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Ordinal Analysis by Transformations
Henry Towsner
Department of Mathematics, Carnegie Mellon University
Abstract
The technique of using infinitary rules in an ordinal analysis has been one of the most productive developments in ordinal analysis. Unfortunately, one of the most advanced variants, the Buchholz Ωμ-rule, does not apply to systems much stronger than Π^11 -comprehension. In this paper, we propose a new extension of the Ω rule using game-theoretic quantifiers. We apply this to a system of inductive definitions with the strength of a recursively inaccessible ordinal.
Key words: Ordinal analysis, inductive definitions 1991 MSC: 03F05, 03F
1 Introduction
Infinitary inference rules have been a key tool in ordinal analysis since their introduction by Sch¨utte [1]. The appropriate infinitary rule for Peano Arith- metic, the ω rule, is reasonably straightforward—it simply branches over the natural numbers—but suitable infinitary rules for stronger systems are less clear.
The first type proposed, Buchholz’s Ωμ rule [2], branches not over numbers, but over a particular class of derivations. Subsequently, Pohlers proposed the method of local predicativity [3], in which infinitary rules branch over infinite ordinals. Rules branching over ordinals have almost entirely replaced the Ωμ rule, in large part because they led to productive generalizations, culminating in an analysis of Π^12 -comprehension [4], while the Ωμ rule seemed limited to iterated systems of Π^11 -comprehension.
In the method of local predicativity, ordinals are built directly into the system, since they are necessary to even describe the system cut-elimination will take
Email address: [email protected] (Henry Towsner).
Preprint submitted to Annals of Pure and Applied Logic 3 June 2008
place in. This integration with ordinals is different from earlier analyses, in which the cut-elimination process came first and the ordinals could be “read off” from the reduction procedure; in local predicativity, the crucial collapsing step is justified by reference to the properties of the ordinals, which have, naturally, been defined just so as to make this possible. Unfortunately, as the systems get more complex, this leads to the appearance that the proof proceeds by “magic”, obscuring the underlying structure of the argument. This problem isn’t intrinsic to infinitary techniques—the most advanced finitary methods, as in [5],[6], and [7], also require systems defined in terms of ordinals, and face the same problems as a result.
In the author’s opinion, reductions which can be defined independently of ordinals are clearer and have a greater potential for extracting combinatorial consequences. Unfortunately, ordinal based methods have been the only option for going beyond iterations of Π^11 -comprehension. In this paper, we propose an alternate method of analyzing strong subsystems of analysis, based on a “game-theoretic” extension of the Ωμ rule, and apply it to a lightface version of the system μ 2 described by Mollerfeld [8]. The exact strength of this system, as characterized by subsystems of analysis or recursively large ordinals, is not known to the author, but Mollerfeld’s work implies that it is at least as strong as the system with a recursively inaccessible ordinal, although it may be stronger. Systems with the strength of a recursively inaccessible ordinal were first analyzed using finitary methods in the form of (∆^12 − CA) 0 + BI [9] and later using infinitary methods in the form of KP i [10]. For the equivalence between these systems, see [11]. (There is also an analysis using Ωμ rules, [12], but this uses local predicativity-style ordinal indices to obtain sufficiently large iterations of the Ωμ rule.)
Systems of inductive definitions are relatively susceptible to ordinal analysis, so we will extend the particularly elegant analysis of ID<ω given in [13]. We will work with the simplest fixed point operator which can’t be analyzed by that method, namely a fixed point of the form
μxX.A(x, X, μyY.B(x, X, y, Y ))
where Z appears negatively in A(x, X, Z).
Such definitions contain objects of a corecursive character, so it is not surpris- ing that we use the method of corecursion (as described in [14] and [15]) in a key definition.
While this illustrates the method, it by no means exhausts it. Unsurprisingly, since any proof has to exceed methods available in Π^11 -comprehension—which includes recursion along any easily definable well-ordering—it becomes dif- ficult to even describe the iterated form of the method required to analyze stronger fixed point. Even with this limitation, it appears that, at a mini-
rule introduces ¬μB (m, μA) must be easily converted to a proof of ¬μB (m, F ). This would not be true if we used an ordinary Ω rule, which would face the obstacle that the Ω rule for ¬μB (m, μA) does not even necessarily branch over the right domain to become an Ω rule for ¬μB (m, F ).
We resolve both these problems at once by introducing a new type of Ω rule to be the “small” rule introducing ¬μB (m, μA); this rule will branch over proofs of μB (m, F ) for any F. The difficulty is that such derivations may contain inference rules which more widely than is permitted in a small proof (for instance, the introduction of ¬μA when F is μA).
Such inferences will be converted into non-branching inference rules. We will call these inference rules truncated inferences, since rather than encoding the manner in which the original proof derived ¬F [n], they merely note where such a derivation occurred. The Ω rule will then provide, to each derivation of n ∈ μB (m, F ), a derivation of some G[n, F ] from instances of these truncated inferences, as well as an indication, for each truncated inference in the resulting derivation, a source inference in the original derivation.
Fig. 1. Example Transformation
We cannot be finished, because we have thrown away everything above a widely branching inference in the original derivation. In order to recover it, we must provide, for each truncated inference appearing in our derivation of G[n, F ], not only a truncated inference from the source derivation, but also a new Ω rule which will provide, for each possible premise di, a new derivation F(di) in such a way that {F(di)}ι are valid premises for the widely branching inference.
In order to keep all this information in one place, our Ω rule for n ∈ μB (m, F ) will branch, not over derivations, but over sequences of derivations. Given a derivation d of n ∈ μB (m, F ), we divide this derivation into pieces by chopping it at each introduction of some ¬F [k]. We then build up a new derivation coinductively; the bottommost piece, d 0 , is replaced by some F(〈d 0 〉). Each truncated inference θ appearing in F(d 0 ) is traced to some truncated inference in d 0 , which in turn is traced to some introduction of ¬F [k] in d using an
inference rule I. This introduction rule, whatever it is, has some list of premises {dι}; for each dι there is an inference F(〈d 0 , dι〉) which extends F(〈d 0 〉) at θ. By replacing θ in F(〈d 0 〉) with I, taking for each premises ι the extension in F(〈d 0 , dι〉), we obtain a new valid derivation. We then have new truncated inferences which first appeared in F(〈d 0 , dι〉), and the process repeats.
Fig. 2. A derivation is divided into segments, and (the corresponding portion of) the transformation is applied to each segment in turn.
We may formulate this procedure as a game with two players, a Prover and a Transformer. Prover plays first, and must play a derivation of from our sys- tem of small proofs augmented by truncated inferences (which we wall call a truncated proof system). Transformer must play a derivation with appropriate endsequent from the same system (actually, transformer is given a bit more flexibility, for instance, being allowed to use the cut rule), with the additional property that, for each truncated inference in this derivation, transformer must name a source callback inference in Prover’s play. Prover then chooses some truncated inference in Transformer’s play, and plays this truncated inference together with a new derivation. From here, play continues alternating these last two steps. Transformer wins as long as it is possible to provide derivations with the appropriate endsequent relative to what Prover offers (and an addi- tional condition to be described shortly). The Ω rule is simply an encoding of a winning strategy for Transformer. (The ordinary Ω rule may be viewed as the two step version of this game, where Prover is not permitted an additional play after Transformer has gone once.) Any derivation gives a collection of strategies for Prover, and applying the transformation to some derivation is the result of knitting together the results given by Transformer against all the
Definition 3.1. A sequent is a finite set of formulas.
A proof system consists of a set of formal inference symbols (generally denoted by the variable I), and, for each inference symbol:
- A (possibly infinite) set |I| called its arity
- A sequent ∆(I)
- For each ι ∈ |I|, a sequent ∆ι(I)
- A set Eig(I) which is either empty or a singleton {x} where x is a variable not in F V (∆(I)) (in this case we call x the eigenvariable of I)
When we say that a proof system contains an inference rule · · · ∆ι · · · (ι ∈ I) I (^) ∆ !u!
we are declaring I to be an inference symbol with arity I, ∆(I) = ∆, ∆ι(I) = ∆ι, and Eig(I) = {u} (or ∅ if u is omitted). When the arity is finite, we typically list all the premises explicitly.
Definition 3.2. The derivations d of a proof system and the end sequent Γ(d) are defined inductively. If, for each ι ∈ |I|, dι is a derivation and setting Γ := ∆(I) ∪
⋃ ι∈|I| Γ(dι)^ ^ ∆ι(I),^ Eig(I)^ ∩^ F V^ (Γ) =^ ∅^ then^ d^ :=^ I(dι)ι∈|I| is a derivation with Γ(d) := Γ.
If d is a derivation and Γ(d) ⊆ Γ then we write d ` Γ.
Definition 3.3. An expression of the form λx.F is called a predicate, and denoted F. We write F[t] := F (x/t).
3.2 Augmented and Truncated Derivations
We define proof systems with additional rules which serve to mark places where a derivation has been cut off. The rule T runcΓ 7 →Γ,∆ indicates a point where the derivation has been truncated below an inference rule I with ∆(I) = ∆ and
⋃ ι∈|I| Γ(dι)^ ^ ∆ι(I) = Γ.
A CBΥ 7 →∆ inference indicates a point where every branch besides the branch ι of some inference rule I has been cut off, Γ(dι) = Υ and ∆(I)∪Γ(dι)\∆ι(I) = ∆.
Definition 3.4. Let P be a proof system. We define truncated P to consist of P together with inference rules
T runcΓ 7 →Γ,∆ Γ, ∆
We define augmented P to consist of truncated P together with inference rules
CB Υ
We define Θ(d) to be the set of instances of T runc inferences appearing in d.
If θ is a truncated inference T runcΓ 7 →Γ,∆, we set In(θ) := Γ and Out(θ) := ∆.
Note that Θ picks out instances, so it distinguishes two occurrences of the inference rule in different places, even if they have identical parameters.
We will want to be able to talk about systems such as truncated P where P is itself augmented Q; when we speak of truncated inferences in a derivation in augmented P, or refer to Θ(d), we mean to include only those inferences not belonging to P. That is, augmenting and truncating give disjoint unions.
Definition 3.5. We define the exploded derivations of P over Q by induction:
- If d is a derivation in truncated Q and I, E are functions on Θ(d) such that I(θ) is an inference rule from P, In(θ) =
⋃ ι Γ(E(θ, ι))^ ^ ∆ι(I(θ)), Out(θ) = ∆(I(θ)), and each E(θ, ι) is an exploded derivation then 〈d, I, E〉 is an exploded derivation with endsequent Γ(d)
We denote the endsequent of an exploded derivation E by Γ(E). If E = 〈d, I, E〉 is an exploded derivation, we set E 0 := d and call this the main part of the exploded derivation.
Definition 3.6. If 〈d, I, E〉 is an exploded derivation, the unexplosion U(〈d, I, E〉) is given by main induction on E and a side induction on d:
- If d is a T runc inference,
U(〈T runc, I, E〉) := I(θ){U(E(θ, ι))}ι∈|I(θ)|
U(〈J {dι}, I, E〉) := J {U(〈dι, I Θ(dι), E Θ(dι))}ι∈|J |
Definition 3.7. If P, Q are proof systems, we define the explosion EQ(d) of a derivation d in P by:
- If d = I{dι} where I is not an inference of Q,
EQ(d) := 〈T runcΓ(d)\∆(I)→Γ(d), θ 7 → I, (θ, ι) 7 → EQ(dι)〉
- Otherwise d = I{dι} where I is an inference of Q and set, for each ι ∈ |I|, 〈d′ ι, Iι, Eι〉 := EQ(d), and then
EQ(d) := 〈I{d′ ι},
⋃ Iι,
⋃ Eι〉
Proof. The proof is by main induction on E and side induction on T. Let E = 〈d 0 , I, E 0 〉 be given. Then by induction, we produce from any d 0 -well- founded transformation T a d 0 -void transformation T ′. If T is d 0 -void then T ′^ = T. Otherwise, for each d′, θ, T σ 0 _ 〈d′〉, τ 0 _ 〈θ〉 is d 0 -well-founded, and by side IH there is a d 0 -void transformation Tˆd′,θ.
For each d, let d′ σ 0 _ 〈d〉,τ 0 be the result of replacing each θ ∈ Θ(dσ_ 0 〈d〉,τ 0 ) such
that Λσ_ 0 〈d〉,τ 0 (θ) ∈ Θ(d 0 ), with I(Λσ_ 0 〈d〉,τ 0 (θ)) and the premise ι given by T ′ applied to E 0 (Λσ_ 0 〈d〉,τ 0 (θ), ι); this application exists by the main IH.
Then d∗^ := d′ σ 0 _ 〈d 0 〉,τ 0 and Λ := Λσ_ 0 〈d 0 〉,τ 0 Θ(d∗) witness the theorem.
Theorem 3.1. If T is a transformation out of Q from Γ over F with end- sequent Σ and d Γ, Υ for some Υ ⊆ F then there is a derivation T (d) of Σ, Υ.
Proof. Apply the preceding lemma to EQ(d).
Definition 3.10. d′^ broadly extends d if d can be derived from d′^ by replacing subderivations of d′^ ending in callback inferences with truncated inferences. If S is the set of such truncated inferences in d, we say d′^ broadly extends d at S.
Lemma 3.3. Let T be a wellfounded transformation, let {Oi} be a set of operators on derivations, all with the same domain, and for each Oi, let ΛOi be a function with the properties that:
- Each Oi takes wellfounded derivations to wellfounded derivations
- Each Oi preserves extensions in the sense that if d′^ narrowly extends d at θ then Oi(d′) broadly extends Oi(d) at {θ′^ | θ = ΛOi (d)(θ′)}
- For every d in the domain of Oi, ΛOi (d) : Θ(Oi(d)) → Θ(d) with the property that Out(θ) = Out(ΛOi (d)(θ)) and if d′^ In(ΛO(d)(θ)), Υ belongs to the domain then there is an operator Oj such that Oj (d′) In(θ), Υ.
Then each Oi extends to an operator on wellfounded transformations, T 7 → Oi ◦ T , with appropriate domain and range with the property that
(Oi ◦ T )(d) = Oi(T (d))
for any derivation d.
Proof. Follows immediately by applying operators pointwise, using ΛOi to define Oi(Λ).
We call such a system of such operators uniform.
4 The System μ 2
4.1 Language
Definition 4.1. If A(X, x) is a formula, we write A(X) for {x | A(X, x)}; in particular, A(X) ⊆ X means ∀x(A(X, x) → x ∈ X).
As we define our system, we also assign depths to formulas. Depths will be ordinals ≤ ω+ω, although we will immediately restrict ourselves to ω+2. (The use of the ordinal ω + ω is somewhat artificial; we have ω levels corresponding to finitely many iterated inductive definitions, and then three levels above, corresponding to the inaccessible, the negated inaccessible, and an admissible above the inaccessible. The names < I, I, and I + 1 might convey this more clearly.)
Definition 4.2. The language of Lμ 2 is defined as follows:
- 0 is a constant symbol
- S is a unary function constant symbol
- There are infinitely many symbols for variables
- For each n-ary primitive recursive relation, including = and ≤, there is an n-ary predicate constant symbol R
- The logical symbols are ¬, ∧, ∨, ∀, ∃
- If A(x, X) contains no other free variables and contains X positively then μxX.A(x, X) is a unary predicate symbol
- If B(y, Y, Z) contains Y positively and Z negatively and A(x, X, Z) con- tains X positively and Z negatively, and A and B have finite depth then μxX.A(x, X, μyY.B(y, Y, X)) is a unary predicate symbol; we call this a predicate of inaccessible type
The terms are given by:
- 0 is a term
- If t is a term then St is a term
- Each variable is a term
The formulas are given by:
- If R is a symbol for an n-ary primitive recursive relation and for each i ≤ n, ti is a term, then Rt 1... tn is an atomic formula of depth n for any n ≥ 0
- If A(x, X) has depth n and t is a term then t ∈ μxX.A(x, X) is an atomic formula of depth n
- If t is a term then t ∈ μxX.A(x, X, μyY.B(y, Y, X)) is an atomic formula of depth ω
μxX.A(x, X)
∧ A 0 A 1
A 0 ∧A 1 A
0 ∧^ A 1
∧i Ai A 0 ∨A (^1) A 0 ∨ A 1 i ∈ { 0 , 1 }
∧y A(y) ∀xA (^) ∀xA !x!
∨t A(t) ∃xA (^) ∃xA
Cut C^ ¬C C (^) ∅^ Ind
t F (^) ¬F[0], ¬∀x(F[x] → F[Sx]), F[t]
where C is not large A(t, μxX.A(x, X)) Clt∈μxX.A(x,X) t ∈ μxX.A(x, X)
IndμxX.A F (x,X),t ¬(A(F) ⊆ F), t 6 ∈ μxX.A(x, X), F[t]
We say a derivation d belongs to IDn if every formula in every endsequent in d belongs to LIDn.
4.2 Infinitary Derivations
We define an infinitary system μ∞ 2 ; its language is the same language Lμ 2 , but only closed formulas are permitted. This definition will require that a number of weaker systems be defined along the way.
The following, which we will call ID∞ 0 , will be the basis for all the systems we need. Roughly, it is the standard infinitary system for Peano Arithmetic plus a closure rule—but not an induction rule—for μxX.A(x, X) of depth 0.
Definition 4.5. Ax∆^ ∆
where ∆ contains a true primitive recursive formula
∧ A 0 A 1
A 0 ∧A 1 A
0 ∧^ A 1
∨i Ai A 0 ∨A (^1) A 0 ∨ A 1 i ∈ { 0 , 1 }
∧ · · · A(i) · · · (i ∈ N) ∀xA (^) ∀xA
∨n A(n) ∃xA (^) ∃xA n ∈ N
A(n, μxX.A(x, X)) Cln∈μxX.A(x,X) n ∈ μxX.A(x, X)
Cut^ C^ ¬C C (^) ∅
and all formulas have depth 0.
Definition 4.6. If q is a proof and Γ a sequent, ∆Γ q := Γ(q) \ Γ.
The systems ID n∞+1 are defined inductively; as the name suggests, they are essentially the infinitary systems from [13].
Definition 4.7. Given ID n∞ , the language of the system ID n∞+1 is LIDn —that is, formulas with depth ≤ n + 1, and consists of the rules of ID∞ n together with
Ax∆ ∆ where ∆ contains n ∈ μxX.A(x, X), n 6 ∈ μxX.A(x, X) with dp(μxX.A(x, X)) < n + 1
k ∈ μxX.A(x, X)... ∆k q ∈μxX.A(x,X)... (q ∈ |k ∈ μxX.A(x, X)|) Ωk 6 ∈μxX.A(x,X) ∅ where |k ∈ μxX.A(x, X)| is the set of cut-free proofs of ID∞ dp(k∈μxX.A(x,X)) and dp(μxX.A(x, X)) ≤ n, and ∆q(Ωk 6 ∈μxX.A(x,X)) := Υ where q ` k ∈ μxX.A(x, X), Υ
Note that the premise of the Ω rule d defines a function taking proofs of k ∈ μxX.A(x, X) to proofs of Γ(d).
Definition 4.8. Next we define a system μ∞ ω , which extends the union of ID∞ n over n with the closure rule for predicates of inaccessible type.
Note that this doesn’t add any derivations—there’s no way to introduce A(n, μA) since there’s no way to introduce n 6 ∈ μB (μA). We’re including the rule so that it will be present in the extensions we need.
Definition 4.9. The system μ∞ I extends μ∞ ω by the rule
If d ` A(n, F) then en F,A(d) is the derivation
d .. . A(n, F)
dF[n],¬F[n] .. . F[n], ¬F[n] F[n], A(n, F) ∧ ¬F[n] F[n], ¬(A(F) ⊆ F)
or symbolically ∨^ n
¬(A(F)⊆F)
∧
A(n,F)∧¬F[n]
dd¬(F[n]),F[n]
Lemma 5.1. There is a function SUBΠ μxX.A(x,X),F such that if dp(μxX.A(x, X)) < ω and d ` Π(μxX.A(x, X)), Σ is a cut-free proof in ID dp∞(μxX.A(x,X)) then
SUBΠ μxX.A(x,X),F (d) ` Π(F), ¬(A(F) ⊆ F), Σ
is a proof in μ∞.
Proof. By induction on d. We simply proceed up through the proof, adding to Π as we encounter subformulas or new formulas produced by closures rules. A typical case is
B 0 (μxX.A(x, X)) B 1 (μxX.A(x, x)) B 0 (μxX.A(x, X)) ∧ B 1 (μxX.A(x, x))
B 0 (F) B 1 (F)
B 0 (F) ∧ B 1 (F)
where B 0 ∧ B 1 belongs to Π.
The only difficult case is the closure rule, which we handle with the help of e:
A(n, μxX.A(x, X)) n ∈ μxX.A(x, X)
→ (^) A(n, F)
dF[n],¬F[n] .. . F[n], ¬F[n] F[n], A(n, F) ∧ ¬F[n] F[n], ¬(A(F) ⊆ F)
Importantly, we never encounter n 6 ∈ μxX.A(x, X) anywhere; in particular, we do not have to deal with the axiom Axn∈μxX.A(x,X),n 6 ∈μxX.A(x,X).
The full definition is given by
SUBΠ n∈μxX.A(x,X),F (I(dι)ι∈|I|) :=
en F,G (SUBΠ μxX.A∪{∆^0 ((x,XI)}),F (d 0 )) if I = Cln∈μxX.A(x,X)
and n ∈ μxX.A(x, X) ∈ Π
IA(F)
( SUB Π∪{∆ι(I)} μxX.A(x,X),F (dι)ι∈|I|
) if I = IB(μxX.A(x,X)) and B(μxX.A(x, X)) ∈ Π
IA(SUBΠ μxX.A(x,X),F (dι)ι∈|I|) otherwise
Lemma 5.2. Let A(x, X) be a formula. Then there is an operator SUBΠ μxX.A(x,X),F such that if d Π(μxX.A(x, X)), Σ is a cut-free proof in an augmentation of μ∞ <I then SUBΠ μxX.A(x,X),F (d) Π(F), ¬(A(F) ⊆ F), Σ
is a proof in the corresponding augmentation of μ∞. Furthermore, this operator is uniform.
Proof. By induction on d. The proof is essentially the same as in the previ- ous lemma, except that we add an additional case to handle T runc and CB inferences.
SUBΠ μxX.A(x,X),F (I(dι)ι∈|I|) :=
en F (SUBΠ μxX.A∪{∆^0 ((x,XI)}),F (d 0 )) if I = Cln∈μxX.A(x,X)
and n ∈ μxX.A(x, X) ∈ Π
T runcF 7 →Π(F),ΣSUBΠ μxX.A(x,x),F (d 0 ) if I = T runcF 7 →Π(μxX.A(x,X)),Σ CBΠ(F),ΣSUBΠ μxX.A(x,x),F (d 0 ) if I = CBΠ(μxX.A(x,X)),Σ
IA(F)
( SUBΠ μxX.A∪{∆ι((x,XI)}),F (dι)ι∈|I|
) if I = IB(μxX.A(x,X))
and B(μxX.A(x, X)) ∈ Π IA(SUBΠ μxX.A(x,X),F (dι)ι∈|I|) otherwise
Lemma 5.3. Let A(x, X) be a formula. Then there is an operator SUBΠ μxX.A(x,X),F , and a companion Λσ,τ , giving a well-founded transformation from n ∈ μxX.A(x, X) out of μ∞ <I over μxX.A(x, X) positive formulas with endsequent F[n], ¬(A(F) ⊆ F).
Proof. By induction on the length of σ. SUBΠ μxX.A(x,X,G 1 ,Gk ),F (〈d 0 〉, 〈〉) is just
SUBΠ μxX.A(x,X),F (d 0 ) as given by the previous lemma. Given SUBΠ μxX.A(x,X),F (σ, τ ),
SUBΠ μxX.A(x,X),F (σ_〈d〉, τ _〈θ〉) is given by replacing θ in SUBΠ μxX.A(x,X),F (σ, τ )
In any other case, we do nothing:
SUBΠ μxX.A(x,X),F (I(dι)) := IA(SUBΠ μxX.A(x,X),F (dι)ι∈|I|)
Lemma 5.5. If h is a closed μ 2 derivation of ∆ with dp(∆) ≤ ω + 2 then there is a μ∞^ derivation h∞^ so that h∞^ `m Γ(h) for some finite m.
Proof. We define the ·∞^ operation by induction on the proof h:
∀xA(d^0 [n] ∞)n∈N
- (Ind^0 F )∞^ := dF[0],¬F[0]
- (Indn F+1 )∞^ :=
∨n ∃x(F[x]∧¬F[Sx])
∧ F[n]∧¬F[Sn](Ind n F ) ∞d¬F[Sn],F[Sn]
- (Indμ FA ,n)∞^ := Ωn 6 ∈μA Axn∈μA,n 6 ∈μA {SUBn μ∈Aμ,FA (q)} if dp(μA) < ω or has inac- cessible type
- (Indμ FA ,n)∞^ := ¬n 6 ∈μA Axn∈μA,n 6 ∈μA {SUBn μ∈Aμ(FA 1 (,...,F^1 ,...,Fk ))Fk,F^ ) (σ, τ )}σ,τ,F 1 ,...,Fk if dp(μA) ≥ ω, μA(μ 1 ,... , μk) does not have inaccessible type, and the μi are all predi- cates of inaccessible type appearing in A
- Otherwise (Ih 0... hn− 1 )∞^ := Ih∞ 0... h∞ n− 1
6 Cut-Elimination
Definition 6.1. We say that A has
∧ -Form if it is either A 0 ∧ A 1 or ∀xA 0.
We say that A has
∧+ -Form if it has
∧ -Form, is a true primitive recursive formula, or has the form μxX.A(x, X)n. Define
C[k] :=
Ck if C = C 0 ∧ C 1 or C = C 0 ∨ C 1 where k ∈ { 0 , 1 }
A(k) if C = ∀xA or C = ∃xA where k ∈ N
Lemma 6.1. If C is a
∧ -Form then there is a uniform operator J (^) Ck such that whenever d m Γ, C, J (^) Ck (d)m Γ, C[k].
Proof. By induction on d.
J (^) Ck (d) :=
J (^) Ck (dk) if I =
∧ C CBF 7 →Σ,C[k](J (^) Ck (d 0 )) if I = CBF 7 →Σ,C
¬{J (^) Ck ◦ Fq}q if I = ¬{Fq}q I(J (^) Ck (dι))ι∈|I| otherwise
Lemma 6.2. Let rk(C) ≤ m with
∧+ -Form and e m Γ, C. Then there is an operator RC (e, ·) such that whenever dm Γ, ¬C, RC (e, d) `m Γ and such that {RC } ∪ {J (^) Dk}k,D is uniform.
Proof. By induction on d.
RC (e, d) :=
CutC[k]J (^) Ck (e)RC (e, dk) if I =
∨k ¬C e if I = Ax¬C,C
CBF 7 →ΣRC (e, d 0 ) if I = CBF 7 →Σ,¬C ¬{RC ◦ Fq}q if I = ¬{Fq}q
I(RC (e, dι))ι∈|I| otherwise
Lemma 6.3. For each m, there is an operator Em so that whenever d m+1 Γ, Em(d)m Γ and {Em} ∪ {RC }C ∪ {J (^) Dk}k,D is uniform.
Proof. By induction on d.
Em(I(dι)ι∈|I|) :=
RC (Em(d 0 ), Em(d 1 )) if I = CutC , rk(C) = m
and C has
∧+ -Form
R¬C (Em(d 1 ), Em(d 0 )) if I = CutC , rk(C) = m
and ¬C has
∧+ -Form ¬{Em ◦ Fq}q if I = ¬{Fq}q
I(Em(dι))ι∈|I| otherwise
