℓ − ) ∧ a ′ (cid:17) .We can observe that since a j ⊢ () a { n } for j, ≤ j < ℓ , ( a ) . . . ( a ℓ − ) ∧ a ′ ⊢ ( a ) (cid:16) ( a ) . . . ( a ℓ − ) ∧ a ′ (cid:17) These two last inferences can be combined by means of the Rule 4 from whichwe get: a ⊢ ( a )( a ) (cid:16) ( a ) . . . ( a ℓ − ) ∧ a ′ (cid:17) . Iterating this process we get that a ⊢ ( a ) . . . ( a ℓ − ) a ′ and since, a ⊢ () a { n } by Rule 3 we get that a ⊢ ( a { n } ) . . . ( a ℓ − ) a ′ that is, a ⊢ ( a { n } ) . . . ( a ℓ − ) a ′ ( a { n } ) . . . ( a ℓ − )( a ℓ ) . . . ( a k ) . This whole last part of the argument can be iterated, obtaining that a ⊢ (cid:0) ( a { n } ) . . . ( a ℓ − ) (cid:1) n +1 ( a ℓ ) . . . ( a k ) . Thus, a ⊢ (cid:0) ( a { n } ) . . . ( a ℓ − ) (cid:1) n +1 ( a ℓ ) . . . ( a k ) ∧ ( a ) (Rules 4 and 3) a ⊢ ( a ) (cid:0) ( a { n } ) . . . ( a ℓ − ) (cid:1) n +1 ( a ℓ ) . . . ( a k ) (Rule 3) a ⊢ () (cid:0) ( a { n } ) . . . ( a ℓ − ) (cid:1) n +1 ( a ℓ ) . . . ( a k ) Therefore, we can conclude that a { n } ⊳ a .Now, let us define for a ∈ W () and n ∈ N , a new worm a ⦃ n ⦄ ∈ W () recursively by a ⦃ ⦄ = a and a ⦃ n + 1 ⦄ = a ⦃ n ⦄ { n + 1 } . Theorem 42.
For each a ∈ W () , there is i ∈ N such that a ⦃ i ⦄ = ⊤ .Proof. Assume towards a contradiction that there is no i ∈ N such that a ⦃ i ⦄ = ⊤ . Then, Proposition 41 yields a ⦃ i +1 ⦄ ⊳ a ⦃ i ⦄ for all i , and thus (cid:0) o ∗ ( a ⦃ i ⦄ ) (cid:1) i ∈ N defines an infinite descending chain of ordinals, contradicting the well-foundednessof Γ . 15 Independence
We will now show that Theorem 42 is independent of
ATR . The theory ATR is defined in the language of second order arithmetic, which extendsthe language of PA with a new sort of variables X, Y, Z for sets of naturalnumbers with atomic formulas t ∈ X and quantification ∀ Xϕ ( X ) ranging oversets of natural numbers. Using coding techniques, the language of arithmeticsuffices to formalize many familiar mathematical notions such as real numbersand continuous functions on the real line. More relevant to us, (countable) well-orders and Turing jumps can be formalized in this context. The particulars arenot important for our purposes, but these notions are treated in detail in [29].The theory ACA is the second order analogue of Peano Arithmetic, definedby extending Robinson’s Q with the induction axiom stating that every non-empty set has a least element and the axiom scheme stating that { x ∈ N : ϕ ( x ) } is a set, where ϕ does not contain second order quantifiers but possibly containsset-variables. Equivalently, ACA can be defined with an axiom that statesthat for every set X , the Turing jump of X exists. The theory of ArithmeticalTransfinite Recursion, ATR , can then be obtained by extending ACA withan axiom stating that for every set X and well-order α , the α th Turing jump of X exists. This theory is related to predicative mathematics [14], and discussedin detail in [29].The proof-theoretic ordinal of ATR is Γ . In order to make this precise, weneed to study Veblen hierarchies in some more detail; recall that we have definedthem in Section 4. The Veblen normal form of ξ > φ α β + γ = ξ such that γ < ξ and β < φ α β . We will call this the Veblen normalform of ξ and write ξ ≡ VNF φ α β + γ . The order relation between elements of Γ can be computed recursively on their Veblen normal form. Below we consideronly ξ, ζ >
0, as clearly 0 < φ α β + γ regardless of α, β, γ . The following is foundin e.g. [27]. Lemma 43.
Given ξ, ξ ′ < Γ with ξ = φ α β + γ and ξ ′ = φ α ′ β ′ + γ ′ both inVeblen normal form, ξ < ξ ′ if and only if one of the following holds:1. α = α ′ , β = β ′ and γ < γ ′ ;2. α < α ′ and β < φ α ′ β ′ ; 3. α = α ′ and β < β ′ , or4. α ′ < α and φ α β < β ′ . In order to prove independence from
ATR , we also need to review funda-mental sequences based on Veblen notation. Definition 44.
For ξ < Γ and x < ω , define α [ x ] recursively as follows. Firstwe set x / α = x + 1 if α is a successor, x / α = 1 otherwise. Then, define:1. x ] = 0 .2. ( φ α β + γ )[ x ] = φ α β + γ [ x ] if γ > . 3. ( φ x ] = 0 (note that φ ).4. φ ( β + 1)[ x ] = φ β · ( x + 2) . . ( φ α x ] := φ x / α α [ x ] if α > .6. φ α ( β + 1)[ x ] := φ x / α α [ x ] ( φ α β + 1) if α > .7. ( φ α λ )[ x ] := φ α ( λ [ x ]) if λ is alimit. For an ordinal ξ < Γ and n ≥ ξ J K = ξ and ξ J n +1 K = ξ J n K [ n + 1]. The system of fundamental sequences satisfies the Bachmannproperty [28]:
Proposition 45. If α, β < Γ and k < ω satisfy α [ k ] < β < α , then α [ k ] ≤ β [1] . This property is useful because it allows us to appeal to the following, provenin [20].
Proposition 46. If ( α i ) i ∈ N is a sequence of elements of Γ such that for all i , α i [ i + 1] ≤ α i +1 ≤ α i , it follows that for all i , α i ≥ α J i K . This allows us to establish new independence results by appealing to the factthat
ATR does not prove that the process of stepping down the fundamentalsequences below Γ always reaches zero. By recursion on n define γ n < Γ asfollows, γ := 0 and γ n +1 := φ γ n . Then, ∀ m ∃ ℓ ( γ m J ℓ K = 0) is not provable in ATR . In fact, the least ℓ such that γ m J ℓ K = 0 grows more quickly than anyprovably total computable function.Let us make this precise. For our purposes, a partial function f : N → N is computable if there is a Σ formula ϕ f ( x, y ) in the language of first orderarithmetic (with no other free variables) such that for every m, n , f ( m ) = n if and only if ϕ f ( m, n ) holds. The function f is provably total in a theory T if T ⊢ ∀ x ∃ yϕ f ( x, y ) (more precisely, f is provably total if there is at least onesuch choice of ϕ f ). Then, the function F such that F ( m ) is the least ℓ with γ m J ℓ K = 0 is computable and, by the well-foundedness of Γ , total. Moreover, itgives an upper bound for all the provably total computable functions in ATR . Theorem 47 ([28]) . If f : N → N is a computable function that is provably totalin ATR , then ∃ N ∀ n > N ( f ( n ) < F ( n )) . So, our goal will be to show that the witnesses for Theorem 42 grow at leastas quickly as F , from which we obtain that the theorem is unprovable in ATR .We begin with a straightforward technical lemma, which will be useful below. Lemma 48.
Let a , . . . , a k , b , . . . b k ∈ W () . If for any j with ≤ j ≤ k , wehave that a j E b j , then ( a ) . . . ( a k ) E ( b ) . . . ( b k ) .Proof. By a simple induction on k applying Rule 3. For the base, we have that ⊤ ⊢ ⊤ a E b ( b ) ⊢ ( a ) The inductive step follows from the I.H. with an analogous reasoning.17t will be useful to extend the ↑ operation of Definition 6 to elements of W () using the ι operation of Definition 33. Definition 49.
Let a ∈ W () , α < Γ . By α ↑ a we denote the expression ( α ↑ a ∗ ) ι ∈ W () . Lemma 50.
For any a ∈ W () , k < ω and α < Γ : o ∗ (cid:16) ( α ↑ a ) { k } (cid:17) ≥ o ∗ (cid:16) α ↑ ( a { k } ) (cid:17) . Proof.
By induction on the complexity of a with the base case being trivial. Forthe inductive step, let a := ( a )( b ) . . . ( b j ) . Observe that α ↑ a = ( α ↑ a )( α ↑ b ) . . . ( α ↑ b j ) and so,( α ↑ a ) { k } = (cid:0) ( ( α ↑ a ) { k } )( α ↑ b ) . . . ( α ↑ b i ) (cid:1) k +1 ( α ↑ b i +1 ) . . . ( α ↑ b j ) By the I.H. we have that o ∗ (cid:0) ( α ↑ a ) { k } (cid:1) ≥ o ∗ (cid:0) α ↑ ( a { k } ) (cid:1) and so, α ↑ ( a { k } ) E ( α ↑ a ) { k } . Therefore, applying Lemma 48 we can conclude that: (cid:0) ( α ↑ ( a { k } ) )( α ↑ b ) . . . ( α ↑ b i ) (cid:1) k +1 ( α ↑ b i +1 ) . . . ( α ↑ b j ) E ( α ↑ a ) { k } Thus with the help of Theorem 16 and Proposition 17 together with Definition49, o ∗ (cid:0) ( α ↑ a ) { k } (cid:1) ≥ o ∗ (cid:16)(cid:0) ( α ↑ ( a { k } ) )( α ↑ b ) . . . ( α ↑ b i ) (cid:1) k +1 ( α ↑ b i +1 ) . . . ( α ↑ b j ) (cid:17) = e α (cid:16) o ∗ (cid:0) ( ( a { k } )( b ) . . . ( b i ) (cid:1) k +1 ( b i +1 ) . . . ( b j ) (cid:1)(cid:17) = e α (cid:0) o ∗ ( a { k } ) (cid:1) = o ∗ (cid:0) α ↑ ( a { k } ) (cid:1) .The following is the key lemma in showing that Theorem 42 implies thatstepping down the fundamental sequences eventually reaches zero. Lemma 51. If a ∈ W () , a ′ = () a and < k < ω then o ∗ ( a ) J k K ≤ o ∗ ( a ′ ⦃ k + 1 ⦄ ) . Proof.
It suffices to show that o ∗ ( a )[ k ] ≤ o ∗ ( a { k + 1 } ) for every a ∈ W () . Thelemma then follows from the Bachmann property (Proposition 45), since thenwe can apply Proposition 46 to the sequence ( α i ) i ≤ ω with α = o ∗ ( a ) and α i +1 = a ′ ⦃ i + 2 ⦄ to obtain o ∗ ( a ) J k K < o ∗ ( a ′ ⦃ k + 1 ⦄ ). We proceed by inductionon o ∗ ( a ). If o ∗ ( a ) = 0, the claim is trivially true, so we assume otherwise. Write o ∗ ( a ) = φ δ β + γ in Veblen normal form and consider the following cases.18 ase 1 ( γ > o ∗ ( a ) is additively decomposable. By inspection onTheorem 16, a is of the form a () b and o ∗ ( a )[ k ] = o ∗ ( b )+ o ∗ ( a )[ k ]. By the I.H., o ∗ ( a )[ k ] ≤ o ∗ ( a { k + 1 } ), therefore o ∗ ( b ) + o ∗ ( a )[ k ] ≤ o ∗ ( b ) + o ∗ ( a { k + 1 } ) = o ∗ ( a { k + 1 } () b ) = o ∗ ( a { k + 1 } ). Case 2 ( γ = 0). Then, o ∗ ( a ) = ϕ δ β , with δ, β < o ∗ ( a ). We distinguish severalsub-cases. Case 2 .1 ( δ = 0 and β = β ′ + 1). Then, inspection of Theorem 16 and Propo-sition 17 (which we will no longer mention in subsequent cases) shows that o ∗ ( a ) = φ β ′ + 1 = e ( β ′ + 1) = e (cid:0) o ∗ ( () b ) (cid:1) where o ∗ ( b ) = β ′ . Thus, a = (()) (1 ↑ b ). We can observe that a { k + 2 } = (cid:16) () (cid:0) ↑ b (cid:1)(cid:17) k +2 and so we have that: o ∗ ( a { k + 1 } ) = o ∗ (1 ↑ b ) · ( k + 2) + 1 = φ (cid:0) o ∗ ( b ) (cid:1) · ( k + 2) + 1 > φ ( β ′ ) · ( k + 2) = φ ( β ′ + 1)[ k ] . Case 2 .2 ( δ > β = 0). Let d be such that o ∗ ( d ) = δ . Then, we have that a = ( b ) with o ∗ ( b ) = ω δ = e ( δ ) = e ( o ∗ ( d )) = o ∗ (1 ↑ d ) . Therefore, a = ( (1 ↑ d ) ) , and so we have that o ∗ ( a { k + 1 } ) = o ∗ (cid:0) ( (1 ↑ d ) { k + 1 } ) k +2 (cid:1) = e o ∗ (cid:0) (1 ↑ d ) { k +1 } (cid:1) ( k + 2) ≥ e ω δ [ k ] ( k + 2) , where the last inequality uses the induction hypothesis on δ < o ∗ ( a ). We claimthat e ω δ [ k ] ( k + 2) ≥ φ k / δ δ [ k ] (1);indeed, if δ = δ ′ + 1 is a successor, then ω δ [ k ] = ω δ ′ · ( k + 2), so e ω δ [ k ] ( k + 2) = e ω δ ′ · ( k +2) ( k + 2) ≥ φ k +2 δ ′ ( k + 1) > φ k / δ δ [ k ] (1) , while, if δ is a limit, e ω δ [ k ] ( k + 2) = e ω δ [ k ] ( k + 2) ≥ φ δ [ k ] ( k + 1) ≥ φ k / δ δ [ k ] (1) . Hence, o ∗ ( a )[ k ] ≤ o ∗ ( a { k + 1 } ). Case 2 .3 ( δ > β = β ′ + 1). By Definition 44, Item 6, we have that o ∗ ( a )[ k ] = φ k / δ δ [ k ] (cid:0) φ δ ( β ′ ) + 1 (cid:1) . Since φ δ ( β ′ ) is infinite, we have that 1 + φ δ ( β ′ ) =19 δ ( β ′ ), and hence φ iδ [ k ] (cid:0) φ δ ( β ′ ) + 1 (cid:1) = e ω δ [ k ] · i (cid:0) e ω δ ( β ′ ) + 1 (cid:1) for all i . Then, a = (( d )) ω δ ↑ b , where o ∗ ( b ) = 1 + β ′ and o ∗ ( d ) = δ , and we have that o ∗ ( a )[ k ] = o ∗ (cid:0) ω δ [ k ] ↑ () ( ω δ ↑ b ) (cid:1) . By the I.H. and Lemma 48, o ∗ (cid:0) ω δ [ k ] ↑ () ( ω δ ↑ b ) (cid:1) ≤ o ∗ (cid:0) o ∗ ( d { k + 1 } ) ↑ () ( ω δ ↑ b ) (cid:1) = o ∗ (cid:0) (( d { k + 1 } )) ( ω δ ↑ b ) (cid:1) ≤ o ∗ ( a { k + 1 } ) . Case 2 .4 ( δ > β ∈ Lim). Then, a = ω δ ↑ b , where o ∗ ( b ) = β . ByDefinition 44, Item 7, we get that φ δ ( β )[ k ] = φ δ (cid:0) o ∗ ( b )[ k ] (cid:1) and since by the I.H. o ∗ ( b )[ k ] ≤ o ∗ ( b { k + 1 } ), we have that φ δ (cid:0) o ∗ ( b )[ k ] (cid:1) ≤ φ δ (cid:0) o ∗ ( b { k + 1 } ) (cid:1) . On theother hand, φ δ (cid:0) o ∗ ( b { k + 1 } ) (cid:1) = e ω δ (cid:0) o ∗ ( b { k + 1 } ) (cid:1) = o ∗ (cid:16) ω δ ↑ ( b { k + 1 } ) (cid:17) . By Lemma 50, o ∗ (cid:0) ω δ ↑ ( b { k + 1 } ) (cid:1) ≤ o ∗ (cid:0) ( ω δ ↑ b ) { k + 1 } (cid:1) . Thus, o ∗ ( a )[ k ] ≤ o ∗ ( a { k + 1 } ). Theorem 52.
Theorem 42 is not provable in
ATR .Proof. Let γ m be as in Theorem 47 and define recursively a := ⊤ , a := () , and a n +2 := ((( a n ))) . Then, set a ′ n = () a n . We can observe that o ∗ ( a m ) = γ m .For m < m ≥
2, it follows from a simple inductionon m together with the fact that o ∗ ( a m ) = h ( m + 1), as given in Definition 22.Thus, Lemma 51 yields γ m J k K = o ∗ ( a m ) J k K ≤ o ∗ ( a ′ m ⦃ k + 1 ⦄ )for all k . It follows that, if a ′ m ⦃ k +1 ⦄ = ⊤ for some k , then γ m J k K = o ∗ ( a m ) J k K =0 for some k .Recall that we had defined F ( m ) to be the least ℓ so that γ m J ℓ K = 0. Defining G ( m ) to be the least k such that a ′ m ⦃ k + 1 ⦄ = ⊤ , it follows that F ( m ) ≤ G ( m )for all m , and hence, by Theorem 47, ATR does not prove that G ( m ) is total;in other words, ATR
6⊢ ∀ m ∃ k ( a ′ m ⦃ k +1 ⦄ = ⊤ ), and Theorem 42 is unprovablein ATR .
10 Concluding remarks
Beklemishev’s ‘brackets’ provided an autonomous notation system for Γ basedon worms, but did not provide a method for comparing different worms withoutfirst translating into a more traditional notation system. Our calculus BC showsthat this is not necessary, and indeed all derivations may be carried out entirely20ithin the brackets notation. To the best of our knowledge, this yields the firstordinal notation system presented as a purely modal deductive system.Our analysis is purely syntactical and leaves room for a semantical treatmentof BC . As before one may first map BC into RC Γ and then use the Kripkesemantics presented in [4, 12], but we leave the question of whether it is possibleto define natural semantics that work only with BC expressions and do notdirectly reference ordinals.The independence of Theorem 42 provides a relatively simple combinatorialstatement independent of the rather powerful theory ATR . In particular, ourfundamental sequences for worms enjoy a more uniform definition than thosebased on Veblen functions. It is of interest to explore whether this can be ex-tended to provide statements independent of much stronger theories. In [16], wesuggest variants of the brackets notation for representing the Bachmann-Howardordinal and beyond. Sound and complete calculi for these systems remain tobe found, as do natural fundamental sequences leading to new independentcombinatorial principles. References [1] L. D. Beklemishev. Provability algebras and proof-theoretic ordinals, I.
Annals of Pure and Applied Logic , 128:103–124, 2004.[2] L. D. Beklemishev. Veblen hierarchy in the context of provability alge-bras. In P. H´ajek, L. Vald´es-Villanueva, and D. Westerst˚ahl, editors,
Logic,Methodology and Philosophy of Science, Proceedings of the Twelfth Inter-national Congress , pages 65–78. Kings College Publications, 2005.[3] L. D. Beklemishev. Calibrating provability logic. In T. Bolander,T. Bra¨uner, T. S. Ghilardi, and L. Moss, editors,
Advances in Modal Logic ,volume 9, pages 89–94, London, 2012. College Publications.[4] L. D. Beklemishev. Positive provability logic for uniform reflection princi-ples.
Annals of Pure and Applied Logic , 165(1):82–105, 2014.[5] L. D. Beklemishev, D. Fern´andez-Duque, and J. J. Joosten. On provabilitylogics with linearly ordered modalities.
Studia Logica , 102(3):541–566, 2014.[6] L. D. Beklemishev and D. Gabelaia. Topological completeness of the prov-ability logic
GLP . Annals of Pure and Applied Logic , 164(12):1201–1223,2013.[7] L.D. Beklemishev. Another pathological well-ordering.
Bulletin of SymbolicLogic , 7(4):534–534, 2001.[8] L.D. Beklemishev. On the reflection calculus with partial conservativityoperators. In
WoLLIC 2017 , volume 10388 of
Lecture Notes in ComputerScience , pages 48–67, 2017. 219] L.D. Beklemishev. Reflection calculus and conservativity spectra.
RussianMathematical Surveys , 73(4):569–613, 2018.[10] L.D. Beklemishev. A universal algebra for the variable-free fragment of RC ∇ . In Logical Foundations of Computer Science, International Sympo-sium, LFCS 2018 , volume 10703 of
Lecture Notes in Computer Science ,pages 91–106, Berlin, Heidelberg, 2018. Springer.[11] G. S. Boolos.
The Logic of Provability . Cambridge University Press, Cam-bridge, 1993.[12] E. V. Dashkov. On the positive fragment of the polymodal provability logic
GLP . Mathematical Notes , 91(3-4):318–333, 2012.[13] A. de Almeida Borges and J.J. Joosten. The worm calculus. In G. Bezhan-ishvili, G. D’Agostino, G. Metcalfe, and T. Studer, editors,
Advances inModal Logic , volume 12. College Publications, 2018.[14] S. Feferman. Systems of predicative analysis.
Journal of Symbolic Logic ,29:1–30, 1964.[15] D. Fern´andez-Duque. The polytopologies of transfinite provability logic.
Archive for Mathematical Logic , 53(3-4):385–431, 2014.[16] D. F´ernandez-Duque. Worms and spiders: Reflection calculi and ordinalnotation systems.
Journal of Applied Logics – IfCoLoG Journal of Logicsand their Applications , 4(10):3277–3356, 2017.[17] D. Fern´andez-Duque and J. J. Joosten. Hyperations, Veblen progressionsand transfinite iteration of ordinal functions.
Annals of Pure and AppliedLogic , 164(7-8):785–801, 2013.[18] D. Fern´andez-Duque and J. J. Joosten. The omega-rule interpretation oftransfinite provability logic.
ArXiv , 1205.2036 [math.LO], 2013.[19] D. Fern´andez-Duque and J. J. Joosten. Well-orders in the transfiniteJaparidze algebra.
ArXiv , 1212.3468 [math.LO], 2013.[20] D. Fern´andez-Duque and A. Weiermann. Ackermannian goodstein se-quences of intermediate growth. In
Computability in Europe , 2020.[21] E. Hermo-Reyes and J. J. Joosten. Relational semantics for the TuringSchmerl calculus. In G. Bezhanishvili, G. D’Agostino, G. Metcalfe, andT Studer, editors,
Advances in Modal Logic , volume 12, pages 327–346,London, 2018. College Publications.[22] T. F. Icard III. A topological study of the closed fragment of
GLP . Journalof Logic and Computation , 21:683–696, 2011.[23] K. N. Ignatiev. On strong provability predicates and the associated modallogics.
The Journal of Symbolic Logic , 58:249–290, 1993.2224] G. K. Japaridze.
The modal logical means of investigation of provability .PhD thesis, Moscow State University, 1986. In Russian.[25] Thomas Jech.
Set theory, The Third Millenium Edition, Revised and Ex-panded . Monographs in Mathematics. Springer, 2002.[26] G. Kreisel. Wie die beweistheorie zu ihren ordinalzahlen kam undkommt.
Jahresbericht der Deutschen Mathematiker-Vereinigung , 78:177–224, 1976/77.[27] W. Pohlers.
Proof Theory, The First Step into Impredicativity . Springer-Verlag, Berlin Heidelberg, 2009.[28] D. Schmidt. Built-up systems of fundamental sequences and hierarchies ofnumber-theoretic functions.
Arch. Math. Log. , 18(1):47–53, 1977.[29] S. G. Simpson.