Proof–theoretic investigations on Kruskal’s theorem
Michael Rathjen
Department of Mathematics, Ohio State University
Columbus, Ohio 43210, USA
Andreas Weiermann
Institut für mathematische Logik und Grundlagenforschung
der Westfälischen Wilhelms–Universität Münster
Einsteinstraße 62, D–W–4400 Münster, Germany
Abstract
In this paper we calibrate the exact proof–theoretic strength of Kruskal’s theorem, thereby
giving, in some sense, the most elementary proof of Kruskal’s theorem.
Furthermore, these investigations give rise to ordinal analyses of restricted bar induction.
Introduction
S.G. Simpson in his article [10], “Nonprovability of certain combinatorial properties of finite
trees”, presents proof–theoretic results, due to H. Friedman, about embeddability properties
of finite trees. It is shown there that Kruskal’s theorem is not provable in ATR0 . An exact
description of the proof–theoretic strength of Kruskal’s theorem is not given. On the assumption
that there is a bad infinite sequence of trees, the usual proof of Kruskal’s theorem utilizes the
existence of a minimal bad sequence of trees, thereby employing some form of Π11 comprehension.
So the question arises whether a more constructive proof can be given. The need for a more
elementary proof of Kruskal’s theorem is especially felt due to the fact that this theorem figures
prominently in computer science, because it is the main tool for showing that sets of rewrite
rules are terminating (see [3], p. 258, where this challenge is offered).
Our paper gives a complete proof–theoretic characterization of Kruskal’s theorem in terms
of ordinal notation systems, subsystems of second order arithmetic, and subsystems of Kripke–
Platek set theory.
The paper is divided into eleven parts.
In Section 1 we introduce an ordinal notation system TA , which represents the Ackermannordinal (see [2] for a definition) in a natural way. It is shown that within ACA0 , Kruskal’s
theorem implies the well–foundedness of TA .
The reversal of the latter implication constitutes the content of Section 2. Given a bad
sequence of trees, we show how to produce effectivly a strictly descending sequence of ordinals
in TA of the same length.
1
The equivalence of Kruskal’s theorem with the well–foundedness of TA then provides an
upper bound for the order–types of simplification orderings, since Kruskal’s theorem can be
used to proof the well–foundedness of these orderings.
For α ≤ ω let T(α) be the set of finite trees T such that every vertex of T has less than
α immediate successors (so KT (ω) is just Kruskal’s theorem). T(α) is quasi-ordered by the
natural tree-embeddability relation. Let KT (α) be the statement ”T(α) is well-quasi-ordered.”.
We also show that ∀nKT (n) and KT (ω) are equivalent over ACA0 .
In [10], p. 99, it is stated that Kruskal’s theorem is provable in the formal system T :=
ACA0 +Π12 −BI. The investigations of this paper were mainly prompted by this remark. It turns
out that this is not quite true. Indeed, ACA0 +KT (ω) proves the uniform Π11 reflection principle
of the latter theory, RF NΠ11 (T ), and is therefore a stronger theory. The remainder of this paper
is devoted to proving T := ACA0 + Π12 − BI 0 ∀nKT (n) and ACA0 ` KT (ω) ↔ RF NΠ11 (T ).
Sections 4–10 pursue the ordinal analysis of the system ACA0 + Π12 − BI, thereby showing
that the order type of TA is the proof–theoretic ordinal of the latter system. The methods
used here are perfectly general in that they provide a general framework for analyzing all the
theories ACA0 + Π1n − BI for n ≥ 2. Using results from [5], we also establish the proof–theoretic
equivalence of ACA0 + Π1n − BI and KP ω − + Πn − F oundation for all n ≥ 2, where KP ω −
stands for Kripke-Platek set theory including infinity but with foundation restricted to sets.
An effective version of the ordinal analysis of ACA0 + Π12 − BI is sketched in Section 11,
finally establishing ACA0 ` KT (ω) ↔ RF NΠ11 (T ).
1
An ordinal notation system
Firstly, we need some ordinal–theoretic background. Let On be the class of ordinals. Let
AP := {ξ ∈ On : ∃η ∈ On[ξ = ω η ]} be the class of additive principal numbers and let E := {ξ ∈
On : ξ = ω ξ } be the class of ε–numbers which is enumerated by the function λξ.εξ .
We write α =N F ω β + δ if α = ω β + δ and either, δ = 0 and β < α, or δ = ω δ1 + · · · + ω δk
with β ≥ δ1 ≥ . . . ≥ δk and k ≥ 1.
Note that by Cantor’s normal form theorem, for every α ∈
/ E ∪ {0}, there are uniquely
β
determined ordinals β and δ such that α =N F ω + δ.
Let Ω := ℵ1 . For any α < εΩ+1 we define the set EΩ (α) which consists of the ε–numbers
below Ω which are needed for the unique representation of α in Cantor normal form recursively
as follows:
1. EΩ (0) := EΩ (Ω) := ∅,
2. EΩ (α) := {α}, if α ∈ E ∩ Ω,
3. EΩ (α) := EΩ (β) ∪ EΩ (δ) if α =N F ω β + δ.
Let α∗ := max(EΩ (α) ∪ {0}).
We define sets of ordinals C(α, β), Cn (α, β), and ordinals ϑα by main recursion on α < εΩ+1
and subsidiary recursion on n < ω (for β < Ω) as follows.
2
(C1) {0, Ω} ∪ β ⊆ Cn (α, β),
(C2) γ, δ ∈ Cn (α, β) ∧ ξ =N F ω γ + δ =⇒ ξ ∈ Cn+1 (α, β),
(C3) δ ∈ Cn (α, β) ∩ α =⇒ ϑδ ∈ Cn+1 (α, β),
S
(C4) C(α, β) := {Cn (α, β) : n < ω},
(C5) ϑα := min{ξ < Ω : C(α, ξ) ∩ Ω ⊆ ξ ∧ α ∈ C(α, ξ)}.
Lemma 1.1 ϑα is defined for every α < εΩ+1 .
Proof: Let β0 := α∗ + 1. Then α ∈ C(α, β0 ) via (C1) and (C2). Since the cardinality of
C(α, β) is less than Ω there exists a β1 < Ω such that C(α, β0 ) ∩ Ω ⊂ β1 . Similarly there exists
for each βn < Ω (which is constructed recursively) a βn+1 < Ω such that C(α, βn ) ∩ Ω ⊆ βn+1 .
Let β := sup{βn : n < ω}. Then α ∈ C(α, β) and C(α, β)∩Ω ⊂ β < Ω. Therefore ϑα ≤ β < Ω.2
Lemma 1.2
1. ϑα ∈ E,
2. α ∈ C(α, ϑα),
3. ϑα = C(α, ϑα) ∩ Ω, and ϑα ∈
/ C(α, ϑα),
4. γ ∈ C(α, β) ⇐⇒ γ ∗ ∈ C(α, β),
5. α∗ < ϑα,
6. ϑα = ϑβ =⇒ α = β,
7. ϑα < ϑβ ⇐⇒ (α < β ∧ α∗ < ϑβ) ∨ (β < α ∧ ϑα ≤ β ∗ ),
8. β < ϑα ⇐⇒ ω β < ϑα.
Proof: 1.) and 8.) issue from closure of ϑα under (C2).
2.) and 3.) follow from Lemma 1.1 and the definition of ϑα.
4.): If γ ∗ ∈ C(α, β), then γ ∈ C(α, β) by (C2). On the other hand,
γ ∈ Cn (α, β) =⇒ γ ∗ ∈ Cn (α, β) is easily seen by induction on n.
5.): α∗ ∈ C(α, ϑα) holds by 4.). As α∗ < Ω, this implies α∗ < ϑα by 3.).
6.): Suppose, aiming at a contradiction, that ϑα = ϑβ and α < β. Then C(α, ϑα) ⊆
C(β, ϑβ); hence α ∈ C(β, ϑβ) ∩ β by 2.); thence ϑα = ϑβ ∈ C(α, ϑβ), contradicting 3.).
7.): Suppose α < β. Then ϑα < ϑβ implies α∗ < ϑβ by 5.). If α∗ < ϑβ, then α ∈ C(β, ϑβ);
hence ϑα ∈ C(β, ϑβ); thus ϑα < ϑβ. This shows
(a) α < β =⇒ (ϑα < ϑβ ⇐⇒ α∗ < ϑβ).
By interchanging the roles of α and β, and employing 5.), one obtains
(b) β < α =⇒ (ϑα < ϑβ ⇐⇒ ϑα ≤ β ∗ ).
2
(a) and (b) yield 7.).
The Ackermann ordinal is denoted in this context by ϑΩω .
3
Definition 1.1 Inductive definition of a set OT (ϑ) of ordinals and a natural number Gϑ α for
α ∈ OT (ϑ).
1. 0, Ω ∈ OT (ϑ), Gϑ 0 := Gϑ Ω := 0,
2. α =N F ω β + δ ∧ β, δ ∈ OT (ϑ) =⇒ α ∈ OT (ϑ), Gϑ α := max{Gϑ β, Gϑ δ} + 1,
3. α = ϑα1 ∧ α1 ∈ OT (ϑ) =⇒ ϑα1 ∈ OT (ϑ), Gϑ α := Gϑ α1 + 1.
Observe that according to Lemma 1.2.1.) and 1.2.6.), the function Gϑ is well-defined. Each
ordinal α ∈ OT (ϑ) has a unique normal form using the symbols 0, Ω, +, ω, , ϑ. Furthermore,
if for α, β ∈ OT (ϑ), represented in their normal form, we were to decide α < β, we could do
this by deciding α0 < β0 for ordinals α0 and β0 that appear in these representations and, in
addition, satisfy Gϑ α0 + Gϑ β0 < Gϑ α + Gϑ β. This follows from Lemma 1.2 7.) and the recursive
procedure for comparing ordinals in Cantor normal form. So we come to see the following fact.
Lemma 1.3 After a straightforward coding in the natural numbers, we may consider hOT (ϑ), <
OT (ϑ)i as a primitive recursive ordinal notation system.
S
Lemma 1.4
1. OT (ϑ) = {C(α, 0) : α < εΩ+1 },
2. OT (ϑ) ∩ α = α for α ∈ OT (ϑ) ∩ Ω.
From now on, we presume an effective coding of hOT (ϑ), < OT (ϑ)i in the natural numbers,
so that the latter structure can be dealt with in ACA0 (actually in primitive recursive arithmetic).
Of course, the well–foundedness of hOT (ϑ), < OT (ϑ)i is not provable in ACA0 .
Next, we recall some basic definitions from [10]. A finite tree T is a finite partial order
hT, ≤T i such that,
1. (∃r ∈ T )(∀t ∈ T )[t 6= r → r ≤T t ∧ t 6≤T r],
2. (∀s ∈ T )(∀t ∈ T )(∀u ∈ T )[t ≤T s ∧ u ≤T s → t ≤T u ∨ u ≤T t].
For a finite tree T = hT, ≤T i and t, u ∈ T we denote the ≤T -infimum of t and u by t ∧T u.
The uniquely determined ≤T -minimal element is called the root of T .
A finite tree T = hT, ≤T i is embeddable into a finite tree U = hU, ≤U i if there exists an
one-to-one (embedding-) function f : T → U such that f (t ∧T u) = f (t) ∧U f (u) for all t, u ∈ T.
For a ∈ T let T a := {b ∈ T : a ≤T b} and T a = hT a , ≤T T a i. An immediate subtree U of
T has the form T a where a is an immediate ≤T -successor of the root of T . Every immediate
subtree U of T is embeddable into T .
Theorem 1.1 (Kruskal) For any ω-sequence hTi : i < ωi of finite trees there exist indices i and
j such that i < j < ω and Ti is embeddable into Tj .
In [10] it is shown that Kruskal’s theorem implies the well–foundedness of Γ0 within ACA0 .
This proof can be extended to also yield the well–foundedness of ϑΩω from Kruskal’s theorem in
ACA0 . The proof utilizes a normal form for ordinals < ϑΩω . This normal form can be computed
primitive recursively.
We require some notation. Let Ω · 0 := 0 and Ω · (n + 1) := Ω · n + Ω. Let Ωn · 0 := 0. If
β = ω β1 + · · · + ω βk and β1 ≥ . . . ≥ βk we set Ωn · β := ω Ω·n+β1 + · · · + ω Ω·n+βk .
4
Proposition 1.1 Let α ∈ E ∩ ϑΩω . Then there exists a unique n < ω and unique ordinals
α0 , . . . , αn < α such that α = ϑ(Ωn · αn + · · · + Ω0 · α0 ), and αn 6= 0 if n 6= 0.
Definition 1.2 For any α ≤ ω let T(α) be the set of trees T such that any vertex in T has less
than α immediate successors. We use KT (α) to abbreviate that T(α) is well–quasi–ordered.
For α ∈ OT (ϑ) let WF (α) stand for “< restricted to {β ∈ OT (ϑ) : β < α} is well–founded”.
For ordinals α, β we denote by α ⊕ β the (commutative) natural sum of α and β and by α ⊗ β
the (commutative) natural product of α and β. These operations are defined as follows: Let
α ⊕ 0 := 0 ⊕ α := α and α ⊗ 0 := 0 ⊗ α := 0.
Now suppose α = ω δ0 + · · · + ω δn ≥ δ0 ≥ . . . ≥ δn , n ≥ 0 and β = ω δn+1 + · · · + ω δm ≥ δn+1 ≥
. . . ≥ δm , m ≥ n + 1. Then
α ⊕ β := ω δπ(0) + · · · + ω δπ(m) ,
where π is a permutation of {0, . . . , m} such that δπ(i) ≥ δπ(j) if i < j ≤ m. We define α ⊗ β to
be
(ω δ0 ⊕δn+1 ⊕ · · · ⊕ ω δ0 ⊕δm ) ⊕ · · · ⊕ (ω δn ⊕δn+1 ⊕ · · · ⊕ ω δn ⊕δm ).
Theorem 1.2 ACA0 ` (∀n < ω)KT (n) → W F (ϑΩω )
Proof. We reason within ACA0 . We shall define a primitive recursive mapping
o : T(ω) −→ {α ∈ OT (ϑ) : α < ϑΩω }
by recursion on the number of elements of a tree T , | T |. If T consists only of its root, then
let o(T ) = 0. Otherwise the root of T has finitely many immediate successors a0 , . . . , ak . Let
Ti be the immediate subtree of T determined by ai . Since | Ti | < | T |, we may assume that
αi := o(Ti ) is already defined. We may further assume that α0 ≥ . . . ≥ αk . We set o(T ) = α0
if k = 0, o(T ) = α1 ⊕ α0 if k = 1 (⊕ being defined in Definition 1.2), o(T ) = ω α0 if k = 2 and
ω α0 > α0 , o(T ) = ω α0 +1 if k = 2 and ω α0 = α0 and o(T ) = ϑα0 if k = 3. To deal with the case
k ≥ 4 we introduce some auxiliary notation. For finite sequences of ordinals we define
hβ0 , . . . , βm i <l hγ0 , . . . γn i ⇐⇒
max{β0 , . . . , βm } ≤ max{γ0 , . . . γn }
∧ 2 ≤ m < n ∨ [2 ≤ m = n ∧ (∃j < m)[βj < γj ∧ (∀k < j)βk = γk ] .
For a finite sequence hβ0 , . . . , βm i (m ≥ 1) of countable ordinals let
ϑ(hβ0 , . . . , βm i) := ϑ(Ωm · (1 + βm ) + Ωm−1 · βm−1 + · · · + Ω0 · β0 ).
Then, by Lemma 1.5 below, hβ0 , . . . , βm i <l hγ0 , . . . , γn i implies ϑ(hβ0 , . . . , βm i) < ϑ(hγ0 , . . . , γn i).
Now we can complete the definition of o(T ) for k ≥ 4. Let
S := {hβπ(0) , . . . , βπ(i) i : 1 ≤ i ≤ k and π is a permutation of {0, . . . , i}}.
Let hβ0 , . . . , βi0 i be the (k−3)-th element of S with respect to <l and let o(T ) = ϑ(hβ0 , . . . , βi0 i).
Note that this assignment of ordinals to trees is weakly increasing, i.e, o(Ta ) ≤ o(T ) if Ta is an
immediate subtree of T . This is due to the fact, that for β0 , . . . , βn < Ω we have β0 , . . . , βn <
ϑ(Ωn · βn + · · · + Ω0 · β0 ).
Now we define (for technical reasons) a set of ordinal representations OT 0 ⊆ ϑΩω as follows.
5
1. 0 ∈ OT 0 .
2. γ = ω α + β & α, β ∈ OT 0 ⇒ γ ∈ OT 0 .
3. α0 , . . . , αn ∈ OT 0 & card{α0 , . . . , αn } = n + 1 ⇒ ϑ(Ωn · αn + · · · Ω0 · α0 ) ∈ OT 0 ,
where card(X) stands for the number of elements of a finite set X.
Then the order type of OT 0 is equal to ϑΩω . Moreover the well-foundedness of ϑΩω follows
(provably in ACA0 ) from the well-foundedness of OT 0 . For, let h : ϑΩω → OT 0 be defined by
h(0) := 0, h(ω α + β) := ω h(α) + h(β) and
h(ϑ(Ωn · αn + · · · + Ω0 · α0 )) := ϑ(Ωn (ω · h(αn ) + n) + · · · + Ω0 · (ω · h(α0 ) + 0)).
Then γ < δ < ϑΩω implies h(γ) < h(δ) as can be readily verified.
By induction on Gϑ α one easily verifies that for α ∈ OT 0 there is a tree T such that o(T ) = α.
For α ∈ E one has to employ Proposition 1.1.
Given an embedding of trees f : T1 −→ T2 , where o(T1 ), o(T2 ) ∈ OT 0 , we claim that o(T1a ) ≤
f (a)
o(T2 ) for each a ∈ T1 . The proof is by induction on | T1a |; it just springs from the above
considerations and the following observation. Let ξ0 < . . . , ξm < Ω, η0 < . . . < ηn < Ω and
g : {ξ0 , . . . , ξm } → {η0 , . . . , ηn } be a one to one mapping such that g(ξi ) ≥ ξi for 1 ≤ i ≤ m.
Then
Ωn · ξπ(n) + · · · + Ω0 · ξπ(0) ≤ Ωn · g(ξπ(n) ) + · · · + Ω0 · g(ξπ(0) )
holds for every permutation π : {1, . . . , m} → {1, . . . , m}.
If now hαk : k < ωi were an infinitely descending sequence of ordinals from OT 0 , then
we would get a corresponding sequence of finite trees hTk : k < ωi with o(Tk ) = αk for each
k < ω. Since sup{ϑ(Ωn + · · · + Ω0 ) : n < ω} = ϑΩω , there would exist n0 such that for all
k < ω, Tk ∈ T(n0 ). Therefore, by KT (n0 ), we could pick i < j < ω such that Ti ≤ Tj .
Hence o(Ti ) = αi ≤ o(Tj ) = αj , contradicting the assumption that hαk : k < ωi is an infinitely
descending sequence of ordinals.
2
In the next Section we will frequently draw on the following result (we already did in the previous
Theorem).
Lemma 1.5 Suppose α0 , . . . , αm , β0 , . . . , βn ∈ OT (ϑ) ∩ Ω. Assume Ωn · βn + · · · + Ω0 · β0 <
∗ ). Then
Ωm · αm + · · · + Ω0 · α0 and max(β0∗ , . . . , βn∗ ) ≤ max(α0∗ , . . . , αm
ϑ(Ωn · βn + · · · + Ω0 · β0 ) < ϑ(Ωm · αm + · · · + Ω0 · α0 ).
t
u
Proof. This follows from lemma 1.2 7.).
2
An elementary proof of Kruskal’s theorem
From now on we argue in ACA0 (in fact, RCA0 would suffice).
Let TA := {α ∈ OT (ϑ) : α < ϑΩω } and <TA := (< OT (ϑ)) TA . Then hTA , <TA i is
a primitive recursive ordinal notation system for the Ackermann ordinal. We introduce some
6
more notation. A quasi–order X is an ordered pair hX, ≤i where X is a countable set and ≤ is a
binary, reflexive and transitive relation on X. X = ∅ is explicitly allowed. A (finite or countably
infinite) sequence ~x = hx0 , x1 , . . .i of elements in X is called bad if there are no indices i and j
such that i < j < length(~x) and xi ≤ xj . In particular, the empty sequence is bad.
X is a well–quasi–order if all bad sequences in X are finite. Let Bad(X) be the set of all
finite bad sequences in X. A reification of X into α ∈ OT (ϑ) is a mapping f : Bad(X) → α + 1
such that f (~x_ y) < f (~x) for every ~x ∈ Bad(X) and every y ∈ X such that ~x_ y ∈ Bad(X).
Lemma 2.1 The following is provable in ACA0 : If f is a reification of X into α and α is
well–founded, then X is a well–quasi–order.
Proof: An infinite bad sequence in X would give rise to an infinite descending chain below α+1.2
We introduce some more terminology. Let X = hX, ≤i be a quasi-order. For ~x ∈ Bad(X) let
X~x := {y ∈ X : ~x_ y ∈ Bad(X)} and X~x := hX~x , ≤ X~x i.
Let X0 = hX0 , ≤0 i and X1 = hX1 , ≤1 i be quasi–orders.
We define quasi–orders X0 ⊕ X1 and X0 ⊗ X1 as follows. The domain of X0 ⊕ X1 is the disjoint
union X0 ] X1 of X0 and X1 . Therefore X0 ] X1 consists of ordered pairs h0, x0 i and h1, x1 i
where x0 ∈ X0 and x1 ∈ X1 . X0 ] X1 is quasi–ordered by a relation ≤0 ⊕ ≤1 as follows:
hi, ui ≤0 ⊕ ≤1 hj, vi : ⇐⇒ i = j ∧ u ≤i v.
When we write X0 ] X1 , we assume (without loss of generality) that X0 and X1 are disjoint and
we will identify x0 ∈ X0 with h0, x0 i and x1 ∈ X1 with h1, X1 i. The domain of X0 ⊗ X1 is the
cartesian product X0 × X1 of X0 and X1 . X0 × X1 is quasi-ordered by the relation ≤0 ⊗ ≤1 as
follows:
hx0 , x1 i ≤0 ⊗ ≤1 hx00 , x01 i : ⇐⇒ x0 ≤0 x00 ∧ x1 ≤1 x01 .
For a given quasi–order X = hX, ≤i we define the quasi–order X<ω as follows:
The domain of X<ω consists of the set X <ω of all finite sequences in X. X <ω is quasi–ordered
by a relation ≤<ω as follows:
hx1 , . . . , xm i ≤<ω hx01 , . . . , x0n i
if and only if there exists a sub–sequence 1 ≤ i1 < . . . < im ≤ n such that xl ≤ x0il for every
1 ≤ l ≤ m.1
If X0 and X1 are well–quasi–orders then X0 ⊕ X1 , X0 ⊗ X1 and X<ω
are well–quasi–orders,
0
too. According to [8], this can be shown in ACA0 (especially, Higman’s lemma is provable in
ACA0 ).
A finite tree T with labels in a quasi-order X = hX, ≤i is an ordered triple hT, ≤T , lT i such
that hT, ≤T i is a finite tree and lT : T → X. If T = hT, ≤T , lT i and U = hU, ≤U , lU i are finite
trees with labels in a quasi-order X = hX, ≤X i we say that T is embeddable into U if there exists
an embedding-function f : T → U such that lT (t) ≤X lU (f (t)) for every t ∈ T . Let T (X) be the
set of finite trees with labels in X and let ≤T (X) be the corresponding embeddability relation.
Let T(X) := hT (X), ≤T (X) i.
1
To be precise, the empty sequence is the bottom element with respect to the ordering ≤<ω .
7
Theorem 2.1 (Kruskal) T(X) is a well-quasi-order for every well-quasi-order X.
We want to show that the existence of a reification of X into α for appropriate implies the
existence of a reification of T(X) into ϑ(Ωω · α). This will imply Kruskal’s theorem by taking
the domain of X to be a singleton, i.e., α = 1. For technical reasons we introduce the following
terminology, which is due to Schmidt [6].
Definition 2.1 Let Xi = hX i , ≤i i (i = 0, . . . , n) be pairwise disjoint quasi–orders and let
α0 , . . . , αn ordinals such that 0 < α0 < . . . < αn ≤ ω. Let X := X0 ⊕ . . . ⊕ Xn . Let
0
X . . . Xn
T
α0 . . . α n
be the set of all finite trees T = hT, ≤T , lT i in T (X) such that for every vertex t ∈ T , if the label
lT (t) of t is in X i , then t has strictly fewer than αi immediate successors in T . Let ≤T (X,~
~ α) be
0
n
X ... X
the restriction of the tree-embeddability-relation to T
and let
α0 . . . αn
0
0
X . . . Xn
X . . . Xn
T
:= hT
, ≤T (X,~
~ α) i.
α0 . . . αn
α0 . . . α n
An ordinal ρ is said to be additively closed if α + β < ρ holds whenever α, β < ρ. The first three
ordinals that are additively closed are 0, 1, ω. The additively closed ordinals > 0 are exactly
the ordinals of the form ω δ for some δ. From the latter it immediately follows that the product
of two additively closed ordinals is additively closed, too. Also note that every ε-number is
additively closed.
Theorem 2.2 The following is provable in ACA0 : Let Xi (i = 0, . . . , n) be pairwise disjoint
quasi–orders Let 0 < α0 < . . . < αn ≤ ω and β0 , . . . , βn ∈ OT (ϑ) ∩ Ω. Suppose that for each
i ∈ {0, . . . , n}, fi is a reification of X i into βi , where βi is additively closed and, moreover, the
range of fi consists entirely of additively closed ordinals. Then there exists a reification of
0
X . . . Xn
T
α0 . . . αn
into ϑ(Ωαn · βn + . . . + Ωα0 · β0 ).
The proof requires some preparations. Let X0 , . . . , Xn be names, that means appropriate Gödelnumbers, for pairwise disjoint quasi–orders X0 , . . . , Xn and let 0 < α0 < . . . < αn ≤ ω. Let
0
X . . . Xn
Comp
α0 . . . α n
be the least collection (of names for quasi–orders) which is closed under the following rules:
1. For every i ≤ n:
i
X ∈ Comp
is a name for Xi .
8
X0 . . . Xn
α0 . . . α n
2. For every i ≤ n and every x~i ∈ Bad(Xi ) :
Xix~i
X0 . . . Xn
α0 . . . αn
∈ Comp
is a name for Xix~i .
3. If
0
r
Y , . . . , Y ∈ Comp
X0 . . . Xn
α0 . . . αn
are names for Y0 , . . . , Yr then
l
X0 . . . Xn
α0 . . . αn
l
⊕{Y : l ≤ r}, ⊗{Y : l ≤ r} ∈ Comp =
are names for ⊕{Yl : l ≤ r} and ⊗ {Yl : l ≤ r}, respectively.
4. If
X ∈ Comp
X0 . . . Xn
α0 . . . α n
is a name for X then
X
<ω
∈ Comp
X0 . . . Xn
α0 . . . αn
is a name for X<ω .
5. If
0
Y ,...,Y
m
∈ Comp
X0 . . . Xn
α0 . . . αn
are names for pairwise disjoint quasi–orders Y0 , . . . , Ym and
0 < γ0 < . . . < γm ≤ ω and γm ≤ αn , then
0
0
Y . . . Ym
X . . . Xn
T
∈ Comp
γ0 . . . γ n
α0 . . . α n
is a name for
T
Y0 . . . Ym
γ0 . . . γm
.
For the remainder of this section we will assume that Xi (i = 0, . . . , n) are pairwise disjoint
quasi–orders and that we have ordinals β0 , . . . , βn ∈ OT (ϑ)∩Ω and reifications fi (i ∈ {0, . . . , n})
of X i into βi , where βi is additively closed and, moreover, the range of fi consists entirely of
additively closed ordinals. The reason for this requirement will become clear in the proof of case
two of Lemma 2.3 below.
9
Definition 2.2 Let X0 , . . . , Xn benames for the quasi–orders X0 , . . ., Xn and let 0 < α0 <
X0 . . . Xn
. . . < αn ≤ ω. For each X ∈ Comp
let o(X) be defined recursively as follows:
α0 . . . αn
1. o(Xi ) := βi .
2. o(Xix~i ) := fi (x~i ).
3. o ⊕ {Yl : l ≤ r} := ⊕{o(Yl ) : l ≤ r} and o ⊗ {Yl : l ≤ r} := ⊗{o(Yl ) : l ≤ r}, where
⊕ and ⊗ featuring in the right hand sides of these equations refer to the natural ordinal
sum and the natural ordinal product, respectively, introduced in Definition 1.2.
4. o(X<ω ) := ϑ(o(X)).
0
Y . . . Ym 5. o T
:= ϑ Ωγm · o(Ym ) + · · · + Ωγ0 · o(Y0 ) .
γ0 . . . γm
Definition 2.3 Let 0 < α0 < . . . < αn ≤ ω. For every
0
X . . . Xn
X ∈ Comp
α0 . . . αn
we define the complexity number c(X) recursively as follows:
1. c(Xi ) := 0,
2. c(Xix~i ) := 0,
3. c(⊕{Yl : l ≤ r}) := c(⊗{Yl : l ≤ r})
:= max{c(Yl ) : l ≤ r} + 1,
4. c(X<ω ) := c(X) + 1,
0
Y . . . Ym 5. c T
:= max{c(Yl ) : l ≤ m} + 1
α0 . . . αm
Definition 2.4 Let X0 = hX0 , ≤0 i and X1 := hX1 , ≤1 i be two quasi–orders.
A function e : X0 → X1 is called a quasi–embedding, if e(x0 ) ≤1 e(x00 ) implies x0 ≤0 x00 for
every x0 , x00 ∈ X.
Lemma 2.2
1. Let X0 = hX0 , ≤0 i, X1 = hX1 , ≤1 i and X2 = hX2 , ≤2 i be quasi–orders and
e0 : X0 → X1 , e1 : X1 → X2 be quasi–embeddings. Then e1 ◦ e0 : X0 → X2 is a
quasi–embedding.
2. Let X0 = hX0 , ≤0 i, X1 = hX1 , ≤1 i be quasi-orders, e : X0 → X1 be a quasi–embedding
and f : X1 → δ be a reification of X1 into δ. Then f and e induce the reification f ◦ e of
X0 into δ.
10
Definition 2.5 Let X = hX, ≤i be a quasi–order and let x ∈ X.
LX (x) := {y ∈ X : ¬ x ≤ y}. LX (x) is quasi–ordered by the restriction of ≤ to LX (x).
Let X(x) := hLX (x), ≤ LX (x)i.
Note that a quasi–order X is a well–quasi–order if X(x) is a well–quasi–order for every x ∈ X .
Theorem 2.2 will follow from the next lemma. (Our proof of this lemma is partly based on [6].)
Lemma 2.3 Let 0 < α0 < . . . < αn ≤ ω,
X ∈ Comp
X0 . . . Xn
α0 . . . αn
be a name for a quasi–order X = hX, ≤i, and x ∈ X. Then there exists a name
0
X . . . Xn
x
X ∈ Comp
α0 . . . αn
for a quasi–order Xx = hX x , ≤x i and a quasi–embedding e(X, x) : LX (x) → X x such that
o(Xx ) < o(X).
We show how the theorem follows from this lemma. Let
0
X . . . Xn
hs0 , . . . , sk i ∈ Bad T
α0 . . . αn
and define recursively (through the use of Lemma 2.3)
f (hs0 , . . . sk i) := o . . . T
X0 . . . X n
α0 . . . αn
!tk
t0 !
...
,
X0 . . . Xn
, e0 := e(T0 , t0 ) , and, for i < k, ei+1 := e(Ti , ti ) ◦ ei ,
α 0 . . . αn
= ei+1 (si+1 ) and Ti+1 := Ttii . Then this function f is a reification of
0
X . . . Xn
T
into ϑ Ωαn · βn + . . . + Ωα0 · β0 .
α0 . . . αn
where t0 := s0 , T0 := T
ti+1
Proof of Lemma 2.3 by induction on c(X). Our proof strategy is to define e(X, x) by primitive
recursion in previously defined functions.
Case 1: c(X) = 0.
i
i
Assume X = X~ixi and ~xi ∈ Bad(Xi ). Then LX i (x) = Xh~
xi .
xi ,xi since x ∈ X~
~
xi
Then e(X, x) := id LX i (x) is quasi–embedding of X(x) into Xih~xi ,xi . Put Xx := Xih~xi ,xi and let
~
xi
Xx be a name for Xx . Then o(Xx ) = fi (h~xi , xi) < fi (~x) = o(Xih~xi i ).
Case 2: X = ⊕{Yl : l ≤ r}, X = ⊕{Yl : l ≤ r} or X = (Y)<ω . In these cases the assertion
follows by arguments employed in [8]. The proof is actually similar to the proof of Case 3
but much simpler. However, we will present more details to clarify the role of the standing
assumption that the βi are additively closed and the range of fi consists of additively closed
11
ordinals. Mainly due to distributivity of ⊗ over ⊕, i.e., X0 ⊗ (X2 ⊕ X3 ) being isomorphic to
(X0 ⊗X1 )⊕(X0 ⊗X2 ) and o(X0 ⊗(X2 ⊕X3 )) = o((X0 ⊗X1 )⊕(X0 ⊗X2 )), every X of either form
X = ⊕{Yl : l ≤ r} or X = ⊕{Yl : l ≤ r} is a name of a quasi-order isomorphic to a quasi-order
with name
M
{Yi0 ⊗ . . . ⊗ Yini : i ≤ m}
L
such that o(X) = o( {Yi0 ⊗ . . . ⊗ Yini : i ≤ m}), c(Yij ) < c(X) and o(Yij ) is additively
closed. This is due to the fact that the ‘atomic’ names of Definition 2.3 (introduced in the first
two clauses) get assigned additively closed ordinals and that every ordinal of the form ϑα is
additively closed.
L i0
Now assume x ∈
{Y ⊗ . . . ⊗ Y ini : i ≤ m}. Without loss of generality suppose x =
(x0 , . . . , xn0 ) ∈ Y 00 ⊗ . . . ⊗ Y 0n0 . Then put
Xx :=
((Y00 )x0 ⊗ Y01 ⊗ . . . ⊗ Y0n0 )
⊕ (Y00 ⊗ (Y01 )x1 ⊗ . . . ⊗ Y0n0 )
..
.
⊕ (Y00 ⊗ Y01 ⊗ . . . ⊗ (Y0n0 )xn0 )
⊕ (Y10 ⊗ Y11 ⊗ . . . ⊗ Y1n1 )
..
.
⊕ (Ym0 ⊗ Ym1 ⊗ . . . ⊗ Ymnm ).
By the induction hypothesis, o((Y0j )xj ) < o(Y0j ), and thus
o(((Y00 )x0 ⊗ Y01 ⊗ . . . ⊗ Y0n0 ) ⊕ . . . ⊕ (Y00 ⊗ Y01 ⊗ . . . ⊗ (Y0n0 )xn0 )) < o(Y00 ⊗ Y01 ⊗ . . . ⊗ Y0n0 )
since the latter ordinal, being a product of additively closed ordinals, is additively closed. As a
result, o(Xx ) < o(X). It is also clear how to construct the quasi-embedding e(X, x) : LX (x) → X x
utilizing the inductively available quasi-embeddings e(Y0j , xj ) : LY 0j (xj ) → Y 0j (see [8]).
Case 3:
X=T
Y0 . . . Ym
γ0 . . . γm
.
Let X be the domain of X and let x ∈ X. Then x is a finite tree with labels in ⊕{Yl | l ≤ m}.
At this stage of the proof we employ a subsidiary induction on the number of vertices (elements)
of x.
Subcase 3.1: Assume x is a tree consisting only of a root with a label y ∈ Y i . By the
main induction hypothesis we can pick a quasi–embedding e(Yi , y) : LY i (y) → (Y i )y such that
o((Yi )y ) < o(Yi ). The quasi-embedding e(Yi , y) induces a natural quasi–embedding of
. . . Yi (y) . . .
T
...
γi
...
. . . (Yi )y . . .
into T
=: Xx .
...
γi
...
12
(Here the
are those
“. . .” parts
which remain unchanged.) Since X(x) is trivially quasi-embeddable
i
. . . Y (y) . . .
into T
we get a quasi-embedding from X(x) into Xx . The ordinal of the
...
γi
...
name of Xx is strictly less than o (X) by Lemma 1.5.
Subcase 3.2: Assume now that x is a tree with root–label y ∈ Y i where the root has exactly
N < γi immediate subtrees x0 , . . . , xN −1 ∈ X.
Assume first that N = γk for some k < i. By the subsidiary induction hypothesis we get
quasi–embeddings e(X, xj ) : LX (xj ) → X xj such that o (Xxj ) < o (X).
By our main induction hypothesis we can pick a quasi–embedding
i y
e(Y, y) : LY i (y) → (Y
that o ((Yi )y ) < o (Yi ). L) such
i
x
<ω
0
Let Z := Y ⊗
(X ) ⊗ . . . ⊗ (Xxj )<ω : j < N
and let Z = hZ, ≤Z i be the corresponding quasi–order. Set
. . . Yk ⊕ Yi ⊕ Z . . . (Yi )y . . .
x
X := T
.
...
γk
...
γi
...
Then Xx is a name for a quasi–order Xx and o(Xx ) < o(X) holds by Lemma 1.2,(7).
If N 6= γk for all k < i, put
. . . Yi ⊕ Z . . . (Yi )y . . .
x
,
X := T
...
N
...
γi
...
where the new column has to be inserted at the place which is determined by the ordering of
{γ0 , . . . , γi , N }.
We shall focus on the case N = γk since the other case is similar. We construct a quasi–
embedding e(X, x) of X(x) into Xx by primitive recursion from e(X, x0 ), . . . , e(X, xN −1 ) and
e(Y, y).
Let z ∈ LX (x). We define e(X, x)(z) by recursion on the number of vertices of z.
Assume first that z is a tree consisting only of a root which carries a label v ∈ Yj .
If j 6= i define e(X, x)(z) := z. Then e(X, x)(z) ∈ X x .
If v ∈ Y i , then v ∈ Y k ] Y i ] Z. So let e(X, x)(z) := z ∈ X x .
Assume now that z is a tree with root-label v ∈ Y j and z0 , . . . , zl−1 are the immediate subtrees, where l < αj . We can assume inductively that e(X, x)(z0 ),. . .,e(X, x)(zl−1 ) are defined.
If j 6= i or l < N let e(X, x)(z) be the tree with root–label v and the immediate subtrees
e(X, x)(z1 ), . . . , e(X, x)(zl−1 ). Then e(X, x)(z) ∈ X x .
Now assume j = i and l ≥ N .
If v ∈ LY i (y) let e(X, x)(z) be the tree with root label e(Y, y)(v) and the immediate subtrees
e(X, x)(z1 ), . . . e(X, x)(zl−1 ).
Assume also y ≤Y i v, where ≤Yi is the quasi–ordering on Y i . Let ≤X be the quasi–ordering
on X. Since z ∈ LX (x) and y ≤Y i v,
hx0 , . . . xN −1 i ≤<ω
X hz0 , . . . , zl−1 i
does not hold. (Otherwise we would have x ≤X z.) So there exists a minimal s < N such that
<ω
hx0 , . . . xs−1 i ≤<ω
hz0 , . . . , zl−1 i.
X hz0 , . . . , zl−1 i does not hold. Then hx0 , . . . xs−2 i ≤X
13
Let j0 < l be minimal such that x0 ≤X zj0 . Let j1 < l minimal such that j0 < j1 and x1 ≤ zj1 .
Let finally js−2 < l be minimal such that js−3 < js−2 and xs−2 ≤ zjs−2 . Then zj0 , . . . , zjs−2 ∈ X,
hz0 , . . . , zj0 −1 i ∈ LX (x0 )<ω , hzjs−3 +1 , . . . zjs−2 −1 i ∈ LX (xs−2 )<ω , and hz(js−2 )+1 , . . . , zl−1 i ∈
LX (xs−1 )<ω . Define e(X, x)(z) to be the tree determined by the immediate subtrees e(X, x)(zj0 ),. . .,e(X, x)(zjs−2 ) and the root labeled with
hv, hhe(X, x)(z0 ), ..., e(X, x)(zj0 −1 )i, ..., he(X, x)(z(js−2 )+1 ), ..., e(X, x)(zl−1 )iii
which is an element of Y i ] Z. Then e(X, x)(z) ∈ X x since s − 1 < γk .
Next, we show that e(X, x)(z) ≤x e(X, x)(z 0 ) implies z ≤X z 0 by induction on the sum of
vertices of z and z 0 . If e(X, x)(z) is embeddable into an immediate subtree ẑ of e(X, x)(z 0 ), then
ẑ has the form e(X, x)(z 00 ) for some immediate subtree z 00 of z 0 . Then z ≤x z 00 ≤x z 0 by the
induction hypothesis.
Now we assume that e(X, x)(z) is embeddable into e(X, x)(z 0 ) and that e(X, x)(z) is not
embeddable into an immediate subtree of e(X, x)(z 0 ). Then the root-label r of e(X, x)(z) is less
than or equal to the root-label r0 of e(X, x)(z 0 ) with respect to the appropriate quasi-ordering.
Thus, if
r = hv, hhe(X, x)(z0 ), ..., e(X, x)(zj0 −1 )i, ..., he(X, x)(z(js−2 )+1 ), ..., e(X, x)(zl−1 )iii
and
0
r0 = hv 0 , hhe(X, x)(z00 ), ..., e(X, x)(zj0 0 −1 )i, ..., he(X, x)(z(j
0
0
0
s −2
0
)+1 ), ..., e(X, x)(zl0 −1 )ii,
then s = s0 , v ≤Y i v 0 ,
he(X, x)(z0 ), ..., e(X, x)(zj0 −1 )i (≤x )<ω he(X, x)(z00 ), ..., e(X, x)(zj0 0 −1 )i, ...,
0
and finally
0
he(X, x)(z(js−2 )+1 ), ..., e(X, x)(zl−1 )i (≤x )<ω he(X, x)(z(j
0
0
s −2
0
)+1 ), ..., e(X, x)(zl0 −1 )i.
Furthermore, there exists a permutation π of {0, . . . , s − 2} such that e(X, x)(zj )
0
0
≤x e(X, x)(zπ(j)
) for j ≤ s − 2. Therefore, by the inductive assumption, zj ≤X zπ(j)
for j ≤ s − 2.
By combining the above results, we obtain a one-to-one mapping ρ : {0, . . . , l−1} → {0, . . . , l0 −1}
0
such that zi ≤x zl(i)
holds for 0 ≤ i ≤ l − 1. Since v ≤Y i = v 0 , we conclude z ≤X z 0 .
t
u
Corollary 2.1 ACA0 ` ∀nKT (n) ↔ KT (ω) ↔ WF (ϑΩω ).
Proof. Clearly KT (ω) implies ∀nKT (n) and, by Theorem 1.2, we know that ∀nKT (n) implies
WF (ϑΩω ) on the basis of ACA0 .
Now assume WF (ϑΩω ). Let X0 be a quasi-order
0 with
exactly one element e. Then the
X
quasi-order of finite trees can be identified with T
. Also note that the function e 7→ 0 is
ω
a reification of X0 into 1 which
meets
the requirements of Theorem 2.2. Thus by said Theorem
X0
we obtain a reification of T
into ϑ(Ωω · 1), showing that WF (ϑΩω ) implies KT (ω). 2
ω
14
3
A comparison between two ordinal notation systems for the
Howard-Bachmann-ordinal
We have just seen that the ordinal notation system for the Howard-Bachmann-ordinal which
is based on the function ϑ is an appropriate tool for the ordinal analysis of Kruskal’s theorem.
For a perspicious proof-theoretic analysis of Π12 bar induction we need another concept for
representing the Howard-Bachmann-ordinal namely the ψ–function which is due to Buchholz.
See, for example, [1] for a definition.
Let Ω(1, ω) := Ωω and Ω(n + 1, ω) := ΩΩ(n,ω) . We are going to show that ϑ(Ω(n, ω)) =
ψ(Ω(n + 1, ω)) is true for every natural number n. This technical result will be needed for
comparing the proof-theoretic strength of Kruskal’s theorem and Π12 − BI over ACA0 . (This
Section is very technical and may be skipped at first reading.)
Definition 3.1 Inductive definition of sets of ordinals Cn (α), C(α), and of ordinals ψα by main
recursion on α and side recursion on n < ω.
1. {0, Ω} ⊆ Cn (α),
2. β, γ ∈ Cn (α) =⇒ ω β + γ ∈ Cn+1 (α),
3. β ∈ C(β) ∩ α ∩ Cn (α) =⇒ ψβ ∈ Cn+1 (α),
S
4. C(α) := {Cn (α) : n < ω}.
ψα := min{ξ : ξ 6∈ C(α)}.
The following Lemmata can be gathered from [1] or [4].
Lemma 3.1 ψα < Ω.
Lemma 3.2
1. ψα = C(α) ∩ Ω,
2. α ∈ C(α) ∩ β =⇒ ψα < ψβ,
3. α, β < ψγ =⇒ ω α + β < ψγ.
Definition 3.2 Inductive definition of a set T (ψ) of ordinals and a natural number Gψ α for
α ∈ OT (ψ).
1. {0, Ω} ⊆ OT (ψ), Gψ 0 := Gψ Ω := 0,
2. α = α1 +· · ·+αn , α > α1 ≥ . . . ≥ αn , α1 , . . . , αn ∈ OT (ψ)∩AP =⇒ α ∈ OT (ψ), Gψ α :=
max{Gψ α1 , . . . , Gψ αn } + 1,
3. α = ω α1 , α > α1 , α1 ∈ OT (ψ) =⇒ α ∈ OT (ψ), Gψ α := Gψ α1 + 1,
4. α = ψα1 , α1 ∈ C(α1 ), α1 ∈ OT (ψ) =⇒ α ∈ OT (ψ), Gψ α := Gψ α1 + 1.
Lemma 3.3
1. OT (ψ) = C(εΩ+1 ),
15
2. OT (ψ) ∩ Ω ⊂ ψεΩ+1 ,
3. OT (ψ) ∩ α = α for α ≤ ψεΩ+1 .
To see that OT (ψ) may be considered as a primitive recursive ordinal notation system we must
be able to decide the relation α ∈ C(β) for α, β ∈ OT (ψ). For this purpose we introduce the
auxiliary concept of coefficient sets.
Definition 3.3 Inductive definition of a set of ordinals Kα for α ∈ OT (ψ).
1. K0 := KΩ := ∅,
2. K(α1 + · · · + αn ) := Kα1 ∪ . . . ∪ Kαn ,
3. K(ω α1 ) = Kα1 ,
4. Kψα1 := Kα1 ∪ {α1 }.
Let kα := max(Kα ∪ {0}) and hα := kα + ω α (cf. [4]).
Lemma 3.4 α ∈ OT (ψ) =⇒ (Kα < β ⇐⇒ α ∈ C(β)).
Lemma 3.5
kα∗ ,
1. α ∈ OT (ψ) =⇒ α∗ , kα ∈ OT (ψ), Gψ α∗ ≤ Gψ α, Gψ kα ≤ Gψ α, kα =
2. α ∈ OT (ψ) ∧ kα < α =⇒ ψα ∈ OT (ψ).
3. α ∈ OT (ψ) =⇒ ψhα ∈ OT (ψ),
Lemma 3.6 α, β ∈ OT (ψ) =⇒ (α∗ < ψβ ⇐⇒ kα < β).
Definition 3.4 Recursive definition of α̂ for α ∈ OT (ϑ)
1. 0̂ := 0, Ω̂ := Ω,
2. α1 +\
· · · + αn := αˆ1 + · · · + αˆn ,
c1 ,
α1 := ω α
d
3. ω
d1 := ψhc
4. ϑα
α1 .
Lemma 3.7
1. α ∈ OT (ϑ) =⇒ α̂ ∈ OT (ψ),
2. α, β ∈ OT (ϑ) =⇒ (α < β ⇐⇒ α̂ < β̂),
c∗ = α̂∗ .
3. α ∈ OT (ϑ) =⇒ α
Proof by a simultaneous induction on Gϑ α + Gϑ β. We consider only the nontrivial case for 2.).
Let α = ϑα1 and β = ϑβ1 . Assume first that α1 < β1 and α1∗ < ϑβ1 . Then we see α
c1 < βb1 and
∗
c∗ < ψhβˆ1 by the induction hypothesis. Thus kc
α
α1 < hβb1 and therefore hc
α1 = kc
α1 + ω αc1 =
1
c
∗
d1 < ϑβ
d1 . Assume now α1 > β1 and ϑα1 ≤ β ∗ .
kc
α1 + α
c1 < k βb1 + ω β1 = hβb1 . This yields ϑα
1
∗
∗
∗
∗
d
c
b
d
b
b
d
d
Then ϑα1 ≤ β1 . Since β1 ≤ hβ1 we see β1 ≤ ψhβ1 by Lemma 3.6. Thus ϑα1 < ϑβ1 .
16
Corollary 3.1
1. ϑα ≤ ψhb
α,
2. ϑ(Ω(n, ω)) ≤ ψ(Ω(n + 1, ω)).
Proof: This follows from Lemma 3.7 and the well known fact that f β ≥ β holds for any strictly
monotonic ordinal function f .
t
u
We now define the reverse embedding α 7→ α of OT (ψ) into OT (ϑ). Of course the trivial setup
ψα 7→ ϑα will induce an order-preserving mapping of OT (ψ) into OT (ϑ). But this yields only
b and this is not an equality in the interesting cases.
ψα ≤ ϑα ≤ ψhα
Lemma 3.8 For each α ∈ OT (ψ) \ {0} there exist uniquely determined α1 , . . . , αn
∈ OT (ψ) and β1 , . . . , βn ∈ OT (ψ) such that α = Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ), α1 > . . . > αn
and β1 , . . . , βn < Ω.
Definition 3.5 α =Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn )
:⇐⇒ α = Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ) ∧ α1 > . . . > αn ∧ β1 , . . . , βn < Ω.
Lemma 3.9 α =Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn )
=⇒ α∗ = max{α1∗ , . . . , αn∗ , β1∗ , . . . , βn∗ }.
Definition 3.6 Inductive definition of α for α ∈ OT (ψ).
1. 0 := 0, Ω := Ω,
2. α1 + · · · + αn := α1 + · · · + αn ,
3. ω α1 := ω α1 ,
4. ψ0 := ϑ0,
5. ψα := ϑ(Ωαn + · · · + ϑ(Ωα2 + ϑ(Ωα1 + β1 ) + β2 ) · · · + βn ),
if α =Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ).
Lemma 3.10 α ∈ OT (ψ) =⇒ α ∈ OT (ϑ).
The following Lemma is of crucial importance.
Lemma 3.11
1. If α, β ∈ OT (ψ), then (α < β ⇐⇒ α < β).
2. If α =Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ) + Ωαn+1 (1 + βn+1 ) + · · · +
Ωαm (1 + βm ) ∈ OT (ψ),
if β =Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ) + Ωγn+1 (1 + δn+1 ) ∈ OT (ψ),
and if α ∪ Kα < β,
then ϑ(Ωαm + · · · + ϑ(Ωαn+1 + ϑ(Ωαn + (· · · ) + βn ) + βn+1 ) + · · · + βm ) < ϑ(Ωγn+1 +
ϑ(Ωαn + (· · · ) + βn ) + δn+1 ),
3. If β =Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ) + Ωγn+1 (1 + δn+1 )
α < β and α ∈ OT (ψ),
then α∗ < ϑ(Ωγn+1 + ϑ(Ωαn + (· · · ) + βn ) + δn+1 ).
17
∈
OT (ψ),
Proof by main induction on 2Gψ α + 2Gψ β and side induction on m − n.
1.): The critical case is α = ψα0 , β = ψβ0 , Kα0 < α0 , Kβ0 < β0 and α < β. Let
α0 :=Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ) + Ωαn+1 (1 + βn+1 ) + · · · + Ωαm (1 + βm )
and
β0 =Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ) + Ωγn+1 (1 + δn+1 ) + · · · + Ωγr (1 + δr ),
where Ωαn+1 (1 + βn+1 ) < Ωγn+1 (1 + δn+1 ).
Let
γ0 :=Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ) + Ωγn+1 (1 + δn+1 ).
Then α0 < γ0 and Kα0 < γ0 . By the main induction hypothesis applied to 2.) we see that
ψα0 < ϑ(Ωγn+1 + ϑ(Ωαn + (· · · ) + βn ) + δn+1 )
< ϑ(Ωγn+2 + ϑ(Ωγn+1 + (· · · ) + δn+1 ) + δn+2 )
≤ ...
≤ ψβ0 .
2.): The assertion holds for m − n = 0. We assume first that γn+1 > αn+1 is true. Let
α0 :=Ω−N F Ωα1 (1 + β1 ) + · · · + Ωαn (1 + βn ) + Ωαn+1 (1 + βn+1 ) + · · · + Ωαm−1 (1 + βm−1 ).
Then max{α0 , kα0 } ≤ max{α, kα} < β. We conclude by the side induction hypothesis ϑ(Ωαm−1 +
· · · + ϑ(Ωαn+1 + ϑ(Ωαn + (· · · ) + βn ) + βn+1 ) + · · · + βm−1 )
< ϑ(Ωγn+1 + ϑ(Ωαn + (· · · ) + βn ) + δn+1 ) =: γ 0 . The main induction hypothesis for 1.) yields
∗
γn+1 > αn+1 > . . . > αm . The main induction hypothesis for 3.) yields βm < γ 0 and αm ∗ < γ 0 .
Therefore
ϑ(Ωαm + · · · + ϑ(Ωαn+1 + ϑ(Ωαn + (· · · ) + βn ) + βn+1 ) + · · · + βm ) < γ 0 .
Assume now γn+1 = αn+1 and βn+1 < δn+1 . If αm < αn+1 = γn+1 the assertion follows as
before. Assume now αm = αn+1 = γn+1 and βn+1 < δn+1 . The main induction hypothesis for
1.) yields βn+1 < δn+1 . Thus
ϑ(Ωαm + (· · · ) + βn+1 ) < ϑ(Ωγn+1 + (· · · ) + δn+1 ).
3.): The claim is true for α ∈ {0, Ω}. If α is not of the form ψα0 with Kα0 < α0 then the
assertion follows immediately from the main induction hypothesis for 3.). Let α = ψα0 and
Kα0 ∪ {α0 } = Kα < β. The main induction hypothesis for 2.) yields the assertion.
2
Corollary 3.2 ψ(Ω(n + 1, ω)) = ϑ(Ω(n, ω)).
Remark: Let θ be the collapsing function defined in [7]. Then θ(Ω(n, ω))0 = ϑ(Ω(n, ω)) for
every n ∈ N.
18
4
Majorization Relations and Fundamental Functions
In this section we restate for convenience the concepts of majorization relations and fundamental
functions which are developed in [1]. These concepts are needed for carrying through the ordinal
analysis of the restricted bar induction schemata. All missing proofs can be found in [1].
Definition 4.1
1. α µ β means α < β and for all δ, η:
α ≤ δ ≤ min{β, η}, δ, µ ∈ C(η) =⇒ α ∈ C(η).
2. α β : ⇐⇒ α 0 β,
3. α β : ⇐⇒ (α β ∨ α = β).
Lemma 4.1
1. α β =⇒ α µ β,
2. α < β =⇒ α α β,
3. α < β < γ & α µ γ =⇒ α µ β,
4. 0 < β < ε0 =⇒ α α + β,
5. α < β < Ω =⇒ α β,
6. α β =⇒ α + 1 β.
Lemma 4.2 α µ β, β µ γ =⇒ α µ γ.
Lemma 4.3 α µ β, β < ω γ+1 =⇒ ω γ + α µ ω γ + β.
Corollary 4.1 ω α · n ω α · (n + 1).
Lemma 4.4 α µ β =⇒ ω α · n µ ω β .
Lemma 4.5 α µ β, µ ∈ C(α), β ∈ C(β) implies:
1. α ∈ C(α),
2. ψα µ ψβ.
Corollary 4.2 α = α0 + 1 ∈ C(α) =⇒ α0 ∈ C(α) & ψα0 ψα.
Definition 4.2 A function f with the domain dom(f ) ⊆ OT (ψ) is said to be a fundamental
function if the following holds.
F1. If β ∈ dom(f ) and α < β, then α ∈ dom(f ) and f (α) α f (β).
F2. If β ∈ dom(f ) and f (0) ≤ δ < f (β), then there is an α < β such that f (α) ≤ δ < f (α + 1)
and f (α) f (α + 1).
F3. If α ∈ dom(f ) and f (α) ∈ C(η), then α ∈ C(η).
19
Lemma 4.6 If f is a fundamental function and α ∈ dom(f ), then α ≤ f (α).
Definition 4.3 Let Idβ be the function with domain dom(Idβ ) := {α ∈ OT (ψ) : α ≤ β} and
Idβ (α) := α for all α ∈ dom(Idβ ).
Lemma 4.7 Idβ is a fundamental function.
Definition 4.4 Let f be a fundamental function.
1. Let ω γ + f be the function with domain dom(ω γ + f ) := {α ∈ dom(f ) : f (α) < ω γ+1 } and
(ω γ + f )(α) := ω γ + f (α) for all α ∈ dom(ω γ + f ).
2. Let ω f be the function with domain dom(ω f ) := dom(f ) and (ω f )(α) := ω f (α) for all
α ∈ dom(ω f ).
3. Let ψf be the function with domain dom(ψf ) := {α ∈ dom(f ) : α < Ω, f (α) ∈ C(f (α))}
and (ψf )(α) := ψ(f (α)) for all α ∈ dom(ψf ).
Lemma 4.8 If f is a fundamental function, then also ω γ + f, ω f and ψf are fundamental
functions.
Lemma 4.9 If f is a fundamental function with α, Ω ∈ dom(f ), α < β = ψf (α) and f (α) f (Ω), then also f (β) f (Ω).
Corollary 4.3 If f is a fundamental function with Ω ∈ dom(f ), then f (ψ(f (0))) f (Ω).
5
Ordinal analysis of restricted bar induction
In this Section we determine the proof–theoretic strength of the subsystems of second order
arithmetic ACA0 +Π12 −BI which is based on arithmetical comprehension and Π12 bar induction.
From this result we shall gather the unprovability of Kruskal’s Theorem in ACA0 + Π12 − BI
as well as the proof–theoretic equivalence of ACA0 + Π12 − BI and Kripke–Platek set theory
plus infinity axiom but with foundation restricted to set–theoretic Π2 formulas. Our device for
dealing with Π12 bar induction will be Buchholz’ Ω–rule (cf. [1]).
To set the context, we fix some notations. The language of second order arithmetic, L2 ,
consists of free numerical variables a, b, c, d, . . ., bound numerical variables x, y, z, . . ., free set
variables U, V, W, . . . , bound set variables X, Y, Z, . . ., the constant 0, a symbol for each primitive
recursive function, and the symbols = and ∈ for equality in the first sort and the elementhood
relation, respectively. The numerical terms of L2 are built up in the usual way; r, s, t, . . .
are syntactic variables for them. Formulas are obtained from atomic formulas (s = t), (s ∈
U ) and negated atomic formulas ¬ (s = t), ¬ (s ∈ U ) by closing under ∧, ∨ and quantification
∀x, ∃x, ∀X, ∃X over both sorts; so we stipulate that formulas are in negation normal form.
The classes of Π12 – and Σ1n –formulas are defined as usual (with Π10 = Σ10 = ∪{Π0n : n ∈ N}).
¬A is defined by de Morgan’s laws; A → B stands for ¬A ∨ B. All theories in L2 will be
assumed to contain the axioms and rules of classical two sorted predicate calculus, with equality
in the first sort. In addition, it will be assumed that they comprise the system ACA0 . ACA0
20
contains all axioms of elementary number theory, i.e. the usual axioms for 0, 0 (successor), the
defining equations for the primitive recursive functions, the induction axiom
∀X [0 ∈ X ∧ ∀x(x ∈ X → x0 ∈ X) → ∀x(x ∈ X)],
and all instances of arithmetical comprehension
∃Z ∀x[x ∈ Z ↔ F (x)],
where F (a) is an arithmetic formula, i.e. a formula without set quantifiers.
For a 2-place relation ≺ and an arbitrary formula F (a) of L2 we define
P rog(≺, F ) := (∀x)[∀y(y ≺ x → F (y)) → F (x)] (progressiveness)
T I(≺, F ) := P rog(≺, F ) → ∀xF (x) (transfinite induction)
WF (≺) := ∀X T I(≺, X) :=
∀X (∀x[∀y(y ≺ x → y ∈ X) → x ∈ X] → ∀x[x ∈ X]) (well-foundedness).
Let F be any collection of formulas of L2 . For a 2–place relation ≺ we will write ≺ ∈ F, if ≺
is defined by a formula Q(x, y) of F via x ≺ y := Q(x, y). In addition, we will use the notation
≺ ∈ F − to express that Q(x, y) contains no set parameters.
Definition 5.1
1. (Π1n − BI)0 denotes ACA0 extended by the Π1n bar induction scheme, i.e.
all formulas of the form
WF (≺) → T I(≺, F ),
where ≺ ∈ Π10 and F ∈ Π1n .
2. (Π1n − BI)−
0 denotes the modification, where ≺ is required to contain no set variables.
3. (Π1n − BI)− and (Π1n − BI) denote the corresponding theories augmented by the scheme
F (0) ∧ ∀x(F (x) → F (x0 )) → ∀xF (x)
for every L2 –formula F (a).
For technical purposes it it convenient to reformulate (Π12 − BI)0 in a sequent calculus.
Definition 5.2 The sequent calculus version of (Π12 − BI)0 derives finite sets of formulas denoted by Γ, Θ, Ξ, Λ, ... . The intended meaning of Γ is the disjunction of all formulas of Γ. We
use the notation Γ, A for Γ ∪ {A} and Γ, Ξ for Γ ∪ Ξ. Let Γ be an arbitrary finite set of formulas.
The axioms of (Π12 − BI)0 are:
(Ax1) Γ, A, ¬ A for every prime formula A.
W
(Ax2) Γ, ∆ if ∆ is a set of prime and negated prime formulas such that ∆ (i.e. the disjunction of all formulas of ∆) is a tautological consequence of the equality axioms and the defining
equalities of the primitive recursive functions.
21
(IA) Γ, ∀X [0 ∈ X ∧ ∀x(x ∈ X → x0 ∈ X) → ∀x(x ∈ X)].
(Π10 − CA) Γ, ∃Z ∀x[x ∈ Z, ↔ F (x)] where F is arithmetic.
The logical rules of inferences are:
(∧)
` Γ, A and ` Γ, B =⇒ ` Γ, A ∧ B,
(∨)
` Γ, Ai =⇒ ` Γ, A0 ∨ A1
(∀1 )
` Γ, F (a) =⇒ ` Γ, ∀xF (x),
(∀2 )
` Γ, F (U ) =⇒ ` Γ, ∀XF (X),
(∃1 )
` Γ, F (t) =⇒ ` Γ, ∃xF (x),
(∃2 )
` Γ, F (U ) =⇒ ` Γ, ∃XF (X),
if i ∈ {0, 1},
(Cut) ` Γ, A and ` Γ, ¬ A =⇒ ` Γ,
where in (∀1 ) and (∀2 ) the free variable a, respectively U is not to occur in the conclusion.
The only non–logical rule of inference of (Π12 − BI)0 is the Π12 bar induction rule:
` Γ, W F (≺) and ` Γ, ∃y ∃Y ∀Z[y ≺ a ∧ ¬B(y, Y, Z)], ∃Z B(a, U, Z)
=⇒ ` Γ, ∃ZB(b, V, Z),
where ≺ and B are arithmetic; and a and U are not to occur in
Γ, ∀x ∀Y ∃ZB(x, Y, Z).
We shall conceive axioms as inferences with an empty set of premises. The minor formulas
(m.f.) of an inference are those formulas which are rendered prominently in its premises. The
principal formulas (p.f.) of an inference are the formulas rendered prominently in its conclusion.
(Cut) has no principal formula. So any inference has the form
(∗) For all i < k ` Γ, Ξi =⇒ ` Γ, Ξ
(0 ≤ k ≤ 2), where Ξ consists of p.f. and Xi is the set of m.f. in the i–th premise. The
formulas in Γ are called side formulas (s.f.) of (∗).
Derivations of (Π12 − BI)0 are defined inductively, as usual. D, D0 , D0 , . . . range as syntactic
variables over (Π12 − BI)0 derivations. All this is completely standard, and we refer to [9] for
notions like ”length of a derivation D” (abbreviated by | D |), ”last inference of D”, ”direct
subderivation of D ”. We write D ` Γ to mean that D is a derivation of Γ.
The most important feature of sequent calculi is cut–elimination. Our sequent calculus
1
(Π2 − BI)0 admits cut–elimination concerning cuts whose cut formula is neither a principal
formula of a non–logical rule of inference nor a principal formula of an axiom. This is a general
phenomenon which will be exploited next. To state this fact concisely, let us introduce a measure
of complexity, gr(A), the grade of a formula A:
1. gr(A) = 0 if A is a prime formula or negated prime formula.
2. gr(∀XF (X)) = gr(∃XF (X)) = ω if F (U ) is arithmetic.
22
3. gr(A ∧ B) = gr(A ∨ B) = max{gr(A), gr(B)} + 1.
4. gr(∀xH(x)) = gr(∃xH(x)) = gr(H(0)) + 1.
5. gr(∀XG(X)) = gr(∃XG(X)) = gr(G(U )) + 1,
if G is not arithmetic.
The cut–rank, %(D), of a derivation D is also defined by induction: Let Di , i < k, be
the direct subderivations of D. If the last inference of D is (Cut) with m.f. A and ¬A let
%(D) := sup{gr(A) + 1, sup{%(Di ) : i < k}}. Otherwise, let %(D) := sup{%(Di ) : i < k}. By
k
(Π12 − BI)0 % Γ we mean that there is a derivation D ` Γ in (Π12 − BI)0 such that | D |≤ k and
%(D) ≤ %.
Theorem 5.1 (Cut–elimination) Let 2k0 := k and 2km+1 := 2l where l := 2km .
If (Π12 − BI)0
k
ω+n+1
Γ then (Π12 − BI)0
p
ω+1
Γ where p = 2kn .
Proof. Observe that gr(A) < ω + 1 holds for every p.f. A of an axiom or non–logical rule of
− BI)0 . So the result follows by the standard cut–elimination procedure for sequent calculi
(cf. [9]).
t
u
(Π12
Proposition 5.1 (Π12 − BI)0 proves WF (≺) → T I(≺, F ) for any ≺ ∈ Π10 and F ∈ Π12 .
Proof. Let F (x) be ∀Y ∃ZB(x, Y, Z) with B arithmetic.
Then
(Π12 − BI)0 ` ∀y [y ≺ a → ∀Y ∃ZB(y, Y, Z)], ∃y ∃Y ∀Z[y ≺ a ∧ ¬B(y, Y, Z)]
and
(Π12 − BI)0 ` ∃Y ∀Z¬B(a, Y, Z), ∃ZB(A, U, Z)
yield
(Π12 − BI)0 ` C(a), ∃y ∃Y ∀Z[y ≺ a ∧ ¬B(y, Y, Z)], ∃ZB(a, U, Z)
with
C(a) ≡ ∀y[y ≺ a → ∀Y ∃ZB(y, Y, Z)] ∧ ∃Y ∀Z¬B(a, Y, Z).
Here we assume that a and U are ”fresh” variables. By (∃1 ), we get
(Π12 − BI)0 ` ∃xC(x), ∃y∃Y ∀Z[y ≺ a ∧ ¬B(y, Y, Z)], ∃ZB(a, U, Z).
(1)
Using the Π12 bar induction rule on (1) and
(Π12 − BI)0 ` ¬WF (≺), WF (≺),
we obtain
(Π12 − BI)0 ` ¬ WF (≺), ∃xC(x), ∃ZB(a, U, Z);
23
(2)
thence, by (∀2 ), (∀1 ), and (∨), this becomes
(Π12 − BI)0 ` ¬ WF (≺) ∨ (∃xC(x) ∨ ∀x∀Y ∃ZB(x, Y, Z)).
Therefore (Π12 − BI)0 ` WF (≺) → T I(≺, F ).
(Note that A → B ≡ ¬ A ∨ B.)
t
u
T ∗.
(Π12
In the next section we shall embed
− BI)0 into an infinitary calculus
To handle this
with optimal ordinal bounds, we have to resort to very well behaved derivations.
Definition 5.3 Let ∃Σ12 be the collection of formulas of the form
∃y∃X∀Y A(y, X, Y ), where A(0, U, V ) is arithmetic.
A (Π12 − BI)0 derivation D ` Γ is said to be nice if %(D) ≤ ω + 1, and, for every Σ12 -formula
A that serves as the minor formula of an inference (∃1 ) in this derivation, either A is an element
of Γ or A is never a side formula of an inference in D.
Note that if none of the formulas occuring in a derivation D0 has a ∃Σ12 subformula, then D0 is
automatically nice. We say that a formula C is essentially Σ12 (written ess-Σ12 ) if
[
[
C ∈ ∃Σ12 ∪ {Σ1i : i ≤ 2} ∪ {Π1j : j < 2}.
Lemma 5.1 Let Γ be a set of ess-Σ12 formulas,
Ξ = {∃Z1 B1 (t1 , Z1 ), . . . , ∃Zr Br (tr , Zr )} ⊆ Σ12 , and
Θ = {∃y1 ∃Z1 B1 (y1 , Z1 ), . . . , ∃yr ∃Zr Br (yr , Zr )}.
1. If D ` Γ, Ξ, then we can find a nice D+ ` Γ, Θ.
2. If D0 ` Γ, then there is a nice D0+ ` Γ.
Proof: 1.) By Theorem 5.1, we may assume %(D) ≤ ω + 1. We proceed by induction on | D |. If
Γ, Ξ is an axiom, then so is Γ; hence Γ, Θ is an axiom. The derivation consisting merely of this
axiom is of course nice.
Now suppose 0 <| D | . If neither a m. f. nor a p. f. of the last inference (l. i.) of D is Σ12 ,
then the assertion follows by using the induction hypothesis on the premisses and reapplying the
same inference, since this does not change the stock of Σ12 –formulas. Note that ρ(D) ≤ ω + 1.
Next assume that a formula ∃ZB(t, Z) ∈ Σ12 is a m. f. of the last inference of D. Then this
must be an instance of (∃1 ) because of Γ, Ξ ⊂ess–Σ12 . So the p. f. is of the form ∃y∃ZB(y, Z) ∈ Γ,
and the direct subderivation of D takes the form D0 ` Λ, Ξ0 with Λ ⊆ Γ and Ξ0 = Ξ, ∃ZB(y, Z).
Applying the induction hypothesis we get a nice
D+ ` Λ, Θ, ∃y∃ZB(y, Z),
thence D+ ` Γ, Θ.
Finally, suppose that ∃Y C(Y ) ∈ Σ12 is the p. f. of the last inference of D. This then must
be an instance of (∃2 ). So there is a nice derivation D0 ` Γ, Ξ, C(U ) such that %(D0 ) ≤ ω + 1
and | D0 |<| D |. Inductively we find a nice derivation
D0+ ` Γ, Θ, C(U ).
24
By use of (∃2 ), we can continue D0+ to a nice derivation of Γ, Θ, ∃Y C(Y ). If ∃Y C(Y ) ∈ Γ, then
we are done. Otherwise, ∃Y C(Y ) ∈ Ξ; thus an application of (∃1 ) gives us a nice derivation
D+ ` Γ, Θ, since in this case ∃Y C(Y ) does not appear as a s. f. in D0+ .
2.) follows from 1.) with Ξ = ∅.
t
u
6
The infinitary calculus T ∗
The formulas of T ∗ arise from L2 -formulas by replacing free numerical variables by numerals, i.
e. terms of the form 0, 00 , 000 , ... . Especially, every formula A of T ∗ is a L2 -formula, and thus
gr(A) is understood. A formula without second order variables will be called constant. We are
going to measure the length of derivations by ordinals. For technical reasons we are compelled
to diverge from the ordinal notation system OT (ϑ) of Section 1. Instead we are going to use the
set of ordinals OT (ψ) of Section 3.
Definition 6.1
1. A formula B is said to be weak if it belongs to Π10 ∪ Π11 .
2. Two closed terms s and t are said to be equivalent if they yield the same value when
computed.
3. A formula is called constant if it contains no set variables. The truth or falsity of such a
formula is understood with respect to the standard structure of the integers.
4. 0 := 0, m + 1 := m0 .
Definition 6.2 Inductive definition of T ∗
α
%
Γ for α ∈ OT (ψ) and % < ω + ω.
1. If A is a true constant prime formula or negated prime formula and A ∈ Γ, then T ∗
α
%
Γ.
2. If Γ contains formulas A(s1 , . . . , sn ) and ¬A(t1 , . . . , tn ) of grade 0 or ω, where si and
α
ti (1 ≤ i ≤ n) are equivalent terms, then T ∗ % Γ.
3. If T ∗
β
%
Γi
and β α hold for every premiss Γi of an inference (∧), (∨),
(∃1 ), (∀2 ) or (Cut) with a cut formula having grade < %, and conclusion Γ, then T ∗
4. If T ∗
α0
%
α
%
Γ, F (U ) holds for some α0 α and a non-arithmetic formula F (U ) (i.
gr(F (U )) ≥ ω), then
5. (ω-rule). If T ∗
β
%
α
T∗ %
Γ.
e.,
Γ, ∃XF (X) .
Γ, A(m) is true for every m < ω, ∀xA(x) ∈ Γ, and β α, then T ∗
6. (Ω-rule). Let f be a fundamental function satisfying
a. Ω ∈ dom(f ) and f (Ω) α,
25
α
%
Γ.
b. T ∗
f (0)
%
c. T ∗
β
0
Then T ∗
Γ, ∀XF (X) , where ∀XF (X) ∈ Π11 , and
Ξ, ∀XF (X) implies T ∗
α
%
f (β)
%
Ξ, Γ for every set of weak formulas Ξ and β < Ω.
Γ holds.
α
α
Remark 6.1 The derivability relation T ∗ % Γ is modelled upon the relation P B ∗ n F of [1],
the main difference being the sequent calculus setting instead of P – and N –forms and a different
assignment of cut–degrees. The allowance for transfinite cut–degrees will enable us to deal with
arithmetical comprehension.
Lemma 6.1
1. T ∗
α
δ
Γ & Γ ⊆ ∆ & α β & δ ≤ % =⇒ T ∗
2. T ∗
α
%
Γ, A ∧ B =⇒ T ∗
α
%
Γ, A & T ∗
3. T ∗
α
%
Γ, A ∨ B =⇒ T ∗
α
%
Γ, A, B
4. T ∗
α
%
Γ, F (t) =⇒ T ∗
5. T ∗
α
%
Γ, ∀xF (x) =⇒ T ∗
6. If T ∗
α
%
α
%
α
%
β
%
∆,
Γ, B,
Γ, F (s) if t and s are equivalent,
α
%
Γ, F (s) for every term s.
Γ, ∀XG(X) and gr(G(U )) ≥ ω, then T ∗
α
%
Γ, G(U ) .
Proofs by induction on α. These can be carried out straightforwardly. 5.) requires 4.). As to
6.), observe that ∀XG(X) cannot be the main formula of an axiom.
2
Lemma 6.2 T ∗
equivalent terms.
2·α
0
Γ, A(s1 , . . . , sk ), ¬A(t1 , . . . , tk ) if α ≥ gr(A(s1 , . . . , sk )) and si and ti are
t
u
Proof by induction on gr(A).
Lemma 6.3
2. T ∗
ω+5
0
1. T ∗
2m
0
¬(0 ∈ U ), (∃x)[x ∈ U ∧ ¬(x0 ∈ U )], m ∈ U ,
∀X[0 ∈ X ∧ ∀x(x ∈ X → x0 ∈ X) → ∀x(x ∈ X)].
Proof. For 1.) use induction on m. For a detailed proof see [4] 10.17. 2.) is an immediate
consequence of 1.) using Lemma 6.1 1.), the ω-rule, (∨), and (∀2 ).
Definition 6.3 For formulas F (U ) and A(a), F (A) denotes the result of replacing each occurrence of the form (e ∈ U ) in F (U ) by A(e). The expression F (A) is a formula if the bound
variables in A(a) are chosen in an appropriate way, in particular, if F (U ) and A(a) have no
bound variables in common.
26
Lemma 6.4 Assume that α < Ω. Let ∆(U ) = {F1 (U ), . . . , Fk (U )} be a set of weak formulas
such that U doesn’t occur in ∀XFi (X) (1 ≤ i ≤ k). For an arbitrary formula A(a) we then have:
T∗
α
0
∆(U ) =⇒ T ∗
Ω+α
0
∆(A) .
Proof by induction on α. Suppose ∆(U ) is an axiom. Then either ∆(A) is an axiom too, or
ω+ω
Ω+α
T∗ 0
∆(A) can be obtained through use of Lemma 6.2. Therefore T ∗ 0
∆(A) by Lemma
α
6.1 1.). If T ∗ 0 ∆(U ) is the result of an inference, then this inference must be different from
(∃2 ), (Cut), and (Ω−rule). Therefore the assertion follows easily from the induction hypothesis.
2
Lemma 6.5 Let Γ, ∀XF (X) be a set of weak formulas. If T ∗
α
T∗ 0
α
0
Γ, ∀XF (X) and α < Ω, then
Γ, F (U ) .
Proof by induction on α. Note that ∀XF (X) cannot be a principal formula of an axiom, since
∃X¬F (X) does not surface in such a derivation. Also, due to α < Ω, the derivation doesn’t
involve instances of the Ω-rule. Therefore the proof is straightforward.
2
The role of the (Ω − rule) in our calculus T ∗ is enshrined in the next lemma.
Lemma 6.6 T ∗
A(a).
Ω·2
0
∃XF (X), ¬F (A) for every arithmetic formula F (U ) and arbitrary formula
Proof. Let f (α) := Ω + α with dom(f ) := {α ∈ OT (ψ) : α ≤ Ω}. Then
T∗
f (0)
0
∀X¬F (X), ∃XF (X), ¬F (A)
(1)
according to Lemma 6.2. For α < Ω and every set of weak formulas Θ, we have by Lemmata
6.4 and 6.5,
T∗
α
o
Θ, ∀X¬F (X) =⇒ T ∗
f (α)
0
Θ, ¬F (A).
Therefore, by Lemma 6.1 1.),
T∗
α
0
Θ, ∀X¬F (X) =⇒ T ∗
f (α)
0
Θ, ∃XF (X), ¬F (A).
The assertion now follows from (1) and (2) by the Ω-rule.
7
(2)
2
The reduction procedure for T ∗
Lemma 7.1 Let C be a formula of grade %. Suppose C is a prime formula or of either form
∃XH(X), ∃xG(x) or A ∨ B. Let α = ω α1 + · · · + ω αk with δ ≤ ω αk ≤ . . . ≤ ω α1 . Then we have
α
δ
α+δ
T ∗ % ∆, ¬C & T ∗ % Γ, C =⇒ T ∗ % ∆, Γ .
Proof by induction on δ.
1. Let Γ, C be an axiom. Then there are three cases to consider.
α+δ
1.1. Γ is an axiom. Then so is ∆, Γ Hence T ∗ % ∆, Γ .
27
1.2. C is a true constant prime formula or negated prime formula. A straightforward induction
α
α+δ
on α then yields T ∗ % ∆ , and thus T ∗ % ∆, Γ by 6.1 1.).
1.3. C ≡ A(s1 , . . . , sn ) and Γ contains a formula ¬A(t1 , . . . , tn ) where si and ti are equivalent
α
terms. From T ∗ % ∆, ¬A(s1 , . . . , sn ) one receives
α
α+δ
T ∗ % ∆, ¬A(t1 , . . . , tn ) by use of Lemma 6.1 4.). Thence T ∗ % ∆, Γ follows by use of Lemma
6.1 1.), since ¬A(t1 , . . . , tn ) ∈ Γ.
δ0
2. Suppose C ≡ A ∨ B and T ∗ % Γ, C, A0 with A0 ∈ {A, B} and δ0 δ. Inductively we get
T∗
Next use Lemma 6.1 2.) on T ∗
α
%
α+δ0
%
∆, Γ, A0 .
(1)
∆, ¬A ∧ ¬B to obtain
T∗
α+δ0
%
∆, Γ, ¬A0 .
(2)
Whence use a cut on (1) and (2) to get the assertion.
δ0
3. Suppose C ≡ ∃xG(x) and T ∗ % Γ, C, G(t) with δ0 δ. Inductively we get
T∗
α+δ0
%
∆, Γ, G(t) .
(3)
∆, Γ, ¬G(t) ;
(4)
By Lemma 6.1 1.), 5.), we also get
T∗
thus (3) and (4) yield T ∗
α+δ
%
α+δ0
%
∆, Γ by (Cut).
4. Suppose the last inference was (∃2 ) with p. f. C. Then C ≡ ∃XH(X) and T ∗
for some δ0 δ and gr(H(U )) ≥ ω. Inductively we get
T∗
α+δ0
%
δ0
%
Γ, C, H(U )
∆, Γ, H(U ).
(5)
α+δ0
%
∆, Γ, ¬H(U ).
(6)
T∗
α+δ
%
By Lemma 6.1 1.), 6.) we also get
T∗
From (5) and (6) we obtain
δ
∆, Γ.
5. Let T ∗ % Γ, C be derived by the Ω-rule with fundamental function f . Then the assertion
follows from the I. H. by the Ω-rule using the fundamental function α + f .
6. In the remaining cases the assertion follows from the I. H. used on the premises and by
reapplying the same inference.
2
Lemma 7.2 T ∗
α
η+1
Γ =⇒ T ∗
ωα
η
Γ.
28
Proof by induction on α. We only treat the crucial case when T ∗
where α0 α, and gr(D) = η. Inductively this becomes T ∗
ω α0
η
α0
η+1
Γ, D and T ∗
Γ, D and T ∗
ω α0
α0
η+1
Γ, ¬D ,
Γ, ¬D.
η
α0 +ω α0
ω
T∗ η
D or ¬D must be of one of the forms exhibited in Lemma 7.1, we obtain
Lemma 7.1. As ω α0 + ω α0 ω α , we can use Lemma 6.1 1.) to get the assertion.
Since
Γ by
Theorem 7.1 (Collapsing Theorem) If Γ is a set of weak formulas and α ∈ C(α), then we have
T∗
α
ω
Γ =⇒ T ∗
ψα
0
Γ.
Proof by induction on α. Observe that for β < δ < Ω, we always have β δ.
1. If Γ is an axiom, then the assertion is trivial.
α
2. Let T ∗ ω Γ be the result of an inference other than (Cut) and Ω-rule. Then we have
α0
T ∗ ω Γi with α0 α and Γi being the i-th premiss of that inference. α0 α and α ∈ C(α)
imply α0 ∈ C(α0 ) and ψα0 ψα by Lemma 4.5. Therefore T ∗
ψα
T∗ 0
ψα0
0
Γ0 by the I. H., hence
Γ by reapplying the same inference.
α
3. Suppose T ∗ ω Γ results by the Ω-rule with respect to a Π11 -formula ∀XF (X) and a fundamental function f . Then Ω ∈ dom(f ) and f (Ω) α. Also
f (0)
ω
T∗
Γ, ∀XF (X),
(1)
and, for every set of weak formulas Ξ and β < Ω,
T∗
β
0
Ξ, ∀XF (X) =⇒ T ∗
f (β)
ω
Ξ, Γ.
(2)
From Ω ∈ dom(f ) we get f (0) f (Ω) by (F1.); thus f (Ω) α yields f (0) ∈ C(f (0)) and
f (Ω) ∈ C(f (Ω)) using Lemma 4.5. Therefore the I. H. used on (1) supplies us with
T∗
ψ(f (0))
0
Γ, ∀XF (X) .
Hence with Ξ = Γ we get
f (ψ(f (0)))
ω
T∗
Γ
(3)
from (2). Now Corollary 4.3 ensures that f (β)f (Ω), where β = ψ(f (0)); hence f (β) ∈ C(f (β))
by Lemma 4.5 since f (Ω) ∈ C(f (Ω)). So using the I. H. on (3), we obtain
T∗
ψ(f (β))
0
Γ,
(4)
ψα
thus T ∗ o Γ as f (β) α.
α0
4. Suppose T ∗ ω Γ, A and T ∗
get
ψα0
T∗ 0
Γ, A and
ψα0
T∗ 0
α0
ω
Γ, ¬A , where α0 α and gr(A) < ω. Inductively we then
Γ, ¬A. Let gr(A) = n − 1. Then (Cut) yields
T∗
β1
n
Γ
with β1 = (ψα0 ) + 1. Applying Lemmma 7.2, we get T ∗
we arrive at
βn
T∗ 0 Γ ,
(5)
ω β1
n−1
Γ , and by repeating this process
where βk+1 := ω βk (1 ≤ k < n). Since ψα0 < ψα, we have βn < ψα; thus T ∗
29
ψα
0
Γ.
2
8
The Refined Reduction Lemma
We would like to define nice derivations also in the context of T ∗ . A T ∗ -derivation is a tuple
consisting of its direct subderivations (d. s.), the set of minor formulas (m. f.), the set of
principal formulas (p. f.) (possibly empty), the ordinal bounds for the length and cut-rank,
and a symbol indicating the last inference (l. i.). If there were not the Ω-rule, we needn’t to be
α
α
specific about this. We will write D % Γ if D is a derivation witnessing % Γ . The tuple
D = hD0 , T , f, ∀XH(X), α, %, Ωi
α
is said to be a nice T ∗ -derivation if: D0 % Γ, ∀XH(X) is nice; f is a fundamental function with
Ω ∈ dom(f ) and f (Ω) α; ∀XH(X) ∈ Π11 ; and for every set of weak formulas Ξ and β < Ω the
following is valid:
β
If D0 is a T ∗ -derivation satisfying D0 0 Ξ, ∀XH(X), then T (D0 ) is a nice T ∗ -derivation satisfying
f (β)
T (D0 ) % Ξ, Γ. (So T is a function on proofs.) Nice derivations allow us to improve on the
Reduction Lemma 7.1.
Lemma 8.1 (The Refined Reduction Lemma) Let B ≡ ∃y∃ZA(y, Z) be ∃Σ12 . Let α = ω α1 +
· · · + ω αn ≥ α1 ≥ . . . ≥ αn , β = ω β1 + · · · + ω βk ≥ β1 ≥ . . . ≥ βk such that β1 ≤ αn . Suppose
β
α
that D0 ω+1 Γ, ¬B and D ω+1 Λ, B are nice T ∗ -derivations, where Γ, Λ ⊆ ess−Σ12 . Then there
is a nice T ∗ -derivation D∗
α+β
ω+1
Γ, Λ .
Proof by induction on β. If the last inference (l. i.) does not have a Σ12 or ∃Σ12 principal
formula (p. f.), the assertion follows by using the I. H. on the direct subderivations (d. s.) and
reapplying the same inference. (If this is the Ω-rule with fundamental function f , then the new
fundamental function will be α + f .) Note that such an inference does not change the stock of
”critical” formulas.
If the last inference is (∃2 ) with Σ12 p. f. A, then the d. s. D0 might not be nice. But this
problem can be overcome by adding A as a side formula throughout the derivation D0 , which
gives rise to a nice derivation D1 . So we can apply the I. H. on D1 and subsequently (∃2 ) to get
the assertion.
Now suppose that the p. f. of the l. i. is E ≡ ∃x∃ZA(x, Z) and E ∈ ∃Σ12 . Then the d. s. of
β0
D has the form D0 ω+1 Λ0 , C , where β0 β, C ≡ ∃ZA(t, Z), and Λ0 ⊆ Λ, B. Note that D0 is
also nice. Suppose C ∈ Λ. The I. H. provides us with a nice derivation
D+
α+β0
ω+1
Λ0 \ {B}, Γ, C.
α+β
So we get a nice derivation D∗ ω+1 Γ, Λ by weakening. If C 6∈ Λ, then the niceness of D
implies that the l. i. of D0 is (∃2 ) with p. f. C. So the d. s. D1 of D0 then has the form
D1
β1
ω+1
Λ1 , A(t, U ) with Λ1 ⊆ Λ0 and β1 β0 . Using the I. H. on D1 , we find a nice derivation
D2
α+β1
ω+1
Λ1 \ {B}, Γ, A(t, U ) .
30
Suppose that E ≡ B. Then ¬B ≡ ∀y∀Z¬A(y, Z). Using inversion (6.1 2. and 6.1 6.) on D0 , we
get a derivation
α
D3 ω+1 Γ, ¬A(t, U ) ,
which is also nice, since it has the same stock of Σ12 minor and side formulas as D0 . Using (Cut)
on D2 and D3 (and weakening) we get a nice derivation
D∗
α+β
ω+1
Γ, Λ.
Finally, suppose E is different from B. Then E ∈ Γ. Then apply (∃2 ) followed by (∃1 ) to D2 , to
α+β
get a nice derivation D∗ ω+1 Λ1 \ {B}, Γ, E (note that α +β1 α +β0 α +β). As Λ1 \{B} ⊆ Λ
and E ∈ Γ, the result follows by weakening.
2
9
Embedding (Π12 − BI)0 into T ∗ .
The objective of this Section is to embed (Π12 − BI)0 into T ∗ , so as to obtain an upper bound
for the proof-theoretic ordinal of (Π12 − BI)0 by putting to use results of the previous Section.
The treatment of the Π12 bar induction rule requires the following lemma.
Lemma 9.1 Let E(a) and F (a, U, V ) be arithmetic formulas. If Λ is an arbitrary set of T ∗
δ
formulas and T ∗ ω+1 Λ, ∀y[E(y) → ∀Y ∃ZF (y, Y, Z)], then
T∗
Ω·2+δ·4
ω+1
Λ, ∀y ∀Y ∃Z [E(y) → F (y, Y, Z)].
Proof by induction on δ. Let G0 ≡ ∀y[E(y) → ∀Y ∃ZF (y, Y, Z)]. If G0 is not the p.f. of the
last inference, then we get the desired derivation by employing the I. H. on the premises and
reapplying the same inference.
Now let G0 be the p.f. of the last inference, which must be an instance of the ω-rule. Hence
δ0
ω+1
T∗
Λ, G0 , E(n̄) → ∀Y ∃ZF (n̄, Y, Z)
for some δ0 δ and all n. Let G be ∀y ∀Y ∃Z [E(y) → F (y, Y, Z)]. The I. H. yields
T∗
α
ω+1
Λ, G, E(n̄) → ∀Y ∃ZF (n̄, Y, Z) ,
where α = Ω · 2 + δ0 · 4. Hence, by 6.1 (3) followed by 6.1 (6), we get that, for all n,
T∗
α
ω+1
Λ, G, ¬E(n̄), ∃ZF (n̄, U, Z)
(1)
for a ‘fresh’ variable U .
From Lemma 6.6 we obtain
T∗
Ω·2
ω+1
∃Z[E(n̄) → F (n̄, U, Z)], E(n̄) ∧ ¬F (n̄, U, V ),
whence, by Lemma 6.1 (2),
T∗
Ω·2
ω+1
∃Z[E(n̄) → F (n̄, U, Z)], E(n̄)
31
(2)
and
T∗
Ω·2
ω+1
∃Z[E(n̄) → F (n̄, U, Z)], ¬F (n̄, U, V ) .
(3)
An inference (∀2 ) applied to (3) yields
T∗
Ω·2+1
ω+1
∃Z[E(n̄) → F (n̄, U, Z)], ∀Z¬F (n̄, U, Z) .
(4)
Applying (Cut) to (1) and (2) gives
T∗
α+1
ω+1
Λ, G, ∃Z[E(n̄) → F (n̄, U, Z)], ∃ZF (n̄, U, Z)
(5)
and a further (Cut) with (4) yields
T∗
α+2
ω+1
Λ, G, ∃Z[E(n̄) → F (n̄, U, Z)].
(6)
Continuing the derivation, first using (∀2 ) and then the ω-rule, we arrive at
T∗
α+4
ω+1
Λ, G.
(7)
As α + 4 Ω · 2 + δ · 4, this finishes the proof.
2
Theorem 9.1 Let Γ be a set of ess-Σ12 formulas. Suppose that Γ∗ results from Γ by replacing
all the free variables in formulas of Γ with numerals. Then
(Π12 − BI)0
k
Γ =⇒ T ∗
ω+1
Ωf (k)
ω+1
Γ∗ ,
where f (k) := (k + 1) · 4.
Proof by induction on k.
Using Lemma 5.1 2.), we can assume that we have a nice derivation of Γ. During this proof we
shall also ensure that the corresponding T ∗ -derivation will be nice.
1. If Γ is an axiom (Ax1) or (Ax2), then Γ∗ is also an axiom of T ∗ .
2. If Γ is an axiom (IA), then this follows from Lemma 6.3 2.).
3. Suppose Γ is an axiom (Π10 − CA), i. e., Γ contains a formula (∃Z)(∀x)[x ∈ Z ↔ A(x)],
with A arithmetic. Setting F (U ) := (∀x)[x ∈ U ↔ A∗ (x)], we obtain
T∗
ω+ω
0
F (A∗ )
(1)
∃ZF (Z), ¬F (A∗ ).
(2)
using Lemma 6.2.
By Lemma 6.6 we have
T∗
Ω·2
0
Ωf (0)
Ωf (0)
Thus (1) and (2) yield (since gr(F (A∗ )) < ω) T ∗ ω
∃ZF (Z) , thence T ∗ ω+1 Γ∗ .
Observe that in the cases, we have been considering so far, the infinitary derivations didn’t
contain inferences with Σ12 m. f. or p. f.; so they are automatically nice.
32
k
4. Suppose (Π12 − BI)0 ω+n−1 Γ is the result of an inference (∃2 ) with principal formula
∃XH(X) ∈ Γ having grade ω. Then
k0
ω+1
(Π12 − BI)0
Γ0 , H(U )
Ωf (0)
for some k0 < k and Γ0 ⊆ Γ. By I.H., we then have a nice T ∗ -derivation D0 ω+1 Γ∗0 , H ∗ (U ) .
The (∃2 )-rule of T ∗ is not available in this situation since gr(H ∗ (U ) < ω. We have to resort to
Lemma 6.6 to get a nice
Ω·2
D1 0 ∃XH ∗ (X), ¬H ∗ (U ) .
Applying (Cut) gives us a nice
D2
Ωf (k)
ω+1
Γ∗ .
k0
5. Suppose (Π12 − BI)0 ω+1 Γ, A and (Π12 − BI)0
ω + 1. Inductively we then get
T∗
and
T∗
So, by (Cut), we obtain T ∗
6. Let (Π12 − BI)0
k
ω+1
Ωf (k)
ω+1
Ωf (k0 )
ω+1
Ωf (k0 )
ω+1
k0
ω+1
Γ, ¬A for some k0 < k and gr(A) <
Γ∗ , A∗
Γ∗ , ¬A∗ .
Γ∗ niceness preserving.
Γ be the result of (∀1 ). Then
(Π12 − BI)0
k0
ω+1
Γ0 , F (a)
Ωf (k0 )
for some k0 < k and Γ0 , ∀xF (x) = Γ. Inductively we get T ∗ ω+1 Γ∗0 , F ∗ (n̄) for every n. Thus
the assertion follows by the ω-rule. Since (∀1 ) does not violate niceness in the original derivation,
the ω-rule won’t do this in the infinitary derivation.
k
7. Let (Π12 − BI)0 ω+1 Γ be the result of a logical inference other than the ones already
treated. Then the assertion follows using the I.H. on the premises and subsequently reapplying
the same inference in T ∗ . This of course preserves niceness.
8. Finally, we have to deal with the Π12 bar induction rule. So suppose Γ = Λ, ∃ZB(b, V, Z),
(Π12 − BI)0
and
(Π12 − BI)0
k0
ω+1
k0
ω+1
Λ, W F (≺) ,
(3)
Λ, ∃y ∃Y ∀Z[y ≺ a ∧ ¬B(y, Y, Z], ∃ZB(a, U, Z),
(4)
where k0 < k and a, U are ”fresh” variables. Put
C(n̄) :≡ ∃y∃Y ∀Z[y ≺ n̄ ∧ ¬B(y, Y, Z)]∗ ,
and
A(n̄) :≡ ∀Y ∃ZB(n̄, Y, Z)∗ .
33
By I.H. we then have, for β = Ωf (k0 ) and for each m < ω, a nice T ∗ -derivation
β
Dn
ω+1
Λ∗ , C(n̄), ∃ZB(n̄, U, Z)∗ ;
thus, using (∀2 ), we obtain a nice derivation
Dn1
α0
ω+1
Λ∗ , C(n̄), A(n̄)
(5)
with α0 = Ωf (k0 )+1 (note that ∃ZB(n̄, U, Z)∗ 6∈ Σ12 ), and also, by the I. H. used on (3), we get
a nice
α0
D2 ω+1 Λ∗ , W F (≺)∗ .
(6)
Moreover, Lemma 6.6 provides us with a nice
D0
Ω·2
0
¬∀X T I(≺∗ , X), T I(≺∗ , A),
simply because the formulas occuring in this derivation do not have ∃Σ12 subformulas. By Lemma
6.1 3. we get a nice
D3
Ω·2
0
¬∀X T I(≺∗ , X), ∃z[¬C0 (z) ∧ ¬A(z)], ∀zA(z) ,
(7)
where
C0 (b) ≡ ∃y[y ≺ b ∧ ∃Y ∀Z¬B(y, Y, Z)]∗ .
Claim: If β ≤ Ω · 2, Ξ a set of formulas which do not have Σ12 subformulas, and D∗ is a nice
derivation such that
β
D∗ 0 Ξ, ∃z[¬C0 (z) ∧ ¬A(z)], ∀zA(z) ,
then, for all m < ω, there is a nice
∗
Dm
g(β)
ω+1
Ξ, Γ∗ , A(m) ,
where
g(β) = α0 · ω β = Ωf (k0 )+1 · ω β .
We prove the Claim by induction on β. Put
D :≡ ∃z[¬C0 (z) ∧ ¬A(z)].
If neither D nor ∀zA(z) is a principal formula of the last inference, then the assertion is easily
obtained by the induction hypothesis. Suppose now that ∀zA(z) is the principal formula. Then
there is some β0 β such that for all m < ω there is a nice Dm
g(β0 )
g(β)
β0
0
Ξ, D, ∀zA(z), A(m) . Inductively
0
∗
∗ (note
we then can pick a nice Dm
Ξ, Γ∗ , A(m) , thus Dm
Ξ, Γ∗ , A∗ (m) for some nice Dm
ω+1
ω+1
that g(β0 ) g(β)).
Now let D be the principal formula. Then there are β0 β, a term t, and a nice
D0∗
β0
0
Ξ, ∀zA(z), D, ¬C0 (t) ∧ ¬A(t) .
34
Let n be the value of t. Using the I. H. and Lemma 6.1 4.), we get a nice
D1∗
g(β0 )
Ξ, Γ∗ , A(m), ¬C0 (n̄) ∧ ¬A(n̄).
0
By Lemma 6.1 it 2.) we therefore find nice D2∗ and D3∗ such that
g(β0 )
D2∗
ω+1
and
g(β0 )
D3∗
ω+1
Ξ, Γ∗ , A(m), ¬C0 (n̄)
(8)
Ξ, Γ∗ , A(m), ¬A(n̄).
(9)
We would like to replace
¬C0 (n̄) ≡ ∀y[y ≺∗ n̄ → ∀Y ∃Z B(y, Y, Z)∗ ]
with
¬C(n̄) ≡ ∀y ∀Y ∃Z [y ≺∗ n̄ → B(y, Y, Z)∗ ].
This can be done using Lemma 9.1. By inspection of the proof of 9.1, one readily verifies that
this transformation leads from nice derivations to nice derivations. So from (8) we obtain a nice
D4∗ satisfying
η
D4∗ ω+1 Ξ, Γ∗ , A(m), ¬C(n̄),
(10)
where η := g(β0 ) · 4. If we now use the Refined Reduction Lemma 8.1 on (5) and (10), we obtain
a nice
η+α0
D5∗ ω+1 Ξ, Γ∗ , A(m), A(n̄).
(11)
Note that A(n̄) is neither the minor formula of (∃1 ) in D3∗ nor in D5∗ ((9) and (11)). Thence, by
using the Reduction Lemma 7.1 on (9) and (11), we arrive at a nice
D6∗
η+α0 +g(β0 )
ω+1
Ξ, Γ∗ , A(m) .
Since η+α0 +g(β0 ) = Ωf (k0 )+1 ·ω β0 ·4+Ωf (k0 )+1 +Ωf (k0 )+1 ·ω β0 = Ωf (k0 )+1 ·ω β0 ·5Ωf (k0 )+1 ·ω β =
g(β), this furnishes proof of the Claim. Letting Ξ = {¬W F (≺∗ )} and β = Ω · 2, the Claim, used
on (7), yields a nice
∗
Dm
g(Ω·2)
ω+1
¬W F (≺∗ ), Γ∗ , A(m).
(12)
Now, g(Ω · 2) = Ωf (k0 )+3 . So applying (Cut) to (6) and (12), will provide us with a derivation
+
Dm
Ωf (k)
ω+1
Γ∗ , A(m)
(13)
for all m < ω. Since A(m) ≡ ∀Y ∃ZB ∗ (m, Y, Z) and ∃ZB ∗ (m, V, Z) ∈ Γ∗ for some m < ω, the
assertion follows from (13) and Lemma 6.1 (6).
2
Corollary 9.1 Let A be a Π11 sentence such that (Π12 − BI)0
such that T ∗
ψα
0
A.
35
ω
A , then there is some α < ΩΩ
Proof. Employing Theorem 9.1, we find an n < ω such that T ∗
Lemma 7.2. Theorem 7.1 then yields T ∗
n
ψ(ω Ω )
0
Ωn
A ; thus T ∗
ω+1
A.
ωΩ
ω
n
A by
2.
Corollary 9.2 Let ≺ be an arithmetic well-ordering such that
ω
(Π12 − BI)0
WF (≺) , then the order type of ≺ is less than ψΩΩ .
Proof. From a cut–free proof of WF (≺) in T ∗ of length less α one can extract that the order–type
of ≺ is majorized by 2α (cf. [4] 13.10 or [8]).
2
Corollary 9.3 (Π12 − BI)0 does not prove ∀nKT (n).
Proof. Immediate by Theorem 1.1, Corollary 9.2, and Corollary 3.2.
10
2
A Well-ordering Proof in (Π12 − BI)−
0
ω
By Corollary 9.2 and ψΩΩ ≤ ϑΩω (see Corollary 3.2), we know that the well-foundedness of
the primitive recursive well-ordering < ϑΩω cannot be proved in (Π12 − BI)0 . This Section is
WF (< ϑΩn ) for any (meta) n. In the sequel α, β, γ, δ, . . . are
aimed at showing (Π12 − BI)−
0
supposed to range over OT (ϑ). < will be used to denote the ordering on OT (ϑ). We are going
to work informally in second order arithmetic. Especially, variables X, Y, Z, . . . are ranging over
subsets of N. Note also that we consider OT (ϑ) to be a subset of N. The proofs require some
terminology.
Definition 10.1
1. Acc := {α < Ω : WF (< α)},
2. M := {α : EΩ (α) ⊆ Acc},
3. α <Ω β : ⇐⇒ α, β ∈ M ∧ α < β.
Remark: Acc is a Π11 –definable class. Therefore M and <Ω are also Π11 .
Lemma 10.1 α, β ∈ Acc =⇒ α + ω β ∈ Acc.
Proof. Familiar from Gentzen’s proof in PA. The proof just requires ACA0 . (cf. [4], Sec. 15).2
Lemma 10.2 Acc = M ∩ Ω (:= {α ∈ M : α < Ω}.)
Proof. If α ∈ Acc, then EΩ (α) ⊆ Acc as well; hence α ∈ M ∩ Ω. If α ∈ M ∩ Ω, then
EΩ (α) ⊆ M ∩ Ω, so α ∈ Acc follows from Lemma 10.1.
2
Definition 10.2 Let P rogΩ (X) stand for
(∀α ∈ M )[(∀β <Ω α)(β ∈ X) −→ α ∈ X].
Let AccΩ := {α ∈ M : ϑα ∈ Acc}.
Lemma 10.3 P rogΩ (AccΩ ).
36
Proof. Assume α ∈ M and (∀β <Ω α)(β ∈ AccΩ ). We have to show ϑα ∈ Acc. It suffices to
show
β < ϑα =⇒ β ∈ Acc.
(1)
We shall employ induction on Gϑ (β), i.e., the length of (the term that represents) β. If β 6∈ E,
then (1) follows easily by the inductive assumption and Lemma 10.1. Now suppose β = ϑβ0 .
Then we have to distinguish two cases according to Lemma 1.2.
Case 1: β ≤ α∗ . Since α ∈ M, we have α∗ ∈ EΩ (α) ⊆ Acc; therefore β ∈ Acc.
Case2: β0 < α and β0 < ϑα. As the length of β0∗ is less than the length of β, we get β0∗ ∈ Acc;
thus EΩ (β0 ) ⊆ Acc, therefore β0 ∈ M. By the assumption at the beginning of the proof, we then
get β0 ∈ AccΩ ; hence β = ϑβ0 ∈ Acc.
t
u
It should be noted that the proof of Lemma 10.3 only requires complete induction for Π11 –
classes. The next lemma is where we really need (Π12 − BI)−
0 .
Lemma 10.4 Let A(a) be a Π11 formula. Letting Ak be the formula
∀α[(∀β <Ω α)A(β) −→ (∀β <Ω α + Ωk )A(β)];
(Π12 − BI)−
0 proves P rogΩ ({ξ : A(ξ)}) −→ Ak .
Proof. We proceed by outer induction on k. Assume P rogΩ ({α : A(α)}) and (∀β <Ω δ)A(β).
Then also (∀β ≤Ω δ)A(β). For k = 0 this gives the assertion, since γ < δ + Ω0 implies γ ≤ δ.
Now let k = m + 1. So we get Am by the inductive assumption. Let B(η) be the formula
(∀β <Ω δ + Ωm · η)A(β). B(η) is (in ACA0 ) provably equivalent to a Π12 formula. Suppose that
η ∈ Acc and (∀ρ < η)B(η). Clearly, B(0) holds. If η is a limit, then for β <Ω δ + Ωm · η
there exists ρ < η such that β <Ω δ + Ωm · ρ, hence B(η) holds. Now let η be a successor
γ + 1. Then (∀β <Ω δ + Ωm · γ)A(β). Using Am , this implies (∀β < δ + Ωm · γ + Ωm )A(β), thus
(∀β <Ω δ + Ωm · (γ + 1))A(β), hence B(η). By the above considerations, we have
(∀η ∈ Acc)[(∀ρ < η)B(ρ) −→ B(η)],
hence, using Π12 bar induction along < (η + 1), we obtain
(∀η ∈ Acc)B(η).
Now for every β <Ω δ + Ωk , there is some η ∈ Acc such that β <Ω δ + Ωm · η. Therefore
(∀β <Ω δ + Ωk )A(β)
2
follows from (∀η ∈ Acc)B(η).
Theorem 10.1 For any n, (Π12 − BI)−
0
WF (< ϑΩn ) .
Proof. Obviously (∀β <Ω 0)A(β) holds for any formula. Therefore we obtain
(Π12 − BI)−
0
P rogΩ ({ξ : A(ξ)}) −→ (∀β <Ω Ωk )A(β)
37
(1)
for any Π11 –formula A(α) and k ∈ N. Since the formula “ξ ∈ AccΩ ” can be written Π11 , Lemma
10.3 along with (1) implies, for any k ∈ N,
(Π12 − BI)−
0
Therefore, for any n, (Π12 − BI)−
0
(∀β <Ω Ωk )[β ∈ AccΩ ] .
WF (< ϑΩn ) .
t
u
ω
Corollary 10.1 The proof-theoretic ordinal of (Π12 − BI)0 and (Π12 − BI)−
0 is ϑΩ .
t
u
Proof. Theorem 10.1 and Corollary 9.2.
Corollary 10.2 For every n < ω, (Π12 − BI)−
0 ` KT (n̄).
Proof. For fixed n the proof of KT (n) requires only WF (ϑΩm ) for some m < n + 3. This follows
from the proof of Theorem 2.2.
t
u
The methods of the last six sections can also be employed to determine the proof-theoretic
1
1
−
ordinals of the systems (Π1n − BI)0 , (Π1n − BI)−
0 , (Πn − BI), and (Πn − BI) for n > 2.
For T a theory, let | T | denote the proof–theoretic ordinal of T .
Letting Ω(1, α) = Ωα , and Ω(n + 1, α) := ΩΩ(n,α) for n > 1, the following is true.
Theorem 10.2 Let n ≥ 2.
−
1. | (Π1n − BI)0 |=| (Π1n − BI)−
0 |=| KP ω + Πn − F oundation |= ϑΩ(n − 1, ω).
2. | (Π1n − BI) |=| (Π1n − BI)− |= ϑΩ(n − 1, ε0 ).
Proof. The results about the set theories are from [5].
11
t
u
ACA0 ` KT ↔ RF NΠ11 ((Π12 − BI)0 )
In view of Theorem 10.1 one is naturally led to search for a natural strengthening of (Π12 − BI)0 that
proves WF (ϑΩω ) and is not stronger than ACA0 + WF (ϑΩω ). Of course, (Π12 − BI) proves
WF (ϑΩω ), because the outer induction on k in the proof of Lemma 20.4 can be carried out as a
formal induction within (Π12 −BI). Hence (Π12 −BI) ` ∀nWF (ϑΩn ), thus (Π12 −BI) ` WF (ϑΩω ).
However, (Π12 − BI) has proof–theoretic ordinal ϑΩε0 ; so this is not the right candidate. It turns
out that the Uniform Reflection Principle for (Π12 − BI)0 , RFNΠ11 ((Π12 −BI)0 ), provides the appropiate strengthening. This scheme asserts the Π11 soundness (with parameters) of (Π12 − BI)0 .
Definition 11.1 RFNΠ11 ((Π12 − BI)0 ) is the scheme
∀n[P r(Π12 −BI)0 (pF (n̄)q) → F (n)]
for Π11 formulas F (a) with at most one free variable a. Here P r(Π12 −BI)0 denotes the Σ01 provability
predicate for (Π12 − BI)0 and pF (n̄)q signifies the Gödel number of F (a) when a is replaced by
the nth numeral (for details see [11]).
38
The proof of Theorem 10.1 can be formalized in ACA0 . Therefore ACA0 ` ∀xP r(Π12 −BI)0 (WF (ϑΩx )).
So we come to see the following.
Theorem 11.1 ACA0 + RFNΠ11 ((Π12 − BI)0 ) ` WF (ϑΩω ).
The remainder of the Section will be devoted to outlining the proof of the next theorem.
Theorem 11.2 ACA0 + WF (ϑΩω ) ` RFNΠ11 ((Π12 − BI)0 ).
We first give the rough idea of that proof. So let D ` F (n̄) be a (Π12 − BI)0 proof of a Π11
formula F (n̄) ≡ ∀XA(X, n̄). Using Theorem 9.1, we can pick a T ∗ derivation D∗ such that
Ωm
D∗ ω+1 A(U, n̄) for some m < Ω which is determined by D. Furthermore, we can transform D∗
α
m
into a cut–free derivation D∗∗ 0 A(U, n̄) with α = ψΩΩ . Since α < Ω, the derivation D∗∗ does
not contain instances of the Ω–rule. Especially, all the formulas occurring in D∗∗ have to be
subformulas of A(U, n̄). Thence, by induction on α one verifies that A(X, n) is true for any set
X of natural numbers. Since the formula A(U, n̄) is Σ0k for some k, this very last step requires
only a truth predicate for Σ0k formulas (with parameters) which is available in ACA0 .
A minor problem with the above sketch is that we didn’t exactly specify the amount of
transfinite induction that it requires.. Closer inspection reveals that we can restrict ourselves to
ω
ω
ω
ordinals from C(ΩΩ ). Since the order–type of < C(ΩΩ ) is not much bigger than σ := ψΩΩ ,
σ·ω
ω
ω
namely ω ω , the well–foundedness of < C(ΩΩ ) is still provable in ACA0 + WF (ψΩΩ ).
A considerably greater challenge is offered by the question: How can we formalize infinitary
ω
derivations in ACA0 + WF (ψΩΩ )? Obviously we have to use an effective counterpart of the
above construction, where we work with codes for T ∗ –derivations instead of using the T ∗ –
derivations themselves. The main idea here is that we can do everything with recursive proof–
trees instead of arbitrary derivations. A proof–tree is a tree, with each node labeled by: A
sequent, a rule of inference or the designation “Axiom”, two sets of formulas specifying the
set of principal and minor formulas,respectively, of that inference, and two ordinals (length and
cut–rank) such that the sequent is obtained from those immediately above it through application
of the specified rule of inference. The well-foundedness of a proof–tree is then witnessed by the
(first) ordinal “tags” which are in reverse order of the tree order.
If the inference is an instance of (Ω − rule), the label should also provide an index for the
fundamental function f and an index for the functional T that gives the transformations on
proof–trees (cf. Section 8); hence both are required to be recursive.
We then have to show that none of our manipulations on T ∗ –derivations leads us beyond
this class of recursive proof–trees. The latter is guaranteed by the fact that for the embedding
of (Π12 − BI)0 into T ∗ we only need instances of the (Ω − rule), where the transformation on
proof–trees is given by a recursive functional, and by the fact that all the operations in the
cut–elimination procedure are of a local nature, i.e. they give rise to recursive functionals.
To carry out all the details of this constructivization would mean to produce another lengthy
paper. But it is high time that we finished this paper; so we simply quit at this point.
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40