Download The superjump in Martin-Löf type theory

The superjump in Martin-Löf type theory
Michael Rathjen
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
Abstract
Universes of types were introduced into constructive type theory by MartinLöf [11]. The idea of forming universes in type theory is to introduce a universe
as a set closed under a certain specified ensemble of set constructors, say C.
The universe then ’reflects’ C.
Several gadgets for generating universes have been introduced into MartinLöf type theory (cf. [14, 15, 16]). Among them are the ordinary universe
operator U which acts on families of sets and produces a universe above the
given family which is closed under the standard ensemble of set constructors. A
yet stronger operator is the superuniverse operator which produces a universe
above a given family of sets which is also closed under the universe operator
U. It is plain how to iterate the latter way of devising new universe operators.
This paper introduces a system of Martin-Löf type theory which allows for
the introduction of new universe operators in an autonomous way in that each
operator F : Fam → Fam from families of sets to families of sets gives rise
to a new universe operator S(F ) : Fam → Fam. When applied to a family of
sets, S(F ) produces a universe above it which, in addition to the standard set
constructors, is closed under F . Let MLF denote this system of type theory.
The paper provides characterizations of MLF in terms of two intuitionistic
set theories which are extensions of Aczel’s constructive Zermelo-Fraenkel set
theory, CZF, via large set axioms as well as a characterization via a classical
set theory which describes a recursively Mahlo universe of sets.
The last section ventures to explore the boundaries of Martin-Löf type theory.
1
Introduction
Martin-Löf, in 1975 [11] and in his 1984 monograph [12] on an intuitionistic theory of
types, gave a framework for a theory of constructive types or sets. The role of universes
in this type theory is to allow for the formation of sets of sets which are themselves
closed under certain set forming operations that have been introduced at ’earlier’
stages, and thereby reflect these earlier stages. In this regard universes in constructive
type theory resemble the introduction of reflection cardinals in classical set theory.
Universes have greatly contributed to expanding the realm of constructivism. Strength
and expressiveness of Martin-Löf type theory are obtained through the use of inductive
data types and reflection, i.e. universes.
Several gadgets for generating universes have been introduced into Martin-Löf
type theory (cf. [14, 15, 16]). Among them are the ordinary universe operator U
1
which acts on families of sets and produces a universe above the given family which
is closed under the standard ensemble of set constructors. A yet stronger operator is
the superuniverse operator which produces a universes above a given family of sets
which is also closed under the universe operator U. It is plain how to iterate the
latter way of devising new universe operators. In this paper we introduce a system of
Martin-Löf type theory which allows for the introduction of new universe operators
in an autonomous way in that each operator F : Fam → Fam from families of sets
to families of sets gives rise to a new universe operator S(F ) : Fam → Fam. When
applied to a family of sets, S(F ) produces a universe above it which, in addition to
the standard set constructors, is closed under F . Let MLF denote this system of
type theory.
In thisSpaper we give characterizations of MLF in terms of two intuitionistic set
theories, n CZFn and CZF + (M),S which are extensions of Aczel’s constructive
Zermelo-Fraenkel set theory,
CZF.
n CZFn and CZF + (M) have canonical inS
terpretations in MLF. n CZFn is an extension of Aczel’s constructive set theory
CZF by axioms asserting the existence of many n-inaccessible sets for all external n,
whereas (M) is a reflection rule which forces any theorem to hold in arbitrary large
inaccessible sets.
The proof-theoretic strength of MLF from above is delineated by an extension
of classical Kripke-Platek set theory, KPMr , which formalizes a recursively Mahlo
universe of sets. MLF can be simulated in KPMr via a recursive realizability interpretation.
S
The proof-theoretic equivalence of n CZFn and KPMr has been established via
ordinal analysis, whence all four systems have the same strength.
1.1
Outline of the paper
The paper is organized as follows:
Section 2 recalls the type-theoretic universe operator U and the superuniverse V.
Section 3 introduces the rules S
for the superjump S which define the theory MLS.
The intuitionistic set theories n CZFn and CZF + (M) are introduced in section
4. Their interpretation in MLF is also given in that section. The realizability interpretation of MLF in a classical set theory describing a recursively Mahlo universe is
presented in section 5. The study of ever stronger systems of Martin-Löf type theory
are investigations that contribute to finding the boundaries of Martin-Löf type theory.
These boundaries are addressed in Section 6.
2
A brief history of universes
In [11, 12] Martin-Löf considered an infinite, externally indexed tower of universes
U0 ∈ U1 ∈ · · · ∈ Un ∈ . . . all of which are closed under the same standard ensemble
of set forming operations. The next natural step was to implement a universe operator
into type theory which takes a family of sets and constructs a universe above it. Such
a universe operator was formalized by Palmgren while working on a domain-theoretic
interpretation of the logical framework with an infinite sequence of universes (cf. [15]).
Aiming at extensions of type theory with more powerful axioms, Martin-Löf then
suggested finding axioms for a universe V which itself is closed under the universe
2
operator. The type-theoretic formalization of the pertinent rules appeared first in
Griffor and Palmgren [10] and are, in their final form, due to Palmgren [14], where
the universe was referred to as a superuniverse for intuitionistic type theory.
The formalisation of universes for intuitionistic type theory we use in this paper is
that referred to as the Tarski formulation in Martin-Löf’s monograph [12]. It involves
the simultaneous definition of a universe U and of a mapping T which, given an
element of U, produces the set coded by that element. The codes are built up using
codes for ground sets as well as codes for the set constructors. The mapping T on
codes for ground sets gives them as its values and on codes for constructors applied
to other codes produces those constructors applied to corresponding sets. Thus U
together with T give a family of sets.
In what follows we assume familiarity with type theory as presented in MartinLöf’s 1984 monograph [12] or in Beeson’s book [7].
2.1
The universe operator
In this subsection we introduce the universe operator U. Given a family of sets
A : Set
B(x) : Set [x ∈ A],
(1)
U and the decoding functional T produce a universe of sets U(A, B) whose decoding
functional is T(A, B). In addition to being closed under the standard set formers
Π, Σ, , . . . , the universe U(A, B) contains codes for the sets A, B(a), where a ∈ A.
2.1.1
U(P)-formation
The typing of U and T is spelled out in the introduction rules. Let A abbreviate the
standing assumptions of (1). We shall use the abbreviation P := A, B.
A
z ∈ U(P)
.
T(P, z) : Set
A
U(P) : Set
2.1.2
U(P)-introduction
That A, B(a) are in U(P) (though only via codes) is expressed by:
A
?P ∈ U(P)
A
a∈A
`P (a) ∈ U(P)
A
T(P, ?P ) = A : Set
A
a∈A
.
T(P, `P (a)) = B(a) : Set
Thus ?P is a code for A in U(P). `P (a) (for a ∈ A) provides a code for B(a) in U(P).
Furthermore, U(P) has rules for reflecting the constructors N,N0 , N1 , Π, Σ, +, I,
W:
A
A
A
b ∈ U(P)
N
b ∈ U(P)
N
0
b ∈ U(P)
N
1
3
A
b =N
T(P, N)
A
A
b
T(P, N0 ) = N0
A
b
T(P, N1 ) = N1
a ∈ U(P)
b ∈ U(P) [x ∈ T(P, a)]
b (a, (x)b) ∈ U(P)
2
A
a ∈ U(P)
b ∈ U(P) [x ∈ T(P, a)]
b (a, (x)b)) = 2(T(P, a), (x)T(P, b))
T(P, 2
where 2 is any of the constructors Π, Σ, W,
A
a ∈ U(P)
b ∈ U(P)
b ∈ U(P)
a+b
A
a ∈ U(P)
b ∈ U(P)
b = T(P, a) + T(P, b)
T(P, a+b)
A
a ∈ U(P)
x ∈ T(P, a)
bI(a, x, y) ∈ U(P)
y ∈ T(P, a)
A
a ∈ U(P)
x ∈ T(P, a)
y ∈ T(P, a)
.
T(P, bI(a, x, y)) = I(T(P, a), x, y)
Definition 2.1 MLU is the type theory which extends basic Martin-Löf type theory
via the above rules for the universe operator U.
2.2
The superuniverse
V denotes the superuniverse noted above. The formation rules for V are
a∈V
.
E(a) Set
V Set
The V-introduction rules stating the closure of V under universe formation have the
form:
a∈V
b ∈ V [x ∈ E(a)]
a∈V
b ∈ V [x ∈ E(a)]
u(a, (x)b) ∈ V
E(u(a, (x)b)) = U(E(a), (x)E(b))
a∈V
b ∈ V [x ∈ E(a)]
c ∈ E(u(a, (x)b))
t(a, (x)b, c) ∈ V
a∈V
b ∈ V [x ∈ E(a)]
c ∈ E(u(a, (x)b))
.
E(t(a, (x)b, c)) = T(E(a), (x)E(b), c)
In addition, V has the same rules as U(P) for reflecting the ordinary constructors
N,N0 , N1 , Π, Σ, +, I, W.
Definition 2.2 MLS is the extension of MLU via the foregoing rules for V and E.
The superuniverse construction can be parametrised by an arbitrary family of
sets, giving a new operator, a superuniverse operator. One may then form a super
superuniverse closed under the superuniverse operator. Iteration of the previous step
gives rise to successively stronger operators. Already here there is, at least informally,
an analogy with Mahlo’s π-numbers. The essence of the fundamental step in giving
a new universe operator is collecting a set constructors defined thus far and asserting
the existence of a universe closed under each constructor in this collection.
4
3
The type-theoretic superjump
The aim is to formalize a type theory that allows for the autonomous construction
of ever stronger universe operators and thereby naturally extends ideas leading to
theories like MLU and MLS. For the formalisation we use an extension of the
logical framework (Set, EL(.)) (cf. [13]) with dependent product types.
3.1
An extension of the logical framework
The logical framework, LF, is a finite dependent type structure over a collection of
ground types. The only ground types we will be concerned with here are Set and the
b for sets A. The logical framework’s elementary statements are
type of elements, A,
the judgements that something is a type, that two given types are equal, that some
object is of a given type, and that two objects the same objects of a given type. For
each A such that we have made the judgement A Set in constructive set theory we
have in the logical framework the judgement A : Set. The notation used for the four
judgement forms of LF is: α : type, α = β : type, a : α, and a = b : α, which are
read α is a type, the types α and β are identical, a is of type α, a and b are identical
objects of type α, respectively. In the logical framework every set A, i.e. A : Set, gives
b which is the type of A’s elements, and that a is an element of A is
rise to a type A
b However, we shall continue to use the expression a ∈ A (as in the
expressed by a : A.
b The rules of the logical framework have
theory of constructive sets) instead of a : A.
a form similar to those for the theory of constructive sets. We declare the ground type
Set by:
Set : type
A : Set
b : type
A
A = B : Set
b=B
b : type
A
Other types are formed using the type formation rules for dependent function types
and dependent product types:1
α : type
β : type [x : α]
(x : α)β : type
α : type
β : type [x : α]
.
(x : α) × β : type
The elements of the dependent function type (x : α)β are formed by abstraction
(x)t with application written in the form t(s). When β does not depend on x, we
sometimes emphasize this by writing α → β instead of (x : α)β.
The elements of the dependent product type (x : α) × β are formed by pairing
hx, yi with left and right projections pr0 and pr1 , respectively.
For more details on the logical framework we refer to the books [13], Part III and
[17], sect. 8.
3.2
Formalizing the superjump
Definition 3.1 Let Fam := (A : Set) × (A → Set). The objects of Fam → Fam
will be called operators.
1
The judgements within brackets are the discharged assumptions.
5
For F : Fam → Fam and a family of sets B : A → Set we shall use the following
abbreviations:
UF (A, B) := pr0 (F (hA, Bi)),
TF (A, B, z) := pr1 (F (hA, Bi))(z).
Let A be the assumptions F : Fam → Fam, A : Set, B : A → Set and Q := F, A, B.
The superjump
S : (Fam → Fam) → (Fam → Fam)
is defined as
S := (F )(C) hU (Q), (x)T(F, pr0 (C), pr1 (C), x)i
where U and T are determined by the following rules:
A
U (Q) : Set
A
a ∈ U (Q)
TQ (a) : Set
where TQ (a) := T(Q, a). The introduction rules for U (Q) include the usual ones for
ground sets and closure under the usual set constructors (as for U(P)). That A is in
U (Q) and B(a) (for a ∈ A) is in U (Q) is expressed by:2
A
?Q ∈ U (Q)
A
a∈A
`Q (a) ∈ U (Q)
A
TQ (?Q ) = A : Set
A
a∈A
.
TQ (`Q (a)) = B(a) : Set
Furthermore, we have rules that express closure of U (Q) under F . Here we shall
suppress the premiss A.
a ∈ U (Q)
b ∈ U (Q) [x ∈ TQ (a)]
b
UQ (a, (x)b) ∈ U (Q)
a ∈ U (Q)
b ∈ U (Q) [x ∈ TQ (a)]
b (a, (x)b)) = UF (TQ (a), (x)TQ (b)) : Set
TQ (U
Q
a ∈ U (Q)
b Q (a, (x)b))
b ∈ U (Q) [x ∈ TQ (a)]
z ∈ TQ (U
b (a, (x)b, z) ∈ U (Q)
T
Q
b Q (a, (x)b))
a ∈ U (Q)
b ∈ U (Q) [x ∈ TQ (a)]
z ∈ TQ (U
b (a, (x)b, z)) = TF (TQ (a), (x)TQ (b), z) : Set
TQ (T
Q
Example 3.2 (Universe Operator) Let F : Fam → Fam be a judgement of MLF.
Then MLU can be interpreted in MLF by simulating U, T via U (F, ·, ·), T(F, ·, ·, ·).
Example 3.3 (Super Universe Operator) Let F : Fam → Fam, A : Set, B : A →
Set be judgements of MLF. Let G := S(F ). Then MLS can be interpreted in MLF
by simulating V, E via U (G, A, B), (x)T(G, A, B, x).
b Q (a, (x)b),T
b Q (a, (x)b, c) instead of ?(Q), `(Q, a), U(Q,
b
We’ll write ?Q , `Q (a), U
a, (x)b),
b
T(Q, a, (x)b, c), respectively.
2
6
4
Constructive set theory with reflection and ninaccessibles
In this section we introduce constructive set theories CZF + (M) and CZFn which
are extensions of Aczels set theory CZF (cf. [2, 3, 4]). CZF + (M) has a reflection
rule (M) which forces any theorem of the theory to hold in arbitrary large inaccessible
sets. CZFn has an axiom that asserts the existence of many n-inaccessible sets. It
will be shown that both the S
theories CZF + (M) and CZFn can be interpreted in
MLF. As CZF + (M) and n CZFn also exhausts the proof-theoretic strength of
MLF one obtains interesting characterizations of the strength of MLF in terms of
constructive Zermelo-Fraenkel set theories.
The notion of regularity for sets stems from [4] (also see [21]). The notion of
set-inaccessibility has been introduced and discussed in [21], Definition 2.5.
Definition 4.1 Let A be a set. A is 0-set-inaccessible if A is regular. A is n + 1-setinaccessible if A is a set-inaccessible and
(∀x∈A) (∃y∈A) [x ⊆ y ∧ y is n-set-inaccessible ].
Definition 4.2 CZFn is the theory CZF augmented by the axiom
∀x ∃y [x ∈ y ∧ y is n-set-inaccessible].
Note that CZF0 = CZF + REA.
Definition 4.3 Let (M) be the rule
φ
¤
∀x ∃y x ⊆ y ∧ y is set-inaccessible ∧ φy
£
where φ is arbitrary sentence of CZF and φy is the result of restricting all quantifiers
to y.
n
We use CZF + (M) ψ to convey that the rule (M) has been used at most n
times in deducing ψ.
Lemma 4.4 (CZF) Let I be a regular set such that ∀x, y ∈ I (hx, yi ∈ I). Let A ∈ I.
If
I |= A is n-set-inaccessible
then A is n-set-inaccessible.
Proof : We use induction on n. For n = 0 it suffices to show that for A ∈ I, I |=
A is regular implies that A is regular. So assume A ∈ I and I |= A is regular.
Let a ∈ A and R ∈ mv(a A), where mv(a A) denotes the class of multi-valued functions
from a to A. Then R is a subset of a×A such that for all x ∈ a there is y ∈ A such that
xRy. Let R0 be the set of all hx, hx, yii such that xRy. Then R0 ∈ mv(a I) as I is closed
under Pairing. Hence, as I is regular, there is S ∈ I such that ∀x ∈ a ∃z ∈ S xR0 z ∧
∀z ∈ S ∃x ∈ a xR0 z. Hence S ⊆ R and S ∈ (I ∩ mv(a A)). Since I |= A is regular
there exists b ∈ I such that ∀x ∈ a ∃y ∈ b xSy ∧ ∀y ∈ b ∃x ∈ a xSy. Consequently,
∀x ∈ a ∃y ∈ b xRy ∧ ∀y ∈ b ∃x ∈ a xRy, showing that A is regular.
7
Now let n = k + 1. By induction hypothesis, A is k-set-inaccessible, in particular
A is regular. Also A |= CZF. Let x ∈ A. Then there exists y ∈ A such that
x ⊆ y and I |= y is k-set-inaccessible. Hence y is k-set-inaccessible by induction
hypothesis. The upshot of the above is that A is n-set-inaccessible.
u
t
Corollary 4.5 For every (meta) n,
CZF + (M) ` ∀x ∃y [x ⊆ y ∧ y is n-set-inaccessible].
Proof : Use external induction on n. For n = 0 the assertion follows by applying (M)
to CZF ` ψ with ψ := ∀u u = u. If n = k + 1 let
ψ := ∀u ∃v [u ⊆ v ∧ v is k-set-inaccessible].
By induction hypothesis, CZF + (M) ` ψ. Employing (M) we get
CZF + (M) ` ∀x ∃y [x ⊆ y ∧ y is set-inaccessible ∧ ψ y ].
Using Lemma 4.4, the latter implies
CZF + (M) ` ∀x ∃y [x ⊆ y ∧ y is n-set-inaccessible].
u
t
Corollary 4.6 Every theorem of
S
n
CZFn is a theorem of CZF + (M).
n
Lemma 4.7 If CZF + (M) θ , then
£
¤
CZF ` ∀I I n + 1-set-inaccessible → I |= ∀θ
where ∀θ stands for the universal closure of θ.
Proof : Use induction on n. The assertion is trivial for n = 0 since set-inaccessible
sets satisfy all the axioms of CZF.
£ Now let n = k + 1. If θ is the result of applying
¤
(M), then θ has the form ∀x∃y x ⊆ y ∧ y is set-inaccessible ∧ ψ y with
CZF + (M)
k
ψ . Inductively we get
£
¤
CZF ` ∀A A n-set-inaccessible → A |= ψ .
£
¤
The latter yields CZF ` ∀I I n + 1-set-inaccessible → I |= θ .
If θ is the result of a logical inference then the assertion follows immediately from
the induction hypothesis as validity is preserved under logical inferences.
u
t
Theorem 4.8
S
n
CZFn and CZF + (M) have the same strength.
Proof : From Corollary 4.6 and Lemma 4.7 it follows that both theories prove (at
least) the same Π2 statements (a Π2 formula is of the form ∀x∃yψ with ψ containing
only bounded quantifiers).
u
t
8
4.1
Interpreting CZFn in MLF
We will assume familiarity with Aczel’s interpretation of CZF in Martin-Löf type
theory (cf. [2, 3, 4, 21].
Definition 4.9 Assume G : Fam → Fam, A : Set, and B : A → Set.
Let VG,A,B denote the set of iterative sets over the universe U (G, A, B), i.e.
VG,A,B = (Wx ∈ U (G, A, B))T(G, A, B, x). Thus VG,A,B is inductively generated
by the rule
a ∈ U (G, A, B)
b ∈ T(G, A, B, a) → VG,A,B
.
sup(a, b) ∈ VG,A,B
Until further notice we shall write U, V, T(a) for U (G, A, B), VG,A,B , T(G, A, B, a),
respectively.
Let α, β, γ, . . . range over elements of V.
Associated with the rule inductively specifying the set V is the method of definition
by transfinite recursion on V corresponding to the elimination rule for W-types. If d is
a 3-place function such that d(a, b, e) ∈ C(sup(a, b)) for all a ∈ U, b ∈ T(a) → V and
e ∈ (Πx ∈ A)C(b(x)), where C is a family of types over V, then we have h ∈ Π(V, C)
defined so that
h(sup(a, b)) = d(a, b, (u)h(b(u)))
holds for a ∈ U and b ∈ T(a) → V. An important property of α ∈ V is that
we can always recover its branching-index ᾱ ∈ U and the corresponding mapping
α̃ : T(ᾱ) → V.
Lemma 4.10 There are one-place functions assigning ᾱ ∈ U and α̃ : T(ᾱ) → V to
α ∈ V such that if α = sup(a, b) where a ∈ U and b ∈ T(a) → V, then ᾱ = a ∈ A
and α̃ = b ∈ T(ᾱ) → V. Moreover, α = sup(ᾱ, α̃) ∈ V for α ∈ V.
Proof : [3], Theorem 2.1.
u
t
If B(x) is a proposition for x ∈ V then define
˙ B(β) := ∀x ∈ ᾱ B(α̃(x))
∀β ∈α
˙ B(β) := ∃x ∈ ᾱ B(α̃(x)).
∃β ∈α
(2)
Lemma 4.11 If F is species over V (i.e. a family of propositions over V), then
£
¤
˙
∀α ∀β ∈αF
(β) → F (α) → ∀αF (α)
holds true.
Proof : This is an immediate application of V-induction.
u
t
Lemma 4.12 There is a species on V × V assigning a small proposition (i.e. a
proposition in U) (α=β)
˙
to α, β ∈ V such that
£
¤
˙ ∃δ ∈β
˙ (γ =δ)
˙ ∃γ ∈α
˙ (γ =δ)
(α=β)
˙
= ∀γ ∈α
˙ ∧ ∀δ ∈β
˙
.
(3)
9
Proof : See [3], 2.2.
u
t
In [3, 4] Aczel has shown that V validates CZF + REA when elementhood is interpreted by ∈˙ and equality is interpreted as =.
˙ To be precise, under the above
interpretation each theorem φ of CZF + REA is translated into a proposition φ∗ of
MLF such that MLF ` t ∈ φ∗ for a suitable term t. The latter will be shortened into
˙ =i
hV; ∈;
˙ ² φ.
For the remainder of this subsection we shall assume that F : Fam → Fam is a
judgement of MLF. Let F0 := F and Fn+1 := S(Fn ).
Theorem 4.13 If A : Set, B : A → Set and m ≥ n, then
˙ =i
hVFm ,A,B ; ∈;
˙ ² CZFn .
˙ =i.
Proof : Let VFm ,A,B be a shorthand for the structure hVFm ,A,B ; ∈;
˙
The proof proceeds by (external) induction on n. For n = 0 this is just Aczel’s
result from [4]. Now assume that the claim holds true for n. Let m ≥ n + 1. We have
to show that
VFm ,A,B |= ∀x ∃y [x∈y ∧ y is n + 1-set-inaccessible].
(4)
Let α ∈ VFm ,A,B . Then there exists β ∈ VFm ,A,B such that
VFm ,A,B |= β is the transitive closure of α.
Define
β̂(u) := β̃(u)
for u ∈ TFm (A, B, β̄). Put
C := TFm (A, B, β̄)
and let
D(u) := TFm (A, B, β̂(u))
for u ∈ C. Then C : Set and D : C → Set. Define
h : VFm−1 ,C,D → VFm ,A,B
by recursion on VFm−1 ,C,D via
b Q (β̄, (x)β̂(x), a), (u)h(g(u)))
h(sup(a, g)) = sup(T
with Q := Fm , A, B, and put
δ := sup(b, h)
where b ∈ VFm ,A,B is determined by TFm (A, B, b) = VFm−1 ,C,D , which can be achieved
c U
b Q (β̄, β̂), (x)T
b Q (β̄, β̂, x)).
by letting b := W(
One then verifies
(i) VFm ,A,B |= δ is transitive.
£
¤
(ii) ∀z, w ∈ VFm−1 ,C,D VFm−1 ,C,D |= z = w ↔ VFm ,A,B |= h(z) = h(w) ,
10
£
¤
(iii) ∀z, w ∈ VFm−1 ,C,D VFm−1 ,C,D |= z ∈ w ↔ VFm ,A,B |= h(z) ∈ h(w)
in the same way as [21], Lemma 4.7. From the latter we get
VFm ,A,B |= δ is set-inaccessible.
(5)
By induction hypothesis,
VFm−1 ,A,B |= CZFn ,
and thus, by (5),
VFm ,A,B |= δ is set-inaccessible and δ is a model of CZFn .
As a consequence of Lemma 4.4 we get
VFm ,A,B |= β ∈ δ ∧ δ is n + 1-set-inaccessible.
u
t
Theorem 4.14 If A : Set, B : A → Set, m ≥ n, and CZF + (M)
n
θ , then
˙ =i
hVFm ,A,B ; ∈;
˙ ² θ.
˙ =i
Proof : By Theorem 4.13 we have hVFm ,A,B ; ∈;
˙ ² CZFn . The structure VFm ,A,B can
be embedded into a bigger structure of the form VFk ,A0 ,B 0 for k > m and appropriate
A0 , B 0 so that it is isomorphic to some δ ∈ VFk ,A0 ,B 0 satisfying
VFk ,A0 ,B 0 |= δ is n + 1-set-inaccessible.
From the latter and Lemma 4.7 we get the desired assertion.
5
u
t
Modelling MLF in set theory
To construct a model of MLF that reflects its proof-theoretic strength, we shall resort
to realizability models. First we will lay out an extension of Kripke-Platek set theory,
KPMr , in which the model construction can be formalized.
5.1
The theory KPMr
KPMr is an extension of Kripke-Platek set theory. Its standard models are the sets
Lµ , where µ is a recursively Mahlo ordinal.
In addition to the usual language of set theory with equality = and elementhood
∈, the language of KPMr has a unary predicate symbol Ad to signify that a set is
admissible. For convenience we shall also assume that the language of KPMr has a
constant ω for the first infinite ordinal.
We use φ, ψ, ..., φ(a), ψ(a), .. as meta-variables for formulae. We shall write b =
{y ∈ a : φ(y)} for (∀y ∈ b)[y ∈ a ∧ φ(y)] ∧ (∀y ∈ a)[φ(y) → y ∈ x]. A formula which
contains only bounded quantifiers, i.e. quantifiers of the form (∀x ∈ b), (∃x ∈ b), is said
to be a ∆0 –formula.
11
For a formula ψ the formula ψ b is the result of restricting all unbounded quantifiers
in ψ to b.
The logical rules and axioms of KPMr are the usual ones for classical predicate
logic with equality.
The set–theoretic axioms of KPMr are the universal closures of the following
formulae:
Extensionality:
Foundation Axiom:
Infinity:
(rec-Mahlo):
∀x (x ∈ a ↔ x ∈ b) → a = b.
£
¤
∀x x 6= ∅ → (∃y ∈ x) y ∩ x = ∅ .
£
¤
∃z ∅ ∈ z ∧ ∀y ∈ z y ∪ {y} ∈ z .
£
¤
∀x∃yφ(x, y) → ∃z Ad(z) ∧ (∀x ∈ z)(∃y ∈ z)φ(x, y)
for all ∆0 –formulae φ(a, b).
Ad-Linearity:
∀u ∀v [Ad(u) ∧ Ad(v) → u ∈ v ∨ u = v ∨ v ∈ u].
(Ad1):
Ad(a) → ω ∈ a ∧ ∀x ∈ a ∀z ∈ x z ∈ a.
(Ad2):
Ad(a) → ψ a ,
where the sentence ψ is a universal closure of one of the following
axioms:
Pairing:
Union:
∆0 –Separation:
∆0 –Collection:
∃x (x = {a, b}).
S
∃x (x = a).
¡
¢
∃x x = {y ∈ a : φ(y)} for all ∆0 –formulae φ(b)
(∀x ∈ a)∃yψ(x, y) → ∃z(∀x ∈ a)(∃y ∈ z)ψ(x, y)
for all ∆0 –formulae ψ(a, b).
Remark 5.1 Note that KPMr does not have the full schema of foundation, i.e.
∃u φ(u) → ∃u [φ(u) ∧ (∀v∈u)¬φ(v)]
for all formulae φ. The latter is part of Kripke-Platek set theory (cf. [5]). To be
able to define functions and predicates by Σ recursion (cf. [5]) one needs the above
schema for Π1 formulae. Fortunately the combination of the foundatiom axiom and
the Mahlo scheme give KPMr enough strength for carrying out such definitions.
Lemma 5.2 In KPMr one can define functions and predicates by Σ recursion in the
sense of [5], I.6.4.
Proof : Suppose ∀xy ∃!z φ(x, y, z), where φ is Σ. The aim is to exhibit a Σ definable
class function F such that
∀x φ(x, F ¹TC(x), F (x))
where TC(x) denotes the transitive closure of x and F ¹a is the restriction of F to a.
Let ~c be the parameters of φ. Given a set b, we can find an admissible set A such that
b ∈ A, ~c ∈ A and ∀x∈A ∀y∈A ∃!z∈A φ(x, y, z)A . The latter follows from (rec-Mahlo).
The axiom of foundation yields that the scheme of foundation holds in the set A.
Hence, as A is admissible, we can define a (unique) function FA on A by Σ recursion
in A such that
∀x∈A φ(x, FA ¹TC(x), FA (x))A .
12
The latter implies ∀x∈A φ(x, FA ¹TC(x), FA (x)). Therefore we obtain the desired function by letting
[
F := {FA : A is admissible ∧ ~c ∈ A}.
u
t
5.2
Modelling MLF in KPMr
In this section KPMr will be the background theory. We will write N for ω. To
simplify the treatment we will assume that KPMr has names 0, 1, 2, 3, . . . for all
(meta) natural numbers. We also assume that KPMr has function symbols for addition and multiplication on N as well as for a primitive recursive bijective pairing function h, iN : N2 → N and its primitive recursive inverses ( )0 , ( )1 , that satisfy (hn, miN )0 = n and (hn, miN )1 = m. KPMr should also have a symbol T for
Kleene’s T -predicate and the result extracting function U . Let [e](n) ' k be a shorthand for ∃m[T (e, n, m) ∧ U (m) = k]. Further, let hn, m, kiN := hhn, miN , kiN ,
hn, m, k, liN := hhn, m, kiN , liN , etc. We use e, d, f, n, m, l, k, s, t, j, i as metavariables for natural numbers. Let
π(n, m)
pl(n, m)
w(n, m)
N̂
sup(n, m)
=
=
=
=
=
h0, hn, miN iN
σ(n, m)
h2, hn, miN iN i(n, m, k)
h4, hn, miN iN u(e, m, k)
h6, 0iN
N̂k
h7, hn, miN iN
=
=
=
=
h1, hn, miN iN
h3, hn, m, kiN iN
h5, he, m, kiN iN
h6, k + 1iN
Definition 5.3 By Σ recursion (cf. [5],I.6.4) along the ordinals we define simultaneously five relations R1α , . . . , R5α on N. To increase intelligibility, we shall write
Vα |=n set
Vα |=n ∈ m
Vα |=n = m ∈ k
Vα |=n = m
Vα |=“f is a family of types over k”
for
for
for
for
for
R1α (n)
R2α (n, m)
R3α (n, m, k)
R4α (n, m)
R5α (f, k)
We use the abbreviation “ V<α |= . . . ” for “∃γ < α Vγ |= . . . ”.
If α = 0, then R10 = · · · = R50 = ∅. The clauses in the definition for successors are
as follows:
(i)
Vα+1 |= h6, jiN set
iff
j∈N
Vα+1 |= m ∈ h6, jiN
iff
j =0 ∨ m+1<j
Vα+1 |= n = m ∈ h6, jiN
iff
n = m ∧ Vα+1 |= n ∈ h6, jiN .
(ii) If Vα |= k set, ∀j[ Vα |= j ∈ k ⇒ Vα |= [e](j) set] and
∀j, i[ Vα |= i = j ∈ k ⇒ Vα |= [e](i) = [e](j)], then
Vα+1 |=“ e is a family of types over k”.
13
(iii) If Vα |= “ e is a family of types over k”, then Vα+1 |= π(k, e) set,
Vα+1 |= σ(k, e) set and, letting Z := Vα+1 ,
Z |= n ∈ π(k, e)
iff ∀i ( Vα |= i ∈ k ⇒ Vα |= [n](i) ∈ [e](i)) and
∀i, j [ Vα |= i = j ∈ k ⇒ Vα |= [n](i) = [n](j) ∈ [e](i)]
Z |= n = m ∈ π(k, e)
iff Z |= n ∈ π(k, e) ∧ Z |= m ∈ π(k, e) and
∀j [ Vα |= j ∈ k ⇒ Vα |= [n](j) = [m](j) ∈ [e](j)]
Z |= n ∈ σ(k, e)
iff
Vα |= (n)0 ∈ k and Vα |= (n)1 ∈ [e]((n)0 )
Z |= n = m ∈ σ(k, e)
iff Z |= n ∈ σ(k, e) ∧ Z |= m ∈ σ(k, e) and
Vα |= (n)0 = (m)0 ∈ k ∧ Vα |= (n)1 = (m)1 ∈ [e]((n)0 ).
(iv) If Vα |= n set and Vα |= m set, then Vα+1 |= pl(n, m) set and
Vα+1 |= i ∈ pl(n, m)
iff
[(i)0 = 0 and Vα |= (i)1 ∈ n] or
[(i)0 = 1 and Vα |= (i)1 ∈ m]
Vα+1 |= i = j ∈ pl(n, m)
iff
[(i)0 = (j)0 = 0 and Vα |= (i)1 = (j)1 ∈ n] or
[(i)0 = (j)0 = 1 and Vα |= (i)1 = (j)1 ∈ m].
(v) If Vα |= n set, then Vα+1 |= i(n, m, k) set and
Vα+1 |= s ∈ i(n, m, k)
iff s = 0 and Vα |= m = k ∈ n,
0
Vα+1 |= s = s ∈ i(n, m, k) iff s = s0 = 0 and Vα |= m = k ∈ n.
(vi) Vα+1 |= e = f iff Vα |= e set, Vα |= f set, ∀s( Vα |= s ∈ e ⇒ Vα |= s ∈ f )
and ∀s, t( Vα |= s = t ∈ e ⇒ Vα |= s = t ∈ f ).
(vii) If Vα |= a set, and Vα |= “ e is a family of types over a ”, then
(a) Vα+1 |= w(a, e) set,
(b) Vα+1 |= x ∈ w(a, e) iff xSx,
(c) Vα+1 |= x = y ∈ w(a, e) iff xSy,
where S is inductively defined (on N) by the rule:
£
¤
if Vα |= x = y ∈ a ∧ ∀i, j Vα |= i = j ∈ [e](x) ⇒ [u](i)S[v](j) ,
then sup(x, u)S sup(y, v).
S
If α is a limit which is not recursively inaccessible, then Riα = β<α Riβ for i = 1, . . . , 5.
S
If α is recursively inaccessible, then Riα ⊇ β<α Riβ for i = 1, . . . , 5. In addition,
the following clauses obtain:
Suppose V<α |= “g0 is a family of types over m0 ” and e satisfies
£
∀f k V<α |= “f is a family of types over k” ⇒
¤
V<α |= “([e](hk, f iN ))1 is a family of types over ([e](hk, f iN ))0 ” .
14
Then Vα |= u(e, g0 , m0 ) set and, for all n, Vα |= n ∈ u(e, g0 , m0 ) iff n belongs to the
smallest set X such that m0 ∈ X, ∀n0 [ V<α |= n0 ∈ m0 ⇒ [g0 ](n0 ) ∈ X], and X is
closed under the clauses:
1. h6, jiN ∈ X for j∈ω.
2. If V<α |= “f is a family of types over k”, k ∈ X and ∀m0 [ V<α |= m0 ∈ k ⇒
[f ](m0 ) ∈ X], then π(k, f ), σ(k, f ), w(k, f ) ∈ X.
3. If V<α |= n set, V<α |= m set, and n, m ∈ X, then pl(n, m), i(n, l, l0 ) ∈ X for
all l, l0 ∈ ω.
4. If V<α |= “f is a family of types over k”, k ∈ X and ∀m0 [ V<α |= m0 ∈ k ⇒
[f ](m0 ) ∈ X], then ([e](hk, f iN ))0 ∈ X and
∀n [ V<α |= n ∈ ([e](hk, f iN ))0 ⇒ [([e](hk, f iN ))1 ](n) ∈ X].
Note that such a set X is Σ-definable over any admissible set above α. Finally put
Vα |= n = n0 ∈ u(e, g0 , m0 ) if Vα |= n ∈ u(e, g0 , m0 ), Vα |= n0 ∈ u(e, g0 , m0 ), and
V<α |= n = n0 .
Lemma 5.4 If α ≤ β and Vα |= k set, then the following hold true:
1. Vβ |= k set;
2. ∀n( Vα |= n ∈ k ⇔ Vβ |= n ∈ k);
3. ∀n, m( Vα |= n = m ∈ k ⇔ Vβ |= n = m ∈ k);
4. If Vα |= n = m, then Vβ |= n = m;
5. If Vα |=“f is a family of types over n”, then Vβ |=“f is a family of types over
n”.
Proof : All the assertions are proved simultaneously by induction on β. Trivial, but
lengthy.
u
t
Definition 5.5 The previous lemma justifies the next definition.
V |= k set
V |= n ∈ k set
V |= n = m
V |= “f is a family of types over n”
:=
:=
:=
:=
∃α
∃α
∃α
∃α
Vα
Vα
Vα
Vα
|= k set
|= n ∈ k
|= n = m
|= “f is a family of types over n”.
Lemma 5.6 Suppose V |= “g0 is a family of types over m0 ” and e satisfies
£
∀f k V |= “f is a family of types over k” ⇒
¤
V |= “([e](hk, f iN ))1 is a family of types over ([e](hk, f iN ))0 ” .
Then V |= u(e, g0 , m0 ) set and and V |= n ∈ u(e, g0 , m0 ) if n belongs to the least
set X such that m0 ∈ X, ∀n0 [ V |= n ∈ m0 ⇒ [g0 ](n) ∈ X] and X is closed
under the clauses (i)–(vii) formulated for “ V |=” instead of “ Vα |=”. Moreover,
V |= n = n0 ∈ u(e, g0 , m0 ) if V |= n ∈ u(e, g0 , m0 ), V |= n0 ∈ u(e, g0 , m0 ), and
V |= n = n0 .
15
Proof : The assumptions yield
£
∀α ∀f k ∃β Vα |= “f is a family of types over k” ⇒
¤
Vβ |= “([e](hk, f iN ))1 is a family of types over ([e](hk, f iN ))0 ” .
As the matrix [. . . ] of this formula is equivalent to a Σ1 formula, we can invoke (recMahlo) to conclude that there exists a recursively inaccessible ordinal π such that
V<π |= “g0 is a family of types over m0 ”
and
£
∀α < π ∀f k ∃β < π Vα |= “f is a family of types over k” ⇒
¤
Vβ |= “([e](hk, f iN ))1 is a family of types over ([e](hk, f iN ))0 ”
and hence
V<π
£
∀f k V<π |= “f is a family of types over k” ⇒
¤
|= “([e](hk, f iN ))1 is a family of types over ([e](hk, f iN ))0 ” .
As a result, V<π |= u(e, g0 , m0 ) set. The remaining assertions follow from this.
5.3
u
t
The interpretation
In order to define its interpretation in KPMr , we need a more detailed account of the
syntax of MLF. Here we will follow [7], Ch.XI; however, for the readers convenience,
we shall recall most of the definitions.
Definition 5.7 (cf. [7], XI.20.3) The constants of MLF are: Π, Σ, I, +, W, N, N0 , N1 ,
b T,?,`.
b
0, sN ,r, λ, ap, E, i,j,D, J,R,R0 ,R1 , Π̂, Σ̂,Î,+̂,N̂, N̂0 , N̂1 , U , T, pr0 , pr1 , p.q,U,
The first slew of constants up to R1 is taken from [12] and we refer to [12] for their
pertinent rules.
The terms are generated by:
1. Every constant and variable is a term;
2. If t and s are terms, then t(s) is a term.
3. If t is a term, then (x)t is a term.
To make expressions more readable a repeated application expression E(E1 ) · · · (En )
will be abbreviated E(E1 , . . . , En ) and a repeated abstraction expression (x1 ) · · · (xn )E
will be abbreviated (x1 , . . . , xn )E.
Free and bound occurrences of variables in terms are defined as usual, letting
abstraction, i.e. the formation of (x1 , . . . , xn )t bind the variables x1 , . . . , xn .
If B is any expression, and x1 , . . . , xn are variables, we form the expression
4
(x1 , . . . , xn )B. The symbol ≡ will be used for the relation on expressions satisfying
4
((x1 , . . . , xn )B)(x1 , . . . , xn ) ≡ B
4
and A ≡ C for expressions A and C which differ only in the renaming of bound
variables (cf. [7], XI6).
16
We now would like to assign to every term t of MLF a corresponding term t∗ of the
theory KPMr by replacing the abstract application of MLF with recursive application. However, the technicality here is that recursive application cannot be expressed
on the level of terms within KPMr . Therefore, we first assign to every term t of
MLF an application term t∗ of the theory EON of [7], VI.2 or equivalently of the
theory APP as described in [27], Ch.9, Sect.3. It is then a straightforward matter to
translate a formula of the form t∗ ∈ X into a legitimate formula of KPMr .
Definition 5.8 The theory EON is defined in [7], VI.2.
EON has two important devices for defining functions, the Abstraction Theorem (cf.
[7], VI.2.2) and the Recursion Theorem (cf. [7], VI.2.7) which we shall employ in the
next definition.
Definition 5.9 We now assign to each term t of MLS an application term t∗ of
the theory EON. Occurrences of λ in the definition of t∗ denote the λ-operator
introduced in [7], VI.2.2. We shall write (x, y) for pxy and, inductively, (x1 , . . . , xk+1 )
for p(x1 , . . . , xk )xk+1 . For constants c we define c∗ by:
0∗
∗
Π̂
∗
Σ̂
∗
+̂
∗
Î
Ŵ∗
T∗
b∗
T
b∗
U
∗
N̂k
s∗N
λ∗
D∗
i∗
J∗
t∗
p.q∗
pr∗1
is
is
is
is
is
is
is
is
is
is
is
is
is
is
is
is
is
is
0
Π∗ is λx.λy.(0, x, y)
λx.λy.(0, x, y)
Σ∗ is λx.λy.(1, x, y)
λx.λy.(1, x, y)
+∗ is λx.λy.(2, x, y)
λx.λy.(2, x, y)
I∗ is λx.λy.λz.(3, x, y, z)
λx.λy.λz.(3, x, y, z)
W∗ is λx.λy.(4, x, y)
λx.λy.(4, x, y)
U ∗ is λx.λy.λz.(5, x, y, z)
λx.λy.λz.λv. v
N∗ is (6, 0)
∗
λe.λx.λy.λz.λu.λv. (p1 (e(z, u)))(v)
N̂ is (6, 0)
λe.λx.λy.λz.λu. p0 (e(z, u))
N∗k is (6, k + 1)
(6, k + 1)
sup∗ is λx.λy.(7, x, y)
sN
r∗ is 0
λx.x (i.e. skk)
ap∗ is λx.λy.yx
λx.λy.λz(0, p0 x, y(p0 x), z(p1 x))
E∗ is λx.λy.y(p0 x, p1 x)
λx.(0, x)
j∗ is λx.(1, x)
λx.λy.y
?∗ is λx.λy.λz.y
λx.λx0 .λy.λy 0 .λz.λz 0 .z 0
`∗ is λx.λy.λz.λv.z(v)
λx.λy.(x, y)
pr∗0 is λx.p0 x
λx.p1 x
R∗k is λm.λx0 · · · λxk−1 .ek (m, x0 , . . . , xk − 1), where ek is chosen so that EON proves
½
xm if m < k
ek (m, x0 , . . . , xk − 1) '
Ω otherwise,
where Ω := λx.xx(λx.xx) (signifying undefinedness). R∗ is an application term of
EON introduced by the Recursion Theorem to satisfy (provably in EON) R∗ (a, b, 0) =
a and R∗ (a, b, sN x) ' b(x, R∗ (a, b, x)). For complex terms of MLS we define:
((x1 , . . . , xn )t)∗ is λx1 · · · λxn .t∗ ;
(t(s))∗ is t∗ s∗ ;
(t, s)∗ is pt∗ s∗ .
17
Definition 5.10 Rather than translating application terms of EON into the settheoretic language of KPMr , we define the translation of expressions of the form
t ' u, where t is an application term of EON and u is a variable. First, we select
indices k̄, s̄, p̄, p̄0 , p̄1 , s̄N , p̄N and d̄ for partial recursive functions so that:
[[k̄](n)](m)
[[[s̄](n)](m)](l)
[[p̄](n)](m)
[p̄0 ](n)
[p̄1 ](n)
[s̄N ](n)
[p̄N ](0)
[p̄N ](n + 1)
'
'
'
'
'
'
'
'
n
[[n](l)]([m](l))
2n · 3m = hn, miN
(n)0
(n)1
n+1
0
n
½
x if n = m
[[[[d̄](n)](m)](x)](y) '
y if n 6= m
The definition of (t ' u)4 proceeds along the way that t was built up.
(v ' u)4 is v = u ∧ v ∈ ω if v is a variable;
(0 ' u)4 is 0 = u;
(c ' u)4 is c̄ = u if c is one of the constants k, s, p, p0 , p1 , sN , pN , d.
A complex application term of EON has the form ts.Let
(ts ' u)4 := ∃x, y ∈ ω[(t ' x)4 ∧ (s ' y)4 ∧ [x](y) ' u].
EON is interpreted in KPMr by setting
(Ap(s, t, r))¦
(s = t)¦
(s ∈ N)¦
(φ2ψ)¦
(∀xφ(x))¦
:=
:=
:=
:=
:=
∃u ∈ ω [(st ' u)4 ∧ (r ' u)4 ]
∃u ∈ ω [(s ' u)4 ∧ (t ' u)4 ]
∃u ∈ ω [(s ' u)4 ]
φ¦ 2ψ ¦ (2 ∈ {∧, ∨, →})
∀x ∈ ω φ(x)¦ .
Definition 5.11 The pseudo type terms of MLF are defined inductively by
1. If t is a pseudo term of MLF, then t̂ is a pseudo type term.
2. Set is a pseudo type term.
3. If A and B are pseudo type terms, then (x : A)B and (x : A) × B are pseudo
type terms.
4
4. If A is a pseudo type term and C ≡ A, then C is a pseudo type term.
Definition 5.12 (Interpretation of MLF in KPMr ) By induction on the complexity
of the pseudo type term A we shall assign to each judgement Φ of MLF of the form
u : A or u = v : A (u, v variables) a formula (Φ)4 of KPMr with the same free
variables. (ux : A)4 and (ux = uy : A)4 will be used as shorthand for ∃z : ω[[u](x) '
z ∧ (z : A)4 ] and ∃z : ω[[u](x) ' z ∧ [u](y) ' z ∧ (z : A)4 ], respectively. Likewise,
18
(u : A(vx))4 abbreviates ∃z : ω[[v](x) ' z ∧ (u : A(z))4 ], etc. The clauses in the
definition are as follows:
(u : Set)4
(u = v : Set)4
(u : t̂ )4
is
is
is
V |= u
V |= u
∃x ∈ ω
set
set ∧ V |= v set ∧ V |= u = v
£ ∗
¤
(t ' x)4 ∧ V |= u ∈ x
(u = v : t̂ )4
(u : (x : A)B)4
is
is
(u = v : (x : A)B)4
is
(u : (x : A) × B)4
(u = v : (x : A) × B)4
is
is
∃x∈ω [ (t∗ ' x)4 ∧ V |= u = v ∈ x ]
∀x ∈ ω[(x : A)4 → (ux : B(x))4 ] ∧
∀x, y ∈ ω[(x = y : A)4 → (ux = uy : B(x))4 ]
∀x ∈ ω[(x : A)4 → (ux = vx : B(x))4 ] ∧
(u : (x : A)B)4 ∧ (v : (x : A)B)4
(p̄0 u : A)4 ∧ (p̄1 u : B(p̄0 u))4
(p̄0 u = p̄0 v : A)4 ∧ (p̄1 u = p̄1 v : B(p̄0 u))4
If s and t are arbitrary terms of MLF and A is a pseudo type term of MLF we set:
(t : A)4
(s = t : A)4
iff ∃u ∈ ω[(t∗ ' u)4 ∧ (u : A)4 ],
iff ∃u, v ∈ ω[(s∗ ' u)4 ∧ (t∗ ' v)4 ∧ (u = v : A)4 ].
For pseudo type terms A and B we define (A = B : type)4 by
∀u : ω[(u : A)4 ↔ (u : B)4 ] ∧ ∀u, v : ω[(u = v : A)4 ↔ (u = v : B)4 ].
Theorem 5.13 (Soundness of the Interpretation of MLF in KPMr .) If Φ is a
judgement of MLF not of the form “A : type”, then KPMr ` (Φ)4 .
Proof : First note that if an expression of the form A : type, s : A, s = t : A, or
A = B : type appears in a derivation of MLF, then A is a pseudo type term in the
sense of Definition 5.11, as is readily seen by induction on derivations in MLF.3 This
ensures that any judgement of MLF gets translated under 4 . Secondly, it should
be clear that the above interpretation replaces the abstract application of MLF by
recursive application in a faithful way, i.e. the equations which the rules of MLF
prescribe for the constants of MLF are satisfied by their translations. Since MLF
derivations may involve hypothetical judgements, 5.13 has to be stated in a more
general form so as to be able to carry out the proof by induction on derivations
in MLF. For a more details we refer to [7], XI.20.3.1, where this is done for the
interpretation of ML0 in EON. For a full treatment of dependent judgements one
can consult [24].
The most important case to look at is the translation of a judgement of the form
t ∈ U(F, A, B). We get
£
(t ∈ U(F, A, B))4 iff ∃x, u, f, a, b ∈ ω (U(F, A, B)∗ ' x)4 ∧ (t∗ ' x)4 ∧
¤
(F ∗ ' f )4 ∧ (A∗ ' a)4 ∧ (B ∗ ' b)4 ∧ x = u(f, a, b) .
Therefore the validity of the rules pertaining to U under the the translation ( )4 is
ensured by Lemma 5.6.
u
t
3
Incidentally, there are more pseudo type terms than those that appear in such derivations.
19
For theories T1 and T2 we use T1 ≤ T2 to signify that T1 is proof-theoretically
reducible to T2 in the sense of [7], Chap. XIII. We use T1 ≡ T2 to express that T1 and
T2 are proof-theoretically equivalent, i.e. T1 ≤ T2 and T2 ≤ T1 .
S
Theorem 5.14 n CZFn ≡ CZF + (M) ≤ MLF ≤ KPMr .
Proof : This follows from 4.8, 4.14 and 5.13.
u
t
It can be shown that proof-theoretic equivalence obtains all over.
S
Theorem 5.15 n CZFn ≡ CZF + (M) ≡ MLF ≡ KPMr .
S
In view of Theorem 5.14 it suffices to show that n CZFn and KPMr have the same
strength. The theory KPMr has been subjected to an ordinal analysis (cf. [18, 9])
with the result that its proof-theoretic ordinal is ψΩ1 (supm Im0), in terms of the ordinal
representation system T(M) of [19]. As well-orderingSproofs for the segments of the
T(M) determined
by ψΩ1 (Im0) can be carried out in n CZFn for all external m, it
S
follows that n CZFn and KPMr have the same strength. Due to space limitations,
the well-ordering proof cannot be given in this paper. Details will be incorporated in
[22].
6
Towards the boundaries of Martin-Löf type theory
Strength and expressiveness of Martin-Löf type theory are obtained through the use
of inductive data types and reflection, i.e. type universes. Reflective expansions of
these ideas have led to ever stronger formal systems of Martin-Löf type theory as
e.g. witnessed by the succession of systems MLU, MLS, and MLF. It is therefore
natural to ask for the boundaries of Martin-Löf type theory. Even stronger systems
than MLF have been proposed in [21] and [16]. The strongest theories are Palmgren’s systems MLn of higher order universe operators (cf. [16]) which allow for the
simultaneous construction of finitely many sets U0 , U1 , . . . , Un and decoding functions T0 , T1 , . . . , Tn such that U0 , T0 is a universe, Ui+1 consists of codes for universe
operators of type i + 1 decoded by Ti+1 , and Ui is closed under the universe operators
T i+1 (a) for a ∈ Ui .
Another natural expansion of MLF consists in allowing for higher type recursions
over W-types. All the foregoing expansions are still in the vein of Martin-Löf’s original
papers [11, 12]. On the other hand, it seems that they also exhaust the strength of
all possible extensions. All these theories can be interpreted in KPM which is the
system KPMr augmented by the schema of foundation (cf. Remark 5.1).
S
Theorem 6.1 Palmgren’s theories of higher universe operators n MLn as well as
the theory MLF + higher type recursion can be interpreted in KPM.
Proof : The interpretations are extensions of the one in subsection 5.2. The details
cannot be incorporated in this paper.
u
t
S
Conjecture 6.2 Palmgren’s theories of higher universe operators n MLn as well as
the theory MLF + higher type recursion have the same strength as KPM.
20
A proof of the conjecture would have to exploit the ordinal analysis of KPM given
in [20] by showing in the above type theories that all initial segments of the ordinal
representation system T(M) are well-ordered.
Before ending this paper it is important to address Setzer’s formalization of a Mahlo
universe in type theory (cf. [26]), dubbed TTM. Setzer’s theory is stronger than
KPM and provides an important step for expanding the realm of Martin-Löf type
theory. The difference between TTM and the systems considered in this paper is that
TTM introduces a new construction principle which is not foreshadowed in MartinLöf’s original papers. This is witnessed by the fact that Setzer’s Mahlo universe is
generated by a non-monotonic inductive definition which is incompatible with elimination rules for the universe. TTM means a paradigm shift to a new Martin-Löf type
theory. Therefore KPM is only a boundary for old Martin-Löf type theory.
References
[1] P. Aczel: The strength of Martin–Löf ’s intuitionistic type theory with one universe,
in: S. Miettinen, S. Väänänen (eds.): Proceedings of Symposia in Mathematical Logic,
Oulu, 1974, and Helsinki, 1975, Report No. 2 (University of Helsinki, Department of
Philosophy, 1977) 1–32.
[2] P. Aczel: The Type Theoretic Interpretation of Constructive Set Theory, in: MacIntyre,
A. Pacholski, L., Paris, J. (eds.), Logic Colloquium ’77 (North-Holland, Amsterdam,
1978).
[3] P. Aczel: The Type Theoretic Interpretation of Constructive Set Theory: Choice Principles, in: Troelstra, A. S., van Dalen, D. (eds.), The L.E.J. Brouwer Centenary Symposium, (North–Holland, Amsterdam, 1982).
[4] P. Aczel: The Type Theoretic Interpretation of Constructive Set Theory: Inductive
Definitions, in: Marcus, R. B. et al. (eds), Logic, Methodology, and Philosophy of
Science VII, (North–Holland, Amsterdam 1986).
[5] J Barwise: Admissible Sets and Structures, Springer, Berlin 1975.
[6] M. Beeson: Recursive Models for Constructive Set Theories, Annals of Math. Logic 23,
126–178 (1982).
[7] M. Beeson: Foundations of Constructive Mathematics, (Springer Verlag, Berlin, 1985)
[8] P. Dybjer: A general formulation of simultaneous inductive-recursive definitions in type
theory (preprint 1994) 21 pages.
[9] T. Glaß: Partielle Modelle von Theorien imprädikativer Mengenlehre (Master Thesis,
University of Münster, 1990).
[10] E. Griffor and E. Palmgren: A Intuitionistic Theory of Transfinite Types (preprint
1990) 22 pages.
[11] P. Martin-Löf: An intuitionistic theory of types: predicative part, in: H.E. Rose and J.
Sheperdson (eds.): Logic Colloquium ’73 (North-Holland, Amsterdam, 1975) 73–118.
[12] P. Martin-Löf: Intuitionistic Type Theory, (Bibliopolis, Naples, 1984).
[13] B. Nordström, K. Petersson and J.M. Smith: Programming in Martin–Löf ’s Type Theory, (Clarendon Press, Oxford, 1990).
21
[14] E. Palmgren: Transfinite hierarchies of universes, Department of Mathematics Report
1991:7 (Uppsala University, 1991), 16 pages.
[15] E. Palmgren: An information system interpretation of Martin-Löf ’s partial type theory
with universes, Information and Computation 106 (1993) 26–60.
[16] E. Palmgren: On universes in type theory. To appear in: Proceedings of ”Twenty-five
years of type theory”, Venice, 1995 (Oxford University Press).
[17] Aarne Ranta: Type-theoretical grammar (Clarendon Press, Oxford, 1994).
[18] M. Rathjen: Untersuchungen zu Teilsystemen der Zahlentheorie zweiter Stufe und der
Mengenlehre mit einer zwischen ∆12 − CA und ∆12 − CA + BI liegenden Beweisstärke.
Publication of the Institute for Mathematical Logic and Foundational Research of the
University of Münster (1989). Extension of 1988 Münster University doctoral thesis.
MR 91m#03062 .
[19] M. Rathjen: Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic 29 (1990) 249–263.
[20] M. Rathjen: Proof-Theoretic Analysis of KPM, Arch. Math. Logic 30 (1991) 377–403.
[21] M. Rathjen, E. Griffor, E. Palmgren: Inaccessibility in constructive set theory and type
theory. Annals of Pure and Applied Logic 94 (1998) 181–200.
[22] M. Rathjen: Well-ordering proofs in constructive set theory. In preparation.
[23] M. Rathjen: The strength of some Martin–Löf type theories. Archive for Mathematical
Logic 33 (1994) 347–385.
[24] A. Setzer: Proof theoretical strength of Martin-Löf type theory with W -type and one
universe (PhD thesis, Universität München, 1993).
[25] A. Setzer: A well-ordering proof for the proof theoretical strength of Martin-Löf type
theory. Annals of Pure and Applied Logic 92 (1998) 113–159.
[26] A. Setzer: Extending Martin-Löf type theory by one Mahlo-universe, Archive for Mathematical Logic 39, Issue 3 (2000) 155-181.
[27] A. S. Troelstra and D. van Dalen: Constructivism in Mathematics: an Introduction,
volume II, North–Holland, Amsterdam 1988.
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