THE GENERALISED TYPE-THEORETIC
INTERPRETATION OF CONSTRUCTIVE SET THEORY
NICOLA GAMBINO AND PETER ACZEL
Abstract. We present a generalisation of the type-theoretic interpre-
tation of constructive set theory into Martin-L¨of type theory. The
original interpretation treated logic in Martin-L¨of type theory via the
prop os itions-as-types interpretation. The generalisation involves replac-
ing Martin-L¨of type theory with a new type theory in which logic is
treated as primitive. The primitive treatment of logic in type theories
allows us to study reinterpretations of logic, such as the double-negation
translation.
Introduction
The type-theoretic interpretation of Constructive Zermelo-Frankel set the-
ory, or CZF for short, provides an explicit link between constructive set the-
ory and Martin-L¨of type theory [1, 2, 3]. This interpretation is a useful tool
in the proof-theoretical investigations of constructive formal systems [17]
and allows us to relate the set-theoretic and type-theoretic approaches to
the development of constructive mathematics [5, 28, 32].
A crucial component of the original type-theoretic interpretation of CZF is
the propositions-as-types interpretation of logic. Under this interpretation,
arbitrary formulas of CZF are interpreted as types, and restricted formulas
as small types. By a small type we mean here a type represented by an
element of the type universe that is part of the type theory in which CZF
is interpreted. The propositions-as-types representation of logic is used in
proving the validity of three schemes of CZF, namely Restricted Separation,
Strong Collection, and Subset Collection. Validity of Restricted Separation
follows from the representation of restricted propositions as small types,
while the validity of both Strong Collection and Subset Collection follow
from the type-theoretic axiom of choice, that holds in the propositions-as-
types interpretation of logic [28]. Another ingredient of the original type-
theoretic interpretation is the definition of a type V, called the type of
iterative sets, that is used to interpret the universe of sets of CZF. In this
way, it is possible to obtain a valid interpretation of CZF in the Martin-L¨of
type theory ML
1
+ W, which has rules for the usual forms of type, for a
Date: Octobe r 12th, 2005.
2000 Mathematics Subject Classification. 03F25 (Relative consistency and interpreta-
tions), 03F50 (Metamathematics of constructive systems).
Key words and phrases. Constructive Set Theory, Dependent Type Theory.
1
2 GAMBINO AND ACZEL
type universe reflecting these forms of type, and for W-type s (see Section 1
for details).
Our first aim here is to present a new type-theoretic interpretation of CZF.
The novelty lies in replacing the pure type theory like ML
1
+ W with a suit-
able logic-enriched type theory. By a logic-enriched intuitionistic type theory
we mean an intuitionistic type theory like ML
1
+ W that is extended with
judgement forms that allow us to express, relative to a context of variable
declarations, the notion of proposition and assertions that one proposition
follows from others. Logic-enriched type theories have a straighforward in-
terpretation in their pure counterparts that is obtained by following the
propositions-as-types idea. When formulating logic-enriched type theories
that extend pure type theories with rules for a type universe, like ML
1
+ W,
it is natural to have rules for a proposition universe to match the type
universe. Elements of the proposition universe should be thought of as rep-
resentatives for propositions whose quantifiers range over small types.
The new interpretation generalises the original type-theoretic interpreta-
tion in that logic is treated as primitive and not via the propositions-as-types
interpretation. In particular, we will introduce a logic-enriched type theory,
called ML(COLL), that has two collection rules, corresponding to the collec-
tion axiom schemes of CZF. Within the type theory ML(COLL) we define a
type V, called the type of iterative small classes, that can be used to interpret
the universe of sets of CZF. The particular definition of V allows us to prove
the validity of Restricted Separation without assuming the propositions-as-
types interpretation of logic, and the collection rules of ML(COLL) allow
us to prove the validity of Strong Collection and Subset Collection. We will
therefore obtain a type-theoretic interpretation of CZF that does not rely
on the propositions-as-types treatment of logic and in particular avoids any
use of the type-theoretic axiom of choice.
A fundamental reason for the interest in the generalised interpretation
is that it allows us to provide an analysis of the original interpretation.
This is obtained by considering a logic-enriched type theory ML(AC + PU)
with special rules expressing the axiom of choice and a correspondence
between the proposition and type universes. These rules are valid under
the proposition-as-types interpretation, so that ML(AC + PU) can be in-
tepreted in the pure type theory ML
1
+ W. Furthermore, the collection
rules of ML(COLL) follow from the special rules of ML(AC + PU), and
therefore it is possible to view the generalised interpretation of CZF as tak-
ing place in ML(AC + PU). We then prove that the original interpretation of
CZF in ML
1
+ W can be seen as the result of composing the generalised in-
terpretation of CZF in ML(AC + PU) followed by the propositions-as-types
interpretation of ML(AC + PU) into ML
1
+ W.
Another goal of this paper is to describe how logic-enriched type theo-
ries with collection rules, like ML(COLL), have a key advantage over logic-
enriched type theories with the axiom of choice, like ML(AC + PU). The
advantage is that the former can accomodate reinterpretations of their logic,
THE GENERALISED TYPE-THEORETIC INTERPRETATION 3
while the latter cannot. We focus our attention on reinterpretations of logic
as determined by a map j that satisfies a type-theoretic version of the prop-
erties of a Lawvere-Tierney local operator in an elementary topos [21] or of a
nucleus on a frame [19]. We will call such a j a local operator, and the rein-
terpretation of logic determined by it will be called the j-interpretation. A
typical example of such an operator is provided by double-negation. In our
development, we consider initially a subsystem ML(COLL
−
) of ML(COLL),
and ML(COLL) at a later stage. There are two main reasons for doing so.
A first reason is that the Strong Collection rule is sufficient to prove the
basic properties of j-interpretations. A second reason is that the Strong
Collection rule is preserved by the j-interpretation determined by any lo-
cal operator j, while the Subset Collection rule is not. In order to obtain
the derivability of the j-interpretation of the Subset Collection rule, we will
introduce a natural further assumption. These results allow us to define
a type-theoretic version of the double-negation translation, which in turn
leads to a proof-theoretic application.
The generalised type-theoretic interpretation is related to the study of
categorical models for constructive set theories [7, 29, 30, 13]. Indeed, one
of our initial motivations was to study whether it was possible to obtain
a type-theoretic version of the results in [30] concerning the interpretation
of CZF in categories whose internal logic does not satisfy the axiom of
choice. An essential difference, however, between the development presented
here and the results in the existing literature on categorical models is that
the interaction between propositions and types is more restricted in logic-
enriched type theories than in categories when logic is treated ass uming
the proposition-as-subobjects approach to prop os itions. In particular, logic-
enriched type theories do not generally have rules that allow us to form
types by separation, something that is instead a direct consequence of the
proposition-as-subobjects representation of logic in categories [29, 30]. The
category-theoretic counterpart to the logic-enrichment of a pure dependent
type theory is roughly a first-order fibration over a category that is already
the base category of a fibration representing the dependent type theory. See,
for example, [18, Chapter 11] or [22, 23].
Extensions of pure type theories that allow the formation of types by
separation have already b ee n studied. One approach is via minimal type
theories [24, 39]. Minimal type theories may be understood as extensions
of logic-enriched type theories with extra rules asserting that each propo-
sition represents a type, that is to be thought of as the types of proofs of
the proposition. Using these rules and the standard rules for Σ-types of
the underlying pure type theory, the formation of types by separation can
be easily obtained. Another approach is offered by the pure type theories
with bracket types [6]. These are obtained by extending pure type theories
with rules for a new form of type, called bracket type, that allows the repre-
sentation of propositions as typ e s with at most one element. This approach
provides essentially a type-theoretic version of the proposition-as-subobjects
4 GAMBINO AND ACZEL
idea. In the development of the generalised type-theoretic interpretation we
preferred however to work within logic-enriched type theories, and avoid the
assumption of extra rules allowing formation of types by separation. In this
respect, the generalised type-theoretic interpretation presented here is more
general than the existing categorical models for CZF.
The results presented here are part of a wider research programme, orig-
inally sketched in [4]. The present paper contributes to that programme in
two respects: first, by giving precise pro ofs of the results announced in [4]
regarding the generalised type-theoretic interpretation of CZF and the rein-
terpretations of logic, and secondly by presenting new results concerning
the analysis of the original type-theoretic interpretation via the generalised
one. We regard these new results as fundamental, since they show that
the generalised intepretation allows us to gain further insight into the orig-
inal intepretation. Section 6 contains an informal disc ussion of the overall
research effort.
For the convenience of the reader, we present here a concise review of the
axiom system of CZF. For a discussion of the development of constructive
mathematics in constructive set theories, see [5]. For proof-theoretical in-
vestigations on constructive set theories we invite the reader to refer also
to [10, 14, 20, 34, 35, 33, 36]. The axioms of CZF are presented below in
an extension of the language of first-order logic with primitive restricted
quantifiers (∀x ∈ α) and (∃x ∈ α). The membership relation can then be
defined by letting
α ∈ β =
def
(∃x ∈ β)(x = α)
A formula is said to be restricted if all the quantifiers in it are restricted.
We use letters u, v, z, x, y, w, . . . for variables of the language, and greek
letters α, β, γ, . . . to denote sets. Other greek letters are used to denote
formulas. For formulas φ and ψ, we write φ ⊃ ψ for their implication and
define φ ≡ ψ =
def
(φ ⊃ ψ) ∧ (ψ ⊃ φ).
The axiom system of CZF includes both logical and set-theoretic axioms
and schemes. We will refer to axioms and schemes collectively as axiom
schemes. The logical axioms schemes include the standard ones for intu-
itionistic logic with equality and the following axiom schemes for restricted
quantifiers:
(∀x ∈ α)φ ≡ (∀x)(x ∈ α ⊃ φ) , (∃x ∈ α)φ ≡ (∃x)(x ∈ α ∧ φ)
The set-theoretic axiom schemes of CZF can be conceptually divided into
three groups: structural, basic set existence, and collection. The structural
axiom schemes of CZF are Extensionality (1) and Set Induction (2).
(∀x)(x ∈ α ≡ x ∈ β) ⊃ (α = β) (1)
(∀x)
(∀y ∈ x)φ[y/x] ⊃ φ
⊃ (∀x)φ (2)
The Extensionality axiom asserts that if two sets have the same elements,
then they are equal. The Set Induction scheme is the intuitionistic counter-
part of the classical Foundation axiom, and it is formulated as a scheme in
THE GENERALISED TYPE-THEORETIC INTERPRETATION 5
which φ is an arbitrary formula. The first basic set existence axioms of CZF
are Pairing (3), Union (4), and Infinity (5).
(∃u)(∀x)(x ∈ u ≡ (x = α ∨ x = β)) (3)
(∃u)(∀x)(x ∈ u ≡ (∃y ∈ α)(x ∈ y)) (4)
(∃u)((∃x)(x ∈ u) ∧ (∀x ∈ u)(∃y ∈ u)(x ∈ y)) (5)
They are formulated as in classical set theory. A further basic set existence
scheme of CZF is Restricted Separation. It is the scheme in (6), where θ is
a restricted formula in which the variable u does not appear free.
(∃u)(∀x)(x ∈ u ≡ x ∈ α ∧ θ) (6)
The Restricted Separation scheme is a weakening of the classical scheme
of Full Separation, obtained by limiting the kind of formulas allowed in
the scheme. To formulate the two collection schemes of CZF we use the
abbreviation
(∀∃
x∈α
y∈β
)
φ =
def
(∀x ∈ α)(∃y ∈ β)φ ∧ (∀y ∈ β)(∃x ∈ α)φ
where φ is an arbitrary formula. Note that in the formula
(∀∃
x∈α
y∈β
)
φ
free occurrences of x and y in φ get bound by the operator (∀∃
x∈α
y∈β
)
. The
Strong Collection (7) and Subset Collection (8) schemes are given below.
(∀x ∈ α)(∃y)φ ⊃ (∃u)(∀∃
x∈α
y∈u
)
φ (7)
(∃v)(∀z)[(∀x ∈ α)(∃y ∈ β)φ ⊃ (∃u ∈ v)(∀∃
x∈α
y∈u
)
φ] (8)
Note that in the Subset Collection scheme (8) the formula φ may have free
occurrences of z which get bound by the universal quantifier (∀z). The
Strong Collection scheme is a mild strengthening of the Collection scheme,
needed in order to derive the Replacement scheme in a set theory without
the Full Separation scheme [10]. The Subset Collection scheme is instead a
weakening of the Power Set axiom and a strengthening of Myhill’s Exponen-
tiation axiom, which asserts that the class of functions between two sets is
again a set [31]. In [20] it is shown that, in the presence of axioms (1) – (7),
the Subset Collection scheme is independent of the Exponentiation axiom.
Outline of the paper. A review of Martin-L¨of pure type theories is presented
in Section 1, which also serves to fix the notation used in the reminder of the
paper, while the list of rules for the type theories used here is contained in
Appendix A. Logic-enriched type theories are introduced in Section 2, where
we also describe their propositions-as-types interpretation. In Section 3 we
define the generalised type-theoretic interpretation of CZF. The relationship
between the original and the generalised type-theoretic interpretations is
then described in Section 4. Section 5 discusses the reinterpretations of
logic. The paper ends in Section 6 with conclusions and a perspective of
future work.
