Download The Formulae-as-Classes Interpretation of Constructive Set Theory

The Formulae-as-Classes Interpretation of
Constructive Set Theory
Michael RATHJEN∗
Department of Pure Mathematics, University of Leeds
Leeds LS2 9JT, England
Department of Mathematics, Ohio State University
Columbus, OH 43210, U.S.A.
[email protected]
Abstract. The main objective of this paper is to show that a certain formulae-asclasses interpretation based on generalized set recursive functions provides a selfvalidating semantics for Constructive Zermelo-Fraenkel Set theory, CZF. It is argued
that this interpretation plays a similar role for CZF as the constructible hierarchy for
classical set theory, in that it can be employed to show that CZF is expandable by
several forms of the axiom of choice without adding more consistency strength.
1
Introduction
The general topic of Constructive Set Theory (CST ) originated in John Myhill’s endeavour
(see [16]) to discover a simple formalism that relates to Bishop’s constructive mathematics as
classical Zermelo-Fraenkel Set Theory with the axiom of choice relates to classical Cantorian
mathematics. CST provides a standard set theoretical framework for the development of constructive mathematics in the style of Errett Bishop [8]. One of the hallmarks of constructive
set theory is that it possesses (due to Aczel [1, 2, 3]) a canonical interpretation in MartinLöf’s intuitionistic type theory (see [12, 13]) which is considered to be the most acceptable
foundational framework of ideas that make precise the constructive approach to mathematics.
The interpretation employs the Curry-Howard ‘propositions as types’ idea in that the axioms
of constructive set theory get interpreted as provably inhabited types.
In constructive or intuitionistic set theories often the questions arises whether adding a particular set-theoretic statement to the given set theory leads to a theory which is equiconsistent
with the original theory. The most famous methods for showing equiconsistency and independence of statements from axiom systems in the classical context are Gödel’s constructible
hierarchy L and Cohen’s method of forcing or the closely related technique of Boolean-valued
models. In classical ZF the constructible hierarchy has been utilized to show that augmenting
ZF by the axiom of choice and the generalized continuum hypothesis gives rise to a theory
that is equiconsistent with ZF. This leads to the question whether L can be fruitfully employed in intuitionistic set theories as well. Robert Lubarsky has established the following
result for intuitionistic Zermelo-Fraenkel set theory, IZF, in [11]:
∗
Research partly supported by NSF Grant DMS-0301162 and United Kingdom Engineering and Physical
Sciences Research Council Grant GR/R 15856/01.
2
The Formulae-as-Classes Interpretation of Constructive Set Theory
Theorem 1.1. Let V = L be the statement that all sets are constructible. For a formula ϕ,
let ϕL be the result of relativizing all quantifiers in ϕ to L. We then have
IZF ` (V = L)L
IZF ` ϕ ⇒ IZF ` ϕL .
While 1.1 is an interesting result, it cannot be utilized to provide similar consequences as in
the classical scenario. Classically, L is a very well-behaved class that is “constructed” from
the ordinals and can be well-ordered via a definable well-ordering, whence AC holds in L.
In intuitionistic set theories, however, the ordinals are rather uncontrollable sets that cannot
be shown to be linearly ordered, that is, given ordinals α, β it is not possible to conclude that
α ∈ β or α = β or β ∈ α. As a result, when working in an intuitionistic context, we have
to seek a different approach if we want to show that certain forms of the axiom of choice
can be added without gaining more consistency strength. In this paper it is argued that the
formulae-as-classes interpretation can do for constructive set theories what the constructible
hierarchy does for classical set theory. The formulae-as-classes interpretation is closely related to Aczel’s formulae-as-types interpretation of set theory in Martin-Löf type theory. In
diverges from the latter, however, in that it is an interpretation of set theory into set theory
and, crucially, in that it provides a semantics for Constructive Zermelo-Fraenkel Set Theory,
CZF, which can be formalized in CZF itself (a self-validating semantics).
1.1
The system CZF
In this subsection we will summarize the language and axioms for CZF. The language of
CZF is the same first order language as that of classical Zermelo-Fraenkel Set Theory, ZF
whose only non-logical symbol is ∈. The logic of CZF is intuitionistic first order logic with
equality. Among its non-logical axioms are Extensionality, Pairing and Union in their usual
forms. CZF has additionally axiom schemata which we will now proceed to summarize.
Infinity: ∃x∀u u ∈ x ↔ ∅ = u ∨ ∃v ∈ x u = v + 1 where v + 1 = v ∪ {v}.
Set Induction: ∀x[∀y ∈ xφ(y) → φ(x)] → ∀xφ(x)
Restricted Separation:
∀a∃b∀x[x ∈ b ↔ x ∈ a ∧ φ(x)]
for all restricted formulae φ. A set-theoretic formula is restricted if it is constructed from
prime formulae using ¬, ∧, ∨, →, ∀x ∈ y and ∃x ∈ y only.
Strong Collection: For all formulae φ,
∀a ∀x ∈ a∃yφ(x, y) → ∃b [∀x ∈ a ∃y ∈ b φ(x, y) ∧ ∀y ∈ b ∃x ∈ a φ(x, y)] .
Subset Collection: For all formulae ψ,
∀a∀b∃c∀u ∀x ∈ a ∃y ∈ b ψ(x, y, u) →
∃d ∈ c [∀x ∈ a ∃y ∈ d ψ(x, y, u) ∧ ∀y ∈ d ∃x ∈ a ψ(x, y, u)] .
The Subset Collection schema easily qualifies as the most intricate axiom of CZF. To explain
this axiom in different terms, we introduce the notion of fullness (cf. [1]).
The Formulae-as-Classes Interpretation of Constructive Set Theory
3
Definition 1.2. As per usual, we use hx, yi to denote the ordered pair of x and y. We use
Fun(g), dom(R), ran(R) to convey that g is a function and to denote the domain and range
of any relation R, respectively.
For sets A, B let A × B be the cartesian product of A and B, that is the set of ordered
pairs hx, yi with x ∈ A and y ∈ B. Let A B be the class of all functions with domain A
and with range contained in B. Let mv(A B) be the class of all sets R ⊆ A × B satisfying
∀u ∈ A ∃v ∈ B hu, vi ∈ R. A set C is said to be full in mv(A B) if C ⊆ mv(A B) and
∀R ∈ mv(A B) ∃S ∈ C S ⊆ R.
The expression mv(A B) should be read as the collection of multi-valued functions from the
set A to the set B.
Additional axioms we shall consider are:
Exponentiation: ∀x∀y∃z z = x y.
Fullness: ∀x∀y∃z z is full in mv(x y).
The next result is an equivalent rendering of [1], 2.2. We include a proof for the reader’s
convenience.
Proposition 1.3. Let CZF− be CZF without Subset Collection.
(i) CZF− ` Subset Collection ↔ Fullness.
(ii) CZF ` Exponentiation.
Proof. (i): For “⇒” let φ(x, y, u) be the formula y ∈ u ∧ ∃z ∈ B (y = hx, zi). Using the
relevant instance of Subset Collection and noticing that for all R ∈ mv(A B) we have
∀x ∈ A ∃y ∈ A × B φ(x, y, R), there exists a set C such that ∀R ∈ mv(A B) ∃S ∈ C S ⊆ R.
A
“⇐”: Let
C be full in mv( B). Assume ∀x ∈ A∃y ∈ Bφ(x, y, u). Define ψ(x, w, u) :=
∃y ∈ B w = hx, yi ∧ φ(x, y, u) . Then ∀x ∈ A∃wψ(x, w, u). Thus, by Strong Collection,
there exists v ⊆ A × B such that
∀x ∈ A ∃y ∈ B hx, yi ∈ v ∧ φ(x, y, u) ∧ ∀x ∈ A ∀y ∈ B hx, yi ∈ v → φ(x, y, u) .
As C is full, we find w ∈ C with w ⊆ v. Consequently, ∀x ∈ A∃y ∈ ran(w)φ(x, y, u) and
∀y ∈ ran(w)∃x ∈ A φ(x, y, u), where ran(w) := {v : ∃z hz, vi ∈ w}.
Whence D := {ran(w) : w ∈ C} witnesses the truth of the instance of Subset Collection
pertaining to φ.
(ii) Let C be full in mv(A B). If now f ∈ A B, then ∃R ∈ C R ⊆ f . But then R = f . Therefore
B = {f ∈ C : f is a function}.
t
u
A
Let TND be the principle of excluded third, i.e. the schema consisting of all formulae of the
form A ∨ ¬A. The first central fact to be noted about CZF is:
Proposition 1.4. CZF + TND = ZF.
4
The Formulae-as-Classes Interpretation of Constructive Set Theory
Proof : Note that classically Collection implies Separation. Powerset follows classically from
Exponentiation.
t
u
On the other hand, it was shown in [18], Theorem 4.14, that CZF has only the strength of
Kripke-Platek Set Theory (with the Infinity Axiom), KP (see [5]), and, moreover, that CZF
is of the same strength as its subtheory CZF− , i.e., CZF minus Subset Collection. To stay in
the world of CZF one has to keep away from any principles that imply TND. Moreover, it
is perhaps fair to say that CZF is such an interesting theory owing to the non-derivability of
Powerset and Separation. Therefore one ought to avoid any principles which imply Powerset
or Separation.
The first large set axiom proposed in the context of constructive set theory was the Regular
Extension Axiom, REA, which Aczel introduced to accommodate inductive definitions in
CZF (cf. [3]).
Definition 1.5. A is inhabited if ∃x x ∈ A. An inhabited set A is regular if A is transitive,
and for every a ∈ A and set R ⊆ a × A if ∀x ∈ a ∃y (hx, yi ∈ R), then there is a set b ∈ A
such that
∀x ∈ a ∃y ∈ b (hx, yi ∈ R) ∧ ∀y ∈ b ∃x ∈ a (hx, yi ∈ R).1
2
The axiom of choice in constructive set theories
Among the axioms of set theory, the axiom of choice is distinguished by the fact that is it the
only one that one finds ever mentioned in workaday mathematics. In the mathematical world
of the beginning of the 20th century, discussions about the status of the axiom of choice
were important. In 1904 Zermelo proved that every set can be well-ordered by employing
the axiom of choice. While Zermelo argued that it was self-evident, it was also criticized
as an excessively non-constructive principle by some of the most distinguished analysts of
the day, notably Borel, Baire, and Lebesgue. At first blush this reaction against the axiom of
choice utilized in Cantor’s new theory of sets is surprising as the French analysts had used
and continued to use choice principles routinely in their work. However, in the context of
19th century classical analysis only the Axiom of Dependent Choices, DC, is invoked and
considered to be natural, while the full axiom of choice is unnecessary and even has some
counterintuitive consequences.
Unsurprisingly, the axiom of choice does not have a unambiguous status in constructive mathematics either. On the one hand it is said to be an immediate consequence of the constructive
interpretation of the quantifiers. Any proof of ∀x ∈ A ∃y ∈ B φ(x, y) must yield a function
f : A → B such that ∀x ∈ A φ(x, f (x)). This is certainly the case in Martin-Löf’s intuitionistic theory of types. On the other hand, it has been observed that the full axiom of
choice cannot be added to systems of extensional constructive set theory without yielding
constructively unacceptable cases of excluded middle (see [9]). In extensional intuitionistic
set theories, a proof of a statement ∀x ∈ A ∃y ∈ B φ(x, y), in general, provides only a function
F , which when fed a proof p witnessing x ∈ A, yields F (p) ∈ B and φ(x, F (p)). Therefore, in
the main, such an F cannot be rendered a function of x alone. Choice will then hold over sets
which have a canonical proof function, where a constructive function h is a canonical proof
1
In particular, if R : a → A is a function, then the image of R is an element of A.
The Formulae-as-Classes Interpretation of Constructive Set Theory
5
function for A if for each x ∈ A, h(x) is a constructive proof that x ∈ A. Such sets having
natural canonical proof functions “built-in” have been called bases (cf. [21], p. 841).
The particular form of constructivism adhered to in this paper is Martin-Löf’s intuitionistic
type theory (cf. [12, 13]). Set-theoretic choice principles will be considered as constructively
justified if they can be shown to hold in the interpretation in type theory. Moreover, looking
at set theory from a type-theoretic point of view has turned out to be valuable heuristic tool
for finding new constructive choice principles. For more information on choice principles in
the constructive context see [19].
2.1
Old acquaintances
In many a text on constructive mathematics, axioms of countable choice and dependent
choices are accepted as constructive principles. This is, for instance, the case in Bishop’s
constructive mathematics (cf. [8] as well as Brouwer’s intuitionistic analysis (cf. [21], Ch. 4,
Sect. 2). Myhill also incorporated these axioms in his constructive set theory [16].
The weakest constructive choice principle we shall consider is the Axiom of Countable Choice,
ACω , i.e. whenever F is a function with domain ω such that ∀i ∈ ω ∃y ∈ F (i), then there exists a function f with domain ω such that ∀i ∈ ω f (i) ∈ F (i).
A mathematically very useful axiom to have in set theory is the Dependent Choices Axiom,
DC, i.e., for all formulae ψ, whenever
(∀x ∈ a) (∃y ∈ a) ψ(x, y)
and b0 ∈ a, then there exists a function f : ω → a such that f (0) = b0 and
(∀n ∈ ω) ψ(f (n), f (n + 1)).
Even more useful is the Relativized Dependent Choices Axiom, RDC. It asserts that for
arbitrary formulae φ and ψ, whenever
∀x φ(x) → ∃y φ(y) ∧ ψ(x, y)
and φ(b0 ), then there exists a function f with domain ω such that f (0) = b0 and
(∀n ∈ ω) φ(f (n)) ∧ ψ(f (n), f (n + 1)) .
2.2
Operations on classes
The interpretation of constructive set theory in type theory not only validates all the theorems
of CZF (resp. CZF + REA) but many other interesting set-theoretic statements, including
several new choice principles which will be described next. To state these principles we need
to introduce various operations on classes.
Remark 2.1. Class notation: In doing mathematics in CZF we shall exploit the use of class
notation and terminology, just as in classical set theory. Given a formula φ(x) there may not
exist a set of the form {x : φ(x)}. But there is nothing wrong with thinking about such
collection. So, if φ(x) is a formula in the language of set theory we may form a class {x :
φ(x)}. We allow φ(x) to have free variables other than x, which are considered parameters
6
The Formulae-as-Classes Interpretation of Constructive Set Theory
upon which the class depends. Informally, we call any collection of the form {x : φ(x)} a
class. However formally, classes do not exist, and expressions involving them must be thought
of as abbreviations for expressions not involving them. Classes A, B are defined to be equal
if ∀x[x ∈ A ↔ x ∈ B].
We may also consider an augmentation of the language of set theory whereby we allow atomic
formulas of the form y ∈ A and A = B with A, B being classes. There is no harm in
taking such liberties as any such formula can be translated back into the official language
of set theory by re-writing y ∈ {x : φ(x)} and {x : φ(x)} = {y : ψ(y)} as as φ(y) and
∀z [φ(z) ↔ ψ(z)], respectively (with z not in in φ(x) and ψ(y)).
Definition 2.2. Let CZFExp denote the modification of CZF with Eponentiation in place of
Subset Collection.
Remark 2.3. In all the results of this paper, CZF could be replaced by CZFExp , that is to
say, for the purposes of this paper it is enough to assume Exponentiation rather than Subset
Collection.
Definition 2.4. (CZF)
Q
If A is a set and Bx are classes for all x ∈ A, we define a class x∈A Bx by:
[
Y
Bx | ∀x∈A(f (x) ∈ Bx )}.
Bx := {f : A →
(1)
x∈A
x∈A
If A is a class and Bx are classes for all x ∈ A, we define a class
X
Bx := {hx, yi | x ∈ A ∧ y ∈ Bx }.
P
x∈A
Bx by:
(2)
x∈A
If A is a class and a, b are sets, we define a class I(A, a, b) by:
I(A, a, b) := {z ∈ 1 | a = b ∧ a, b ∈ A}.
(3)
If A is a class and for each a ∈ A, Ba is a set, then
Wa∈A Ba
is the smallest class Y such that whenever a ∈ A and f : Ba → Y , then ha, f i ∈ Y .
Lemma 2.5. (CZF)
Q
P
If A,B,a,b are sets and Bx are sets for all x ∈ A, then x∈A Bx , x∈A Bx and I(A, a, b)
are sets.
S
Proof. First
S S of all, we need to prove that x∈A Bx is a set. Indeed, g = {{x, {x, Bx }} |x ∈ A},
and so
g = {z, x, Bx | z ∈ x, x ∈ A} is a set by Union. Now
[[
[[
ran(g) = {y ∈
g | ∃x∈
g (hx, yi ∈ g)}
S
S
and x∈A Bx = ran(g) are sets by Bounded
Separation and Union.
S
1: The class of all functions from A to x∈A Bx is a set by Exponentiation and
Y
[
Bx := {f : A →
Bx | ∀x∈A(f (x) ∈ Bx )}
x∈A
x∈A
The Formulae-as-Classes Interpretation of Constructive Set Theory
7
is a set by Bounded Separation, since ∀x∈A(f (x) ∈ Bx ) can be rewritten as
∀x∈A ∃y ∈ran(f )∃y 0 ∈ran(g)(hx, yi ∈ f ∧ hx, y 0 i ∈ g ∧ y ∈ y 0 ).
S
2: Using from above that x∈A Bx is a set, by Pairing, Union and Replacement we obtain a
set
[
[
A×
Bx = {hx, yi | x ∈ A ∧ y ∈
Bx }.
x∈A
Now, the set
X
x∈A
Bx := {hx, yi∈A ×
x∈A
[
Bx | x ∈ A ∧ y ∈ Bx }
x∈A
exists by Bounded Separation, since x ∈ A ∧ y ∈ Bx can be rewritten as
x ∈ A ∧ ∃y 0 ∈ran(g)(hx, y 0 i ∈ g ∧ y ∈ y 0 ).
2
3: I(A, a, b) is a set by Bounded Separation.
Lemma 2.6. (CZF + REA)
If A is a set and Bx is a set for all x ∈ A, then Wa∈A Ba is a set.
2
Proof. This follows from [3], Corollary 5.3.
2.3
Inductively defined classes
In the following we shall introduce several inductively defined classes, and, moreover, we
have to ensure that such classes can be formalized in CZF.
We define an inductive definition to be a class of ordered pairs. If Φ is an inductive definition
and hx, ai ∈ Φ then we write
x
Φ
a
and call xa an (inference) step of Φ, with set x of premisses and conclusion a. For any class
Y , let
ΓΦ (Y ) =
a | ∃x x ⊆ Y ∧
x
a
Φ
.
The class Y is Φ-closed if ΓΦ (Y ) ⊆ Y . Note that Γ is monotone; i.e. for classes Y1 , Y2 ,
whenever Y1 ⊆ Y2 , then Γ(Y1 ) ⊆ Γ(Y2 ).
We define the class inductively defined by Φ to be the smallest Φ-closed class. The main result
about inductively defined classes states that this class, denoted I(Φ), always exists.
Lemma 2.7. (CZF) (Class Inductive Definition Theorem) For any inductive definition Φ
there is a smallest Φ-closed class I(Φ).
Moreover, call a set G of ordered pairs good if
(∗)
ha, yi ∈ G ⇒ y ∈ ΓΦ (G∈a ).
where
G∈a = {y 0 | ∃x∈a hx, y 0 i ∈ G}.
8
The Formulae-as-Classes Interpretation of Constructive Set Theory
Letting J =
S
{G | G is good} and J a = {x | ha, xi ∈ J}, it holds
[
I(Φ) =
J a,
a
and for each a,
J a = ΓΦ (
[
J x ).
x∈a
J is uniquely determined by the above, and its stages J a will be denoted by ΓaΦ .
Proof. [2], section 4.2 or [4], Theorem 5.1.
2.4
t
u
Maximal choice principles
Lemma 2.8. (CZF)
There exists a smallest ΠΣ-closed class, i.e. a smallest class Y such that the following
holds:
(i) n ∈ Y for all n ∈ N;
(ii) ωQ∈ Y;
P
(iii) x∈A Bx ∈ Y and x∈A Bx ∈ Y whenever A ∈ Y and Bx ∈ Y for all x ∈ A.
Likewise, there exists a smallest ΠΣI-closed class, i.e. a smallest class Y∗ , which, in addition
to the closure conditions (i)–(iii) above, satisfies:
(iv) I(A, a, b) ∈ Y∗ whenever A ∈ Y∗ and a, b ∈ A.
Proof. The classes Y and Y∗ are inductively defined, and therefore exist by Lemma 2.7.
To be precise, the respective inductive definitions of these classes are given by the classes
Φ1 , . . . , Φ5 consisting of the following pairs:
(i)
(ii)
n
Φ1
, for all n ∈ N;
ω
Φ2
;
(iii)
{dom(g)} ∪ ran(g)
Q
x∈A g(x)
Φ3
, for all functions g with dom(g) = A;
(iv)
{dom(g)} ∪ ran(g)
P
x∈A g(x)
Φ4
, for all functions g with dom(g) = A;
(v)
{A}
I(A, a, b)
Φ5
, if a, b ∈ A.
(Clause (v) is only needed to define Y∗ .)
2
Lemma 2.9. (CZF + REA)
There exists a least ΠΣW-closed class, i.e. a smallest class Yw that in addition to the
clauses (i),(ii),(iii) of Lemma 2.8 satisfies:
(vi) Wa∈A Ba ∈ Yw whenever A ∈ Yw and Bx ∈ Yw for all x ∈ A.
∗
Likewise, there exists a smallest ΠΣWI-closed class, i.e. a least class Yw
, which, in addition
to the closure conditions above, satisfies clause (iv) of Lemma 2.8.
The Formulae-as-Classes Interpretation of Constructive Set Theory
9
Proof. Virtually the same as for Lemma 2.8.
2
Definition 2.10. The ΠΣ-generated sets are the sets in the smallest ΠΣ-closed class, i.e. Y.
Similarly one defines the ΠΣI, ΠΣW and ΠΣWI-generated sets.
A set P is a base if for any P -indexed family (Xa )a∈P of inhabited sets Xa , there exists a
function f with domain P such that, for all a ∈ P , f (a) ∈ Xa .
ΠΣ−AC is the statement that every ΠΣ-generated set is a base. Similarly one defines the
axioms ΠΣI−AC, ΠΣWI−AC, and ΠΣW−AC.
Lemma 2.11. (i) (CZF) For every A ∈ Y∗ there exists a B ∈ Y with a bijection h : B → A.
∗
there exists a B ∈ Yw with a bijection h : B → A.
(ii) (CZF + REA) For every A ∈ Yw
2
Proof. See the lemma following Theorem 3.7 in [3].
Corollary 2.12. (i) (CZF) ΠΣ−AC and ΠΣI−AC are equivalent.
(ii) (CZF + REA) ΠΣW−AC and ΠΣWI−AC are equivalent.
Proof. ΠΣI−AC obviously implies ΠΣ−AC, since Y ⊆ Y∗ . To prove the converse,
assume ΠΣ−AC, A ∈ Y∗ , and ∀x ∈ A∃yϕ(x, y), where ϕ is a formula of CZF. Take a B
and a bijection h : A → B which exists by the previous Lemma; then ∀x∈B∃y ϕ(h−1 (x), y).
By ΠΣ−AC,
∃f : B → V ∀x∈B ϕ(h−1 (x), f (x)).
This yields
∀x∈A ϕ h−1 ◦ h(x), f ◦ h(x)
so that ∀x∈A ϕ x, f ◦ h(x) .
The proof of (ii) is similar.
3
2
Interpreting bounded formulae as sets
Notation. For sets x and y, we define sup(x, y) as hx, yi. If α = sup(A, f ), where f is a
function with domain A, we define ᾱ := A and α̃ := f .
Definition 3.1. (CZF)
Utilizing Lemma 2.7 we define a class V(Y∗ ) by the following rule:
a ∈ Y∗
f : a → V(Y∗ )
,
sup(a, f ) ∈ V(Y∗ )
(4)
The class V(Y) is defined in the same vein by replacing Y∗ by Y in the foregoing clause.
Definition 3.2. (CZF)
.
The (class) functions = : V(Y∗ ) × V(Y∗ ) → Y∗ and ∈˙ : V(Y∗ ) × V(Y∗ ) → Y∗ are
defined by recursion as follows:
YX .
YX .
.
= (α̃(x), β̃(y)) ×
= (α̃(x), β̃(y)),
(5)
= (α, β) is
x∈ᾱ y∈β̄
y∈β̄ x∈ᾱ
X .
∈˙ (α, β) is
= (α, β̃(y)).
y∈β̄
(6)
10
The Formulae-as-Classes Interpretation of Constructive Set Theory
Definition 3.3. (CZF + REA)
∗
In the same vein as in Definitions 3.1 and 3.2 we define a classes V(Yw
), V(Yw ), and (class)
.
∗
∗
∗
∗
∗
∗
˙
by replacing Y∗
functions = : V(Yw ) × V(Yw ) → Yw and ∈ : V(Yw ) × V(Yw ) → Yw
∗
with Yw .
.
.
Convention. We will write α = β and α ∈˙ β for = (α, β) and ∈˙ (α, β), respectively.
Definition 3.4. (CZF)
For any set A and class B we define:
Q
A → B as
x∈A B.
For any classes A and B we define:
P
A × B as Px∈A B,
A + B as
x∈2 Cx , where C0 = A and C1 = B.
(7)
(8)
Definition 3.5. A V(Y∗ )-assignment is a mapping M : Var → V(Y∗ ), where Var is the set
of variables of the language. M(a) will also be denoted by aM .
If M is a V(Y∗ )-assignment, u is a variable, and d ∈ V(Y∗ ), we define a V(Y∗ )-assignment
M(u|d) by
M(v)
if v is a variable other than u
M(u|d)(v) =
d
if v is u.
Sometimes, when an assignment M is fixed, we will omit the subscript M .
Definition 3.6. (CZF)
To any bounded formula θ ∈ L∈ and V(Y∗ )-assignment M we shall assign a set k θ k M .
Since we have already used the symbol “→” for function spaces we shall denote the conditional by “⊃” and the bi-conditional by “iff ”.
The recursive definition of k θ k M is given in the table below:
θ ∈ L∈
k θ kM
⊥
0
a=b
.
a M = bM
a∈b
aM ∈˙ bM
θ0 ∧ θ1
k θ0 k M × k θ1 k M
θ0 ∨ θ1
k θ0 k M + k θ1 k M
θ0 ⊃ θ1
k θ0 k M → k θ1 k M
Q
x∈a k ψ k M(v|g
aM (x))
P M
x∈αM k ψ k M(v|g
aM (x))
∀v ∈a ψ
∃v ∈a ψ
Lemma 3.7. (CZF)
For every bounded θ ∈ L∈ and V(Y∗ )-assignment M, k θ k M ∈ Y∗ .
Proof. This is proved by induction on θ using Lemma 2.8 and Definitions 3.2 and 3.4.
2
The Formulae-as-Classes Interpretation of Constructive Set Theory
11
Lemma 3.8. (CZF + REA)
∗
A V(Yw
)-assignment is defined similarly as a V(Y∗ )-assignment in Definition 3.5. Like∗
)-assignment M we
wise, as in Definition 3.6, to any bounded formula θ ∈ L∈ and V(Yw
∗
assign a set k θ k M . We then have, for every bounded θ ∈ L∈ and V(Yw
)-assignment M,
∗
k θ k M ∈ Yw .
Proof. This is proved as Lemma 3.7.
4
2
The formulae-as-classes interpretation for arbitrary formulae
In order to reflect within CZF the formulae-as-classes interpretation for arbitrary set-theoretic
formulae, we would need to represent large types ΠΣ-generated on top of V(Y∗ ). The language of CZF is not rich enough to do it in a straightforward way. To remedy this we introduce a notion of set recursive partial functions.
4.1
Extended E-recursive functions
We would like to have unlimited application of sets to sets, i.e. we would like to assign a
meaning to the symbol {a}(x) where a and x are sets. In generalized recursion theory this
is known as E-recursion or set recursion (see, e.g., [17] or [20, Ch.X]). However, we shall
introduce an extended notion of E-computability, christened E℘ -computability, rendering the
function exp(a, b) = a b is computable as well, (where a b denotes the set of all functions
from a to b). Moreover, the constant function with value ω is taken as an initial function in E℘ computability. From a classical standpoint, E℘ -computability is related to power recursion,
where the power set operation is regarded to be an initial function. The latter notion has been
studied by Moschovakis [14] and Moss [15].
There is a lot of leeway in setting up E℘ -recursion. The particular schemes we use are especially germane to our situation. Our construction will provide a specific set-theoretic model
for the elementary theory of operations and numbers EON (see, e.g., [7, VI.2], or the theory
APP as described in [21, Ch.9, Sect.3]). We utilize encoding of finite sequences of sets by
the pairing function h , i.
Definition 4.1. (CZF)
First, we select distinct non-zero natural numbers k, s, p, p0 , p1 , sN , pN , dN , 0̄, ω̄, π, σ,
pl, i, fa, and ab which will provide indices for special E℘ -recursive partial (class) functions.
Inductively we shall define a class E of triples he, x, yi. Rather than “he, x, yi ∈ E”, we shall
write “{e}(x) y”, and moreover, if n > 0, we shall use {e}(x1 , . . . , xn ) y to convey that
{e}(x1 ) he, x1 i ∧ {he, x1 i}(x2 ) he, x1 , x2 i ∧ . . . ∧ {he, x1 , . . . , xn−1 i}(xn ) y.
We shall say that {e}(x) is defined, written {e}(x) ↓, if {e}(x) y for some y. Let N := ω.
12
The Formulae-as-Classes Interpretation of Constructive Set Theory
E is defined by the following clauses (inference steps):
{k}(x, y)
{s}(x, y, z)
{p}(x, y)
{p0 }(x)
{p1 }(x)
{sN }(n)
{pN }(0)
{pN }(n + 1)
{dN }(n, m, x, y)
{dN }(n, m, x, y)
{0̄}(x)
{ω̄}(x)
{π}(x, g)
{σ}(x, g)
{pl}(x, y)
{i}(x, y, z)
{fa}(g, x)
{ab}(e, a)
x
{{x}(z)}({y}(z))
hx, yi
(x)0
(x)1
n + 1 if n ∈ N
0
n if n ∈ N
x if n, m ∈ N and n = m
y if n, m ∈ N and n 6= m
0
ω
Q
Pz∈x g(z) if g is a (set-)function with dom(g) = x
z∈x g(z) if g is a (set-)function with dom(g) = x
x+y
I(x, y, z)
g(x) if g is a (set-)function and x ∈ dom(g)
h if h is a (set-)function with dom(h) = a
and ∀x∈a {e}(x) h(x).
Note that for {s}(x, y, z) to be defined it is required that {x}(z), {y}(z) and
{{x}(z)}({y}(z)) be defined. The clause for s is thus to be read as a conjunction of the
following clauses: {s}(x) hs, xi, {hs, xi}(y) hs, x, yi and, if there exist a, b, c such that
{x}(z) a, {y}(z) b, {a}(b) c, then {hs, x, yi}(z) c.
The constants fa and ab stand for function application and function abstraction, respectively.
Lemma 4.2. (CZF)
E is an inductively defined class and E is functional in that for all e, x, y, y 0 ,
he, x, yi ∈ E ∧ he, x, y 0 i ∈ E ⇒ y = y 0 .
Proof. The inductive definition of E falls under the heading of Lemma 2.7. If {e}(x) y
the uniqueness of y follows by induction on the stages (see Lemma 2.7) of that inductive
definition.
2
Definition 4.3. Application terms are defined inductively as follows:
(i) The constants k,s,p,p0 ,p1 ,sN ,pN ,dN ,0̄,ω̄,π,σ,pl,i,fa,ab singled out in Definition 4.1
are application terms;
(ii) variables are application terms;
(iii) if s and t are application terms then (st) is an application term.
Definition 4.4. Application terms are easily formalized in CZF. However, rather than translating application terms into the set–theoretic language of CZF, we define the translation
of expressions of the form t ' u, where t is an application term and u is a variable. The
translation proceeds along the way that t was built up:
[c ' u]∧ is c = u if c is a constant or a variable;
[(st) ' u]∧ is ∃x∃y([s ' x]∧ ∧ [t ' y]∧ ∧ hx, y, ui ∈ E).
The Formulae-as-Classes Interpretation of Constructive Set Theory
13
Abbreviations. For application terms s, t, t1 , . . . , tn we will use:
s(t1 , . . . , tn )
st1 . . . tn
t↓
(s ' t)∧
{x}(y) = z
as a shortcut for
as a shortcut for
as a shortcut for
as a shortcut for
as a shortcut for
((. . . (st1 ) . . .)tn ); (parentheses associated to the left);
s(t1 , . . . , tn );
∃x(t ' x)∧ ; (t is defined)
s↓ ∨ t↓ ⊃ ∃x((s ' x)∧ ∧ (t ' x)∧ );
hx, y, zi ∈ E.
A closed application term is an application term that does not contain variables. If t is a closed
application term and a1 , . . . , an , b are sets we use the abbreviation
t(a1 , . . . , an ) ' b
for
∃x1 . . . xn ∃y x1 = a1 ∧ . . . ∧ xn = an ∧ y = b
∧ [t(x1 , . . . , xn ) ' y]∧ .
Definition 4.5. Every closed application term gives rise to a partial class function. A partial
n-place (class) function Υ is said to be an E℘ -recursive partial function if there exists a closed
application term tΥ such that
dom(Υ) = {(a1 , . . . , an ) | tΥ (a1 , . . . , an ) ↓}
and for all for all sets (a1 , . . . , an ) ∈ dom(Υ),
tΥ (a1 , . . . , an ) ' Υ(a1 , . . . , an ).
In the latter case, tΥ is said to be an index for Υ.
If Υ1 , Υ2 are E℘ -recursive partial functions, then Υ1 (~a) ' Υ2 (~a) iff neither Υ1 (~a) nor Υ2 (~a)
are defined, or Υ1 (~a) and Υ2 (~a) are defined and equal.
The next two results can be proved in the theory APP and thus hold true in any applicative
structure. Thence the particular applicative structure considered here satisfies the Abstraction
Lemma and Recursion Theorem (see e.g. [10] or [7]).
Lemma 4.6. ( Abstraction Lemma, cf. [7, VI.2.2])
For every application term t[x] there exists an application term λx.t[x] with FV(λx.t[x]) :=
{x1 , . . . , xn } ⊆ FV(t[x])\{x} such that the following holds:
∀x1 . . . ∀xn (λx.t[x]↓ ∧ ∀y (λx.t[x])y ' t[y]).
Proof. (i)λx.x is skk; (ii) λx.t is kt for t a constant or a variable other than x; (iii) λx.uv is
s(λx.u) (λx.v).
t
u
Lemma 4.7. (Recursion Theorem, cf. [7, VI.2.7])
There exists a closed application term rec such that for any f , x,
recf ↓ ∧ recf x ' f (recf )x.
Proof. Take rec to be λf.tt, where t is λyλx.f (yy)x.
t
u
Corollary 4.8. For any E℘ -recursive partial function Υ there exists a closed application term
τf ix such that τf ix ↓ and for all ~a,
Υ(ē, ~a) ' τf ix (~a),
where τf ix ' ē. Moreover, τf ix can be effectively (e.g. primitive recursively) constructed from
an index for Υ.
14
4.2
The Formulae-as-Classes Interpretation of Constructive Set Theory
Arbitrary formulae
Definition 4.9. (CZF)
If B is a class and a, x are sets, we write {a}(x) ∈ B with the following meaning:
{a}(x) ∈ B :⇔ ∃y({a}(x) = y ∧ y ∈ B).
(9)
Q
If A is a class and Bx are classes for all x ∈ A, then we define a class e x∈A Bx in the
Y
following way:
f
Bx := {a | ∀x∈A({a}(x) ∈ Bx )}.
(10)
x∈A
For any classes A and B we define a class A f
→ B by
Y
f
Af
→ B := {a | ∀x∈A({a}(x) ∈ B)} =
x∈A
B.
(11)
Definition 4.10. (CZF)
.
For every formula θ ∈ L∈ and V(Y∗ )-assignment M, we define a class k θ k M . The definition is given by the table below:
.
θ ∈ L∈
k θ kM
⊥
0
a=b
.
a M = bM
a∈b
aM ∈˙ bM
θ0 ∧ θ1
k θ0 k M × k θ1 k M
θ0 ∨ θ1
k θ0 k M + k θ1 k M
θ0 ⊃ θ1
k θ0 k M → k θ1 k M
.
.
.
.
.
.
θ0 k M
θ0 ⊃ θ1
.
θ1 k M
→k
f
∀v ∈a ψ
k
Q
∃v ∈a ψ
P
∀vψ
Q
.
e
x∈V(Y ∗ ) k ψ k M(v|x)
P
.
x∈V(Y ∗ ) k ψ k M(v|x)
∃vψ
if θ0 is bounded
if θ0 is not bounded
.
x∈aM
k ψ k M(v|g
aM (x))
.
x∈aM
k ψ k M(v|g
aM (x))
Lemma 4.11. (CZF)
.
For every bounded formula θ and a V(Y∗ )-assignment M, k θ k M = k θ k M .
Proof. This follows by induction on θ by comparing Definitions 3.6 and 4.10.
2
Definition 4.12. If θ(u1 , . . . , ur ) is a formula of L∈ all of whose free variables are among
u1 , . . . , ur , and α1 , . . . , αr ∈ V(Y∗ ), we shall use the shorthand k θ(α1 , . . . , αr ) k rather than
.
k θ k M , whenever M is an assignment satisfying M(ui ) = αi for 1 ≤ i ≤ r. In the special
case when θ is a sentence we will simply write k θ k. We will also the use the following
abbreviations:
e θ(α1 , . . . , αr )
V(Y∗ ) |= θ(α1 , . . . , αr )
|=∗ θ(α1 , . . . , αr )
iff
iff
iff
e ∈ k θ(α1 , . . . , αr ) k
e θ(α1 , . . . , αr ) for some e
V(Y∗ ) |= θ(α1 , . . . , αr ).
For a set-theoretic formula θ(~u) we say that θ(α1 , . . . , αr ) is validated in V(Y∗ ) if we have
produced a closed application term t such that t(~
α) θ(~
α) holds for all α
~ ∈ V(Y∗ ).
The Formulae-as-Classes Interpretation of Constructive Set Theory
4.3
15
The formulae-as-classes interpretation for CZF
The objective of the remainder of this section is to show the following interpretation.
Theorem 4.13. Let θ(u1 , . . . , ur ) be a formula of L∈ all of whose free variables are among
u1 , . . . , ur . If
CZF + RDC + ΠΣ−AC ` θ(u1 , . . . , ur ),
then one can effectively construct an index of an E℘ -recursive partial function g such that
CZFExp ` ∀α1 , . . . , αr ∈ V(Y∗ ) g(α1 , . . . , αr ) ∈ k θ(α1 , . . . , αr ) k,
where CZFExp denotes the modification of CZF with Exponentiation in place of Subset
Collection.
Proof. This will follow from 4.16, 4.17, 4.26 and 4.31 below.
t
u
The proof of 4.13 is rather long and requires close attention to the definition of indices of
E℘ -recursive functions. The details of an encoding are fascinating to work out and boring to
read. The author wrote the present section for his own benefit and his feelings will not be hurt
if the reader chooses to skip some of it.
Lemma 4.14. For every bounded θ(u1 , . . . , ur ) of L∈ all of whose free variables are among
u1 , . . . , ur one can effectively construct an application term tθ such that
CZFExp ` ∀α1 , . . . , αr ∈ V(Y∗ ) tθ (α1 , . . . , αr ) ' k θ(α1 , . . . , αr ) k.
Proof. We proceed by induction on the build-up of θ. The atomic cases amount to showing
˙
that the functions α, β 7→ =(α,
˙
β) and α, β 7→ ∈(α,
β) of Definition 3.2 are E℘ -recursive
∗
∗
partial functions defined on V(Y ) × V(Y ).
1. Let θ(u, v) be the atomic formula u = v. We define an application term ` by
`(e, α, β) ' π ᾱ, ab ᾱ, λx.σ(β̄, ab(β̄, λy.e(α̃(x), β̃(y))))
' π p0 α, ab p0 α, λx.σ(p0 β, ab(p0 β, λy.e(fa(p1 α, x), fa(p1 β, y))))) .
By the recursion theorem 4.7, we find an application term e such that
e(α, β) ' σ `(e, α, β), ab `(e, α, β), λz.`(e, β, α) .
Now put tu=v := e. Let be the well-founded ordering on V(Y∗ ) × V(Y∗ ) defined by:
hα, βi hγ, δi iff α = γ̃(x) and β = δ̃(y) for some x ∈ γ̄ and y ∈ δ̄. In view of Definition
3.2, one shows by induction on that for all α, β ∈ V(Y∗ ), tu=v (α, β) ↓ and tu=v (α, β) '
=(α,
˙
β), i.e. tu=v (α, β) ' k α = β k.
2. For the other type of atomic formula, where θ(u, v) is u ∈ v, we put
tu∈v := λxy.σ p0 y, ab p0 y, λz.tu=v (x, fa(p1 y, z)) ,
˙
so that tu∈v (α, β) ' ∈(α,
β), whence tu∈v (α, β) ' k α ∈ β k for all α, β ∈ V(Y∗ ).
3. If θ(~u) is θ0 (~u) ∧ θ1 (~u), put
tθ(~u) := λ~x.σ tθ0 (~u) (~x), ab tθ0 (~u) (~x), λz.(tθ1 (~u) (~x)) .
16
The Formulae-as-Classes Interpretation of Constructive Set Theory
4. If θ(~u) is θ0 (~u) ∨ θ1 (~u), put tθ(~u) := λ~x.pl tθ0 (~u) (~x), tθ1 (~u) (~x) .
5. If θ(~u) is θ0 (~u) ⊃ θ1 (~u), put
tθ(~u) := λ~x.π tθ0 (~u) (~x), ab tθ0 (~u) (~x), λz.(tθ1 (~u) (~x)) .
6. If θ(v, ~u) is ∀w∈v θ0 (w, v, ~u), put
tθ(v,~u) := λy~x.π p0 y, ab p0 y, λz.tθ0 (w,v,~u) (z, y, ~x) .
7. If θ(v, ~u) is ∃w∈v θ0 (w, v, ~u), put
tθ(v,~u) := λy~x.σ p0 y, ab p0 y, λz.tθ0 (w,v,~u) (z, y, ~x) .
2
Lemma 4.15. There are closed application terms idr , ids , idt , idt1 , idt2 such that for all
α, β, γ ∈ V(Y∗ ),
idr (α)
ids (α, β)
idt (α, β, γ)
idt (α, β, γ)
α = α.
α = β ⊃ β = α.
α = β ∧ β = γ ⊃ α = γ.
α = β ∧ β = γ ⊃ α = γ.
idt1 (α, β, γ) α = β ∧ β∈γ ⊃ α∈γ.
idt2 (α, β, γ) α = β ∧ γ∈α ⊃ γ∈β.
Proof. What do we need to construct idr ? Suppose we have
e(α̃(j)) ∈ k α̃(j) = α̃(j) k
for all j ∈ ᾱ. Then, letting f with domain ᾱ be defined by f (j) ' hj, e(α̃(j))i, we get
hf, f i ∈ k α = α k (see Definition 3.2), thence we have
p ab p0 α, λx.p(x, e(fa(p1 α, x))) , ab p0 α, λx.p(x, e(fa(p1 α, x))) ∈ k α = α k.
Now choose an index d by the recursion theorem so that, for all α,
dα ' p ab p0 α, λx.p(x, d(fa(p1 α, x))) , ab p0 α, λx.p(x, d(fa(p1 α, x)))
and put idr := d. By ∈-induction on α one easily verifies that idr α ↓ and that idr has the
desired property.
Intuitively, ids is the application term that interchanges the left and right member of any pair.
However, a witness for a bounded conditional has to be a function. This is achieved by letting
ids := λαβ.ab k α = β k, λx.p(p1 x, p0 x)
:= λyz.ab(tu=w (y, z), λx.p p1 x, p0 x) ,
The Formulae-as-Classes Interpretation of Constructive Set Theory
17
noting that the latter is an application term owing to Lemma 4.14.
To construct idt , suppose e ∈ k α = β ∧ β = γ k. Then p0 e ∈ k α = β k and p1 e ∈
k β = γ k. Thus p0 e ' hf, gi for functions f, g with domain ᾱ and β̄, respectively. Likewise, p1 e ' hf 0 , g 0 i for functions f 0 , g 0 with domain β̄ and γ̄, respectively. In view of the
definition of k α = γ k, one easily sees that hf 0 ◦ f, g ◦ g 0 i ∈ k α = γ k. We have to show that
hf 0 ◦ f, g ◦ g 0 i can be computed E℘ -recursively from e, α, β, γ. We have
ab p0 α, λw.(λu.fa(f 0 , u)) (λv.fa(f, v))(w)
' f0 ◦ f
ab p0 γ, λw.(λu.fa(g, u)) (λv.fa(g 0 , v))(w)
' g ◦ g0,
thus letting
τ1 := λxyz.ab p0 x, λw.(λu.fa(z, u)) (λv.fa(y, v))(w) ,
τ2 := λx0 y 0 z 0 .ab p0 x0 λw.(λu.fa(y 0 , u)) (λv.fa(z 0 , v))(w) ,
we get τ1 αf f 0 ' f 0 ◦ f and τ2 γgg 0 ' g ◦ g 0 , so that
p τ1 α(p00 e)(p10 e) τ2 γ(p01 e)(p11 e) ' hf 0 ◦ f, g ◦ g 0 i,
where pi,j := λv.pj (pi v). Thus letting
idt := λαβγ.ab k α = β ∧ β = γ k, λyp τ1 α(p00 y)(p10 y) τ2 γ(p01 y)(p11 y)
:= λz1 z2 z3 .ab tθ(u1 ,u2 ,u3 ) (z1 , z2 , z3 ), λyp τ1 z1 (p00 y)(p10 y) τ2 z3 (p01 y)(p11 y) ,
where “θ(u1 , u2 , u3 )” stands for “u1 = u2 ∧ u2 = u3 ”, we get
idt (α, β, γ) α = β ∧ β = γ ⊃ α = γ.
Next we show how to construct idt1 . To this end suppose d0 ∈ k α = β k and d1 ∈ k β∈γ k.
Then d1 = hi, e0 i for some i, e0 with i∈γ̄ and e0 β = γ̃(i). Hence we get
idt α, β, γ̃(i), hd, e0 i α = γ̃(i),
which yields hi, idt α, β, γ̃(i), hd, e0 i i α∈γ.
In view of previous constructions it is therefore clear that we can cook up an application term
t∗ such that whenever d α = β ∧ β∈γ then t∗ (α, β, γ, d) α∈γ. Whence we can put
idt1 := λαβγ.ab k α = β ∧ β∈γ k, λz.t∗ (α, β, γ, z) .
For idt2 suppose that d0 ∈ k α = β k and d1 ∈ k γ∈α k. Then d1 = hi, e0 i for some i, e0 with
i∈ᾱ and e0 γ = α̃(i). Moreover, d0 = hf, gi for functions f, g, where the domain of f is
ᾱ, and f (i) = hj, ci for some j ∈ β̄ and c α̃(i) = β̃(j). It follows that
idt γ, α̃(i), β̃(j), he0 , ci γ = β̃(j),
and hence
hj, idt γ, α̃(i), β̃(j), he0 , ci i γ∈β.
It is evident, by previous application term constructions that hj, idt γ, α̃(i), β̃(j), he0 , ci i can
be obtained E℘ -recursively from d0 , d1 , α, β, γ.
2
18
The Formulae-as-Classes Interpretation of Constructive Set Theory
Lemma 4.16. If the formula θ(~u) is derivable in intuitionistic predicate logic with identity,
then one can effectively construct an application term t from the deduction such that
t(~
α) θ(~
α)
holds for all α
~ ∈ V(Y∗ ).
We have already shown this for the axioms pertaining to equality. We illustrate the ideas for
logical axioms by carrying out a few examples. If ϕ(~u) and ψ(~u) are unbounded formulas,
then one easily checks that
k ϕ(~
α) ⊃ (ψ(~
α) ⊃ ϕ(~
α)),
s (ϕ(~
α) ⊃ (ψ(~
α) ⊃ χ(~
α))) ⊃ (ϕ(~
α) ⊃ ψ(~
α)) ⊃ (ϕ(~
α) ⊃ χ(~
α)) .
(12)
(13)
In fact, this justifies the combinators s and k. Combinatory completeness of these two combinators is equivalent to the fact that the deductive closure under modus ponens of (12) and
(13) is the full set of theorems of propositional implicational intuitionistic logic.
In case that ϕ or ψ are bounded we need to make crucial changes to the above. In case that
both ϕ and ψ are bounded, we get that
ab k ϕ(~
α) k, λx.ab(k ψ(~
α) k, λy.kxy) ϕ(~
α) ⊃ (ψ(~
α) ⊃ ϕ(~
α)).
It is pivotal to note here that according to Lemma 4.14, both k ϕ(~
α) k and
α) k are E℘ k ψ(~
α) k, λx.ab(k ψ(~
α) k, λy.kxy) is E℘ -computable
computable from α
~ , yielding that ab k ϕ(~
from α
~.
In the case that ϕ is bounded but ψ is unbounded we get
ab k ϕ(~
α) k, λxy.kxy) ϕ(~
α) ⊃ (ψ(~
α) ⊃ ϕ(~
α)).
In the case that ϕ is unbounded but ψ is bounded we get
λx.ab(k ψ(~
α) k, λy.kxy) ϕ(~
α) ⊃ (ψ(~
α) ⊃ ϕ(~
α)).
By the foregoing it should also be obvious what modifications have to be made to (13) if one
of the formulas ϕ or ψ is bounded.
Assuming all formulas are unbounded, one can easily check that:
λxy.pxy ϕ(~
α) ⊃ (ψ(~
α) ⊃ ϕ(~
α) ∧ ψ(~
α)),
λx.x ∀y ∀xθ(x, α
~ ) ⊃ θ(y, α
~)
λx.p0̄x ϕ(~
α) ⊃ ϕ(~
α) ∨ ψ(~
α)
and that the term
λxyz.dN (y(p1 x))(z(p1 ))(p0 x)
witnesses ∨-elimination, that is to say, the axiom scheme
ϕ ∨ ψ ⊃ (ϕ ⊃ χ) ⊃ [(ψ ⊃ χ) ⊃ χ] .
Minor modifications yield witnessing terms if one or more of the formulas are bounded.
The Formulae-as-Classes Interpretation of Constructive Set Theory
19
From Lemma 4.15 in conjunction with the usual laws of intuitionistic connectives and quantifiers one can deduce that ∀x[θ(x, ~y ) ⊃ ∀z(z = x ⊃ θ(z, ~y )] and hence one can effectively
construct a term t=,θ such that
t=,θ ∀x[θ(x, α
~ ) ⊃ ∀z(z = x ⊃ θ(z, α
~ )].
(14)
Since the quantifier logic employed here is slightly unusual in that bounded quantifiers are
treated as quantifiers in their own right, we should check that the usual logical connections
obtain between bounded and unbounded quantifiers. The claim is that one can construct terms
t∀,ϕ and t∃,ϕ such that
t∀,ϕ (β, α
~ ) ∀x∈β ϕ(x, β, α
~ ) iff ∀x[x∈β ⊃ ϕ(x, β, α
~ )]
t∃,ϕ (β, α
~ ) ∃x∈β ϕ(x, β, α
~ ) iff ∃x[x∈β ∧ ϕ(x, β, α
~ )].
(15)
(16)
To prove (15) first suppose that f ∀x∈β ϕ(x, β, α
~ ). Then for all i ∈ β̄, f (i) ϕ(β̃(i), β, α
~ ).
∗
Now suppose in addition that δ ∈ V(Y ) and d δ ∈ β. Then d = hj, d1 i with j ∈ β̄ and
d1 δ = β̃(i). Utilizing (14), we can design an application term τ such that
τ (f, d, j, β̃(j), α
~ , β, δ) ϕ(δ, β, α
~ ).
Moreover, since j and β̃(j) are actually computable from d and β, there is a closed application
term τ 0 such that
τ 0 (f, d, α
~ , β, δ) ϕ(δ, β, α
~ ).
(17)
As a consequence of (17), we get
ab k δ∈β k, λy.τ 0 (f, y, α
~ , β, δ) δ∈β ⊃ ϕ(δ, β, α
~ ).
(18)
λδ.ab k δ∈β k, λy.τ 0 (f, y, α
~ , β, δ) ∀x[x∈β ⊃ ϕ(x, β, α
~ )].
(19)
Thus we have
For the other direction of (15), suppose
e ∀x[x∈β ⊃ ϕ(x, β, α
~ )].
(20)
We need to effectively construct a function f with domain β̄ such that f (i) ϕ(β̃(i), β, α
~ ).
Let i∈β̄. By Lemma 4.15, we have idr (β̃(i)) ∈ k β̃(i) = β̃(i) k, thus
hi, idr (β̃(i)i β̃(i) ∈ β,
so that by (20), e(p(i, idr (β̃(i))) ϕ(β̃(i), β, α
~ ). Letting
f := ab β̄, λx.e(p(x, idr (β̃(x))) ,
we get f ∀x∈β ϕ(x, β, α
~ ). Also, by now it is plain how to design an application term
which shows that f is E℘ -computable from e, β.
The proof of (16) is very similar to (15).
2
20
The Formulae-as-Classes Interpretation of Constructive Set Theory
Lemma 4.17. For each axiom θ(~u) of CZF one can effectively construct an application term
t from the deduction such that
t(~
α) θ(~
α)
holds for all α
~ ∈ V(Y∗ ).
Proof. 1. Extensionality: Suppose
e ∀x [(x ∈ α ⊃ x ∈ β) ∧ (x ∈ β ⊃ x ∈ α)].
By lemma 4.16, in particular (15), there is an application term tex such that
tex (α, β, e) ∀x ∈ α x ∈ β ∧ ∀x ∈ β x ∈ α.
The latter actually amounts to tex (α, β, e) α = β, so that
λx.tex (α, β, x) ∀x [(x ∈ α ⊃ x ∈ β) ∧ (x ∈ β → x ∈ α)] ⊃ α = β.
2. Pair: Let α, β ∈ V(Y∗ ). Put γ := sup(2, f ), where dom(f ) = 2, f (0) = α, and f (1) = β.
As 2 ∈ Y∗ , we get γ ∈ V(Y∗ ). Also γ is computable from α and β as 0̄α ' 0, sN 0 ' 1,
sN 1 ' 2, ab(2, λn.dN (0, n, α, β)) ' f and p2f ' γ. Moreover, one easily constructs a
term tp such that tp (α, β) ∀x(x ∈ γ iff [x = α ∨ x = β]) and hence
p(γ, tp (α, β)) ∃y∀x(x ∈ y iff [x = α ∨ x = β]),
from which we can distill a term witnessing the pairing axiom.
Union: Let
γ := sup(a, f ) = paf
a := σ ᾱ, ab(ᾱ, λz.α̃(z))
^
f := ab a, λx.(α̃(p
0 x))(p1 (x))
One readily checks that a ∈ Y∗ and f : a → V(Y∗ ), thus γ ∈ V(Y∗ ). Also, it is clear
that γ is E℘ -computable from α. We leave it to the reader to construct a term t∪ such that
t∪ (α) ∀v[v ∈ γ iff ∃u ∈ α v ∈ u], so that
p(γ, t∪ (α)) ∃y∀v[v ∈ y iff ∃u ∈ α v ∈ u].
Bounded Separation: Let θ(u, v, w)
~ be a bounded formula and α, β~ ∈ V(Y∗ ). Let A :=
~ k and g : A → V(Y∗ ) be defined by g(z) := α̃(p0 z). Put γ :=
k ∃x ∈ α, θ(x, α, β)
sup(A, g). Note that g = ab(A, λz.α̃(p0 z)). We have A ∈ Y∗ by Lemma 3.7 so that
~ We also
γ ∈ V(Y∗ ). By Lemma 4.14, A and g and hence γ are E℘ -computable from α, β.
have
hi, xi ∈ k η ∈ γ k
iff
iff
~ k
i ∈ A ∧ x ∈ k η = γ̃(i) k ∧ p0 i ∈ k θ(γ̃(i), α, β)
~ ∈ k θ(η, α, β)
~ k
i ∈ A ∧ x ∈ k η = γ̃(i) k ∧ t0 (i, γ, α, β)
for some term t0 obtainable from (14). As a result we can compose a term ts such that
~
~ .
p γ, ts (α, β)
∀u u ∈ γ iff [u ∈ α ∧ θ(u, α, β)]
The Formulae-as-Classes Interpretation of Constructive Set Theory
21
Strong Collection: Suppose
~
f ∀u ∈ α ∃v θ(u, v, α, β).
Then f is a function with domain ᾱ such that, for all i ∈ ᾱ,
p1 (f (i)) θ (α̃(i), p0 (f (i)), β~ .
Let g be the result of ab ᾱ, λx.p0 (f (x)) and put γ = p(ᾱ, g). By definition, γ is E℘ computable from α and f . Furthermore, one readily constructs a term t1 satisfying
~ ∧ ∀v ∈ γ ∃u ∈ α θ(u, v, β),
~
t1 (α, f ) ∀u ∈ α ∃v ∈ γ θ(u, v, β)
from which we can glean a term t++ satisfying
~ ∀u ∈ α ∃v θ(u, v, α, β)
~ ⊃ ∃wθ0 (α, w, β),
~
t++ (α, β)
~ denotes the formula ∀u ∈ α ∃v ∈ w θ(u, v, β)
~ ∧ ∀v ∈ w ∃u ∈ α θ(u, v, β).
~
where θ0 (α, w, β)
Subset Collection: Let α, β ∈ V(Y∗ ) and define A := (ᾱ → β̄) and a function h with
domain A by
h(z) := sup ᾱ, ab ᾱ, λx.β̃(z(x)) .
Put γ := sup(A, h). It is easy to see that γ ∈ V(Y∗ ) and that γ is E℘ -computable from α
and β. Now assume that
~
f ∀u ∈ α ∃v ∈ β ϕ(u, v, α, β, ξ).
~ Then we have
Let θ(u, v) stand for ϕ(u, v, α, β, ξ).
∀i ∈ ᾱ p0 (f (i)) ∈ β̄ ∧ p1 (f (i)) θ(α̃(i), β̃(p0 (f (i)))) .
Thus we have g ∈ A, where g is the function with domain ᾱ and g(i) := p0 (f (i)). Let
δ := sup ᾱ, ab ᾱ, λx.β̃(g(x)) .
Then δ ∈ V(Y∗ ) and δ is clearly E℘ -computable from α and f . Moreover, δ = h(g) so that
p(g, ids (δ)) δ ∈ γ.
(21)
Also ᾱ = δ̄ and
∀i ∈ ᾱ p1 (f (i)) θ(α̃(i), δ̃(i)).
The latter yields
p ᾱ, ab ᾱ, λx.p(x, p1 (f (x)))
∀u ∈ α ∃v ∈ δ θ(u, v)
p ᾱ, ab ᾱ, λx.p(x, p1 (f (x)))
∀v ∈ δ ∃u ∈ α θ(u, v).
In view of (21) and (22), we can construct a term tc such that
tc (α, β, f ) ∃w ∈ γ ∀u ∈ α ∃v ∈ w θ(u, v) ∧ ∀v ∈ u ∃u ∈ α θ(u, v) .
(22)
22
The Formulae-as-Classes Interpretation of Constructive Set Theory
Thence we can construct a term tssc such that
~ ∃z ∀u ∈ α ∃v ∈ β θ(u, v) ⊃
tssc (α, β, ξ)
∃w ∈ γ ∀u ∈ α ∃v ∈ w θ(u, v) ∧ ∀v ∈ u ∃u ∈ α θ(u, v) ,
verifying Subset Collection.
∈-Induction: By the recursion theorem we can effectively construct an index of an E℘ recursive partial function h satisfying
h(e, α) ' (eα) ab(ᾱ, λx.h(e, x)) .
Now suppose e ∀u[∀v ∈ u θ(v) ⊃ θ(u)]. By induction on α, we shall prove that
h(e, α) θ(α).
(23)
Toward this end suppose that for all i ∈ ᾱ, h(e, α) θ(α̃(i)). Then we have ab ᾱ, λx.h(e, x)
∀v ∈ α θ(v) so that (eα) ab ᾱ, λx.h(e, x) θ(α), yielding (23). Finally from (23) we
arrive at
λyz.h(y, z) ∀u[∀v ∈ u θ(v) ⊃ θ(u)] ⊃ ∀uθ(u).
Infinity: First note that the formula 0 ∈ y ∧ ∀w ∈ y w + 1 ∈ y written out in full detail reads
(∃z ∈ y ∀u ∈ z ⊥) ∧ ∀w ∈ y ∃v ∈ y∀u ∈ v [u ∈ w ∨ u = w].
Using the recursion theorem, one effectively constructs an index ê such that
êx ' sup x, ab x, λy.dN pN x, y, sup(pN x, ab(pN x, λz.êz)), êy
.
By induction on n ∈ ω, we show êk ↓ and êk ∈ V(Y∗ ) for all k ≤ n. Note that n ∈ Y∗ for
all n ∈ ω. Clearly we have ê0 ↓ and ê0 ∈ V(Y∗ ). Assume that êk ↓ and êk ∈ V(Y∗ ) for all
k ≤ n. As pN (n + 1) ' n we then have ab(pN (n + 1), λz.êz)) ↓, which yields
dN pN (n + 1), m, sup(pN (n + 1), ab(pN (n + 1), λz.êz)), êm ↓
for all m ≤ n. The latter yields ê(n + 1) ↓ and also ê(n + 1) ∈ V(Y∗ ).
By the above we may define a function hω with domain ω such that ên ' hω (n). As ω ∈
V(Y∗ ) we get
ω ∗ := sup(ω, hω ) ∈ V(Y∗ ).
(24)
By construction, ω ∗ is E℘ -computable. Note that
hω (n + 1) ' sup n + 1, ab n + 1, λy.dN n, y, sup(n, hω n), hω (y)) ,
where hω n denotes the restriction of hω to n. Thus, inductively we have hω (0) = 0,
hω (n + 1) = n + 1, and
hω (n) = sup(n, hω n)
(25)
The Formulae-as-Classes Interpretation of Constructive Set Theory
23
for all n ∈ ω, so that
^
∀i ∈ hω (n) (h
ω (n))(i) = hω (i).
We then get
^
^
∀i ∈ hω (n + 1) (hω^
(n + 1))(i) = (h
ω (n))(i) ∨ (hω (n + 1))(i) = hω (n) .
Defining a function fn by dom(f ) = n + 1 and
^
fn (i) := dN n, i, h0, hi, ids ((h
ω (n))(i))ii, h1, ids (hω (n))i ,
we obtain fn ∀u ∈ hω (n + 1) u ∈ hω (n) ∨ u = hω (n) , that is
fn hω (n + 1) = hω (n) + 1
(26)
in short hand. As a result,
hn + 1, fn i ∃v ∈ ω ∗ ∀u ∈ v u ∈ hω (n) ∨ u = hω (n)
and consequently
f ∗ ∀w ∈ ω ∗ ∃v ∈ ω ∗ ∀u ∈ v u ∈ w ∨ u = w ,
(27)
where f ∗ is the function with domain ω and f ∗ (n) = hn+1, fn i. We also have ab(0, λx.x) ∀u ∈ hω (0) ⊥, and whence
h0, ab(0, λx.x)i ∃z ∈ ω ∗ ∀u ∈ z ⊥.
(28)
Since f ∗ is E℘ -computable we can combine (27) and (28) to extract a term t∗0 such that
t∗0 ∃z ∈ ω ∗ ∀u ∈ z ⊥ ∧ ∀w ∈ ω ∗ ∃v ∈ ω ∗ ∀u ∈ v u ∈ w ∨ u = w ,
in other words
t∗0 0 ∈ ω ∗ ∧ ∀w ∈ ω ∗ w + 1 ∈ ω ∗ .
(29)
Let θ(α) abbreviate the formula
∃z ∈ α ∀u ∈ z ⊥ ∧ ∀w ∈ α ∃v ∈ α∀u ∈ v u ∈ w ∨ u = w
and suppose
e θ(α),
(30)
for some α ∈ V(Y∗ ). In view of (26) one then constructs a term tω such that for all n ∈ ω,
tω (n, α, e) hω (n) ∈ α
so that
ab ω, λx.tω (x, α, e) ∀u ∈ ω ∗ u ∈ α.
Hence
λyz.ab ω, λx.tω (x, y, z)
∀a θ(a) ⊃ ω ∗ ⊆ a .
(31)
Using the witnessing terms from (29) and (31) and the fact that ω ∗ is E℘ -computable we then
get a term t∗ such that
t∗ ∃y θ(y) ∧ ∀a θ(a) ⊃ y ⊆ a .
t
u
24
The Formulae-as-Classes Interpretation of Constructive Set Theory
Remark 4.18. In his interpretation of set theory in Martin-Löf type theory (cf. [1, 2, 3]),
Aczel frequently invokes the axiom of choice in type theory as for instance in the proofs of
the validity of Strong Collection and Subset Collection. To the author’s mind, this casts a
shroud of mystery over these proofs. The axiom of choice is usually invoked, as it were, to
pull a function out of thin air, but due to the propositions-as-types doctrine of Martin-Löf’s
type theory, the axiom of choice is deducible therein since a judgement that a proposition of
the form ∀x : A ∃y : B ϕ(x, y) holds true provides all the information needed to construct
a function f : A → B such that ∀x : A ϕ(x, f (x)). The formulae-as-classes interpretation
via E℘ -recursive partial functions bears out this fact very lucidly. The latter remarks are even
more appropriate to the proofs of the subsection to come.
4.4
Choice principles in V(Y∗ )
In this subsection we show that V(Y∗ ) validates RDC and ΠΣI−AC. We will be retracing
the ground of [2] §5–6.
Definition 4.19. Let OP(x, y) = z be an abbreviation for the set-theoretic formula expressing that z is the ordered pair of x and y. By Lemma 4.17 there exist E℘ -recursive functions
α, β 7→ hα, βiV and op
b such that for all α, β ∈ V(Y∗ ), op(α,
b
β) ↓ and hα, βiV ∈ V(Y∗ )
and
op(α,
b
β) OP(α, β) = hα, βiV .
(32)
If A ∈ Y∗ and f is an E℘ -recursive function which is total on A then we shall use the notation
sup i ∈ A f (i) := sup A, ab A, λx.f (x) .
For α, β ∈ V(Y∗ ) let
S(α, β) :=
sup i ∈ ᾱ hα̃(i), β̃(i)iV .
(33)
Note that S is an E℘ -partial recursive function such that if ᾱ = β̄ then S(α, β) ↓ and, moreover, S(α, β) ∈ V(Y∗ ).
Lemma 4.20. There are E℘ -recursive partial functions g0 , g1 , g2 such that the following
hold:
(i) If α, β ∈ V(Y∗ ) with ᾱ = β̄, then
g0 (α, β) S(α, β) is a relation with domain α and range β.
(ii) If α, γ ∈ V(Y∗ ) such that
e γ is a relation with domain α
then g1 (e, α, γ) ∈ V(Y∗ ) and, letting δ := g1 (e, α, γ), we have δ̄ = ᾱ and
g2 (e, α, γ) S(α, δ) ⊆ γ.
The Formulae-as-Classes Interpretation of Constructive Set Theory
25
Proof. (i): Let α, β ∈ V(Y∗ ) such that ᾱ = β̄ and let γ = S(α, β). Also let A = ᾱ. Then, by
choosing x = y = u,
∀u ∈ A ∃x ∈ A ∃y ∈ A |=∗ γ̃ = hα, βiV ,
so that
h0 (α, β) ∀u ∈ γ ∃x ∈ α ∃y ∈ β OP(x, y) = u,
(34)
where h0 stands for the obvious E℘ -recursive partial function. Also, by choosing u = y = x,
we get
h1 (α, β) ∀x ∈ α ∃u ∈ γ ∃y ∈ βOP(x, y) = u,
(35)
where h1 stands for the obvious E℘ -recursive partial function. As the the set-theoretic statements in (34) and (35) (after the “” symbol) provably in CZF entail that
S(α, β) is a relation with domain α and range β,
we can construct the desired function g0 from h0 and h1 .
(ii): Suppose that e γ is a relation with domain α. Then we obtain E℘ -recursive partial
functions `0 , `1 such that
∀i ∈ A `0 (α, i, γ, e) ∃z ∈ γ OP(α̃(i), δ) = z,
where δ = `1 (α, i, γ, e). Now let f : A → V(Y∗ ) be defined by ab A, λv.`1 (α, v, γ, e) and
put β = sup(A, f ). Then β ∈ V(Y∗ ) and β̄ = ᾱ. Clearly, β can be computed from α, γ, e.
From the above we construct g2 such that g2 (e, α, γ) S(α, δ) ⊆ γ.
2
Definition 4.21. α ∈ V(Y∗ ) is injectively presented if for all i, j ∈ ᾱ, whenever
e α̃(i) = α̃(j)
for some e, then i = j.
Lemma 4.22. There are E℘ -recursive partial functions `∗0 , . . . , `∗5 such that for any injectively
presented α ∈ V(Y∗ ) the following hold:
(i) If β ∈ V(Y∗ ), such that β̄ = ᾱ and δ ∈ V(Y∗ ) then for all i ∈ ᾱ,
`∗0 (α, i, β, δ) ∃z ∈ S(α, β) OP(α̃(i), δ) = z iff δ = β̃(i) .
(ii) If γ ∈ V(Y∗ ) and
e γ is a function with domain α
then
`∗1 (α, γ, e) γ = S(α, β),
with β = `∗2 (α, γ, e) ∈ V(Y∗ ) and β̄ = ᾱ.
Conversely, if β̄ = ᾱ and e0 γ = S(α, β), then
`∗3 (α, β, γ, e0 ) γ is a function with domain α.
26
The Formulae-as-Classes Interpretation of Constructive Set Theory
(iii) If β1 , β2 ∈ V(Y∗ ) such that β1 = β2 = ᾱ, then e S(α, β1 ) = S(α, β2 ) implies
∀i ∈ ᾱ `∗4 (α, i, β1 , β2 ) βe1 (i) = βe2 (i),
and conversely, if ∀i ∈ ᾱ e0 i βe1 (i) = βe2 (i), then
`∗5 (α, β1 , β2 , e0 ) S(α, β1 ) = S(α, β2 ).
Proof. Let α be injectively presented.
(i): Let β ∈ V(Y∗ ) such that β̄ = ᾱ, δ ∈ V(Y∗ ) and i ∈ ᾱ. Note that S(α, β) = ᾱ. Let
µ := S(α, β).
First assume e ∃z ∈ µ OP(α̃(i), δ) = z. Then p0 e = j for some j ∈ ᾱ and p1 e OP(α̃(i), δ) = γ̃(j), so that p1 e OP(α̃(i), δ) = hα̃(j), β̃(j)iV . The latter yields
hr (α, β, δ, i, e) α̃(i) = α̃(j) ∧ δ = β̃(j)
for some E℘ -recursive partial function hr . Since α is injectively presented the latter implies
i = j, and hence p1 (hr (α, β, δ, i, e)) δ = β̃(i).
Conversely assume e0 δ = β̃(i). Then
p i, p ids (α̃(i)) α̃(i) = α̃(i) ∧ δ = β̃(i).
The latter yields
h+ (α, β, δ, i, e0 ) OP(α̃(i), δ) = hα̃(i), β̃(i)iV ,
for some E℘ -recursive partial function h+ , and hence
p i, h+ (α, β, δ, i, e0 ) ∃z ∈ µ OP(α̃(i), δ) = z.
It is now obvious how to distill the desired function `∗0 from hr and h+ .
Let γ ∈ V(Y∗ ). If
e γ is a function with domain α,
then by 4.20 (ii), g1 (e, α, γ) ∈ V(Y∗ ) and, letting δ := g1 (e, α, γ), we have δ̄ = ᾱ and
g2 (e, α, γ) S(α, δ) ⊆ γ.
It is now obvious how to define `∗1 and `∗2 .
For the converse let e0 γ = S(α, β), where β ∈ V(Y∗ ) such that β̄ = ᾱ. By 4.20
g0 (α, β) S(α, β) is a relation with domain α and range β.
Also, for i, j ∈ ᾱ, if |=∗ hα̃(i), β̃(j)iV ∈ γ then, owing to (i), |=∗ α̃(i) = α̃(j) since
|=∗ hα̃(i), β̃(i)iV ∈ S(α, β), so that i = j because α is injectively presented. Hence, there is
a E℘ -recursive partial function `∗3 such that
`∗3 (α, β, γ, e0 ) γ is a function with domain α.
The Formulae-as-Classes Interpretation of Constructive Set Theory
27
(iii): Let β1 , β2 ∈ V(Y∗ ) such that β1 = β2 = ᾱ. Then f S(α, β1 ) ⊆ S(α, β2 ) is equivalent
to
∀i ∈ ᾱ f (i) hα̃(i), βe1 (i)iV ∈ S(α, β2 ).
By (ii) there is an E℘ -recursive partial function hs such that the latter implies
hs (α, β1 , β2 , f, i) βe1 (i) = βe2 (i)
for all i ∈ ᾱ. On the other hand, one easily constructs an E℘ -recursive partial function hq such
that ei βe1 (i) = βe2 (i) for all i ∈ ᾱ implies
hq (α, β1 , β2 , e) S(α, β1 ) ⊆ S(α, β2 ).
As the roles of β1 and β2 can be interchanged, by the above it is obvious how to construct the
desired functions `∗4 and `∗5 .
2
Theorem 4.23. There is an E℘ -recursive partial function ı such that whenever α ∈ V(Y∗ )
is injectively presented,
ı(α) α is a base.
Proof. Let α ∈ V(Y∗ ) be injectively presented and let γ ∈ V(Y∗ ) satisfy
e γ is a relation with domain α.
Then by 4.20 (ii), g1 (e, α, γ) ∈ V(Y∗ ) and, letting δ := g1 (e, α, γ), we have δ̄ = ᾱ and
g2 (e, α, γ) S(α, δ) ⊆ γ.
Let η := S(α, δ). By 4.22 (ii) we also have
`∗3 α, β, η, ids (η) η is a function with domain α.
Hence, in view of the foregoing we can compose a function ı∗ such that
ı∗ (α, γ, e) ∃f Fun(f ) ∧ dom(f ) = α ∧ f ⊆ γ
so that
ı(α) α is a base,
2
where ı := λxyz.ı∗ (x, y, z).
Lemma 4.24. ω ∗ and hω (n) for n ∈ ω are injectively presented.
Proof. Recall that ω ∗ = sup(ω, hω ).
Suppose |=∗ hω (n) = hω (k) for some n, k ∈ ω. If n ∈ k, then, by (25), |=∗ hω (n) ∈ hω (k),
so that |=∗ hω (n) ∈ hω (n). But by (4.17) we have |=∗ ∀x x ∈
/ x, and hence, as 6|=∗ ⊥, we
conclude n ∈
/ k. Likewise we can conclude k ∈
/ n. Since
∀x, y ∈ ω (x ∈ y ∨ x = y ∨ y ∈ x),
we arrive at n = k.
Note that by (25), hω (n) = n and
^
(h
ω (n))(k) = hω (k)
for k ∈ n, so that from
^
^ 0
|=∗ (h
ω (n))(k) = (hω (n))(k )
for k, k 0 ∈ n we obtain hω (k) = hω (k 0 ), and thus k = k 0 by the foregoing.
2
28
The Formulae-as-Classes Interpretation of Constructive Set Theory
Lemma 4.25. Let θ0 (x, ~u) and ψ0 (x, y, ~u) be formulas of set theory. There is an E℘ -recursive
partial function hd such that whenever ξ~ ∈ V(Y∗ ) and
~ ⊃ ∃y θ0 (y, ξ)
~ ∧ ψ0 (x, y, ξ)
~
e ∀x θ0 (x, ξ)
(36)
and
~
e0 θ0 (α, ξ)
(37)
then
hd (e, e0 , ξ, α) ∃f f is a function with domain ω ∧
(38)
~ ∧ ψ0 (f (n), f (n + 1), ξ)
~ .
f (0) = α ∧ ∀n ∈ ω θ0 (f (n), ξ)
~ and ψ0 (x, y, ξ),
~ respectively. Suppose that (36) and
Proof. Let θ(x) and ψ(x, y) be θ0 (x, ξ)
(37) hold true. For the sake of simplicity we shall also assume that θ(x) is an unbounded
formula.
Let ai be pi a and aij be pj (pi a). By the recursion theorem we can construct an index c such
that
c(u, x, y, z) ' dN 0, u, px(pxy), z c(pN u, x, y, z) 0 , c(pN u, x, y, z) 11 . (39)
For n ∈ ω put
g(n) := c(n, α, e0 , e)
and f (n) := (g(n))0 . Note that f (0) = α and (g(0))11 = e0 , so that by (37),
(g(0)11 θ(f (0))
By induction on n > 0 we shall show that g(n) ↓, f (n) ↓, and
g(n) ∃y θ(y) ∧ ψ(f (n − 1), y)) .
By (39),
g(1) ' e c(0, α, e0 , e) 0 , c(0, α, e0 , e) 11
' e (g(0))0 , (g(0))11
' e f (0), (g(0))11
so that by (41) and (36) we have g(1) ↓, f (1) ↓, and
g(1) ∃y θ(y) ∧ ψ(f (0), y)) .
Now assume (41) to be true, where n > 0. By (39),
g(n + 1) ' e c(n, α, e0 , e) 0 , c(n, α, e0 , e) 11
' e (g(n))0 , (g(n))11
' e f (n), (g(n))11
(40)
(41)
The Formulae-as-Classes Interpretation of Constructive Set Theory
29
so that by (41) and (36) we have g(n + 1) ↓, f (n + 1) ↓, and
g(n + 1) ∃y θ(y) ∧ ψ(f (n), y)) .
Hence by induction on n, (41) holds. Note that (41) also entails that
(g(n))1 θ(f (n)) ∧ ψ(f (n − 1), f (n))) .
(42)
Let η := sup(ω, f ). Then η ∈ V(Y∗ ) with η̄ = ω, so that if δ = S(ω ∗ , η) then δ ∈ V(Y∗ )
and by 4.22 (ii), as ω ∗ is injectively presented, we get
|=∗ δ is a function with domain ω ∗ .
As δ̃(0) = hhω (0), η̃(0)iV = hhω (0), f (0)iV it follows that |=∗ hhω (0), αiV ∈ δ. Finally, let
k ∈ ω and γ, ρ ∈ V(Y∗ ) such that |=∗ hhω (k), ρiV ∈ δ and |=∗ hhω (k + 1), γiV ∈ δ. Then
by 4.22 (i), |=∗ ρ = f (k) and |=∗ γ = f (k + 1) and hence, from (40) and (42),
|=∗ θ(ρ) ∧ ψ(ρ, γ).
Note also that via its definition, δ is E℘ -computable from α, e, e0 and, moreover, all transformations involving “|=∗ ” can be made E℘ -computable, so that we can extract hd from the
above.
2
Theorem 4.26. RDC is validated in V(Y∗ ).
Proof. This follows from Lemma 4.25.
t
u
The aim of the remainder of this section is to show that internally in V(Y∗ ), every ΠΣIgenerated set is a base. In the following definitions we shall define Σ(α, β) ∈ V(Y∗ ) and
Π(α, β) ∈ V(Y∗ ) for α, β ∈ V(Y∗ ) such that ᾱ = β̄. We then show that when α is injectively presented then in V(Y∗ ) these are the disjoint union and cartesian product, respectively, of the family of sets S(α, β) indexed by α.
Definition 4.27. Let α, β ∈ V(Y∗ ) with ᾱ = β̄. As ᾱ ∈ Y∗ and β̃(i) ∈ Y∗ for i ∈ ᾱ it
follows that C ∈ Y∗ , where
C := Σi∈ᾱ β̃(i).
For z ∈ C, p0 z ∈ ᾱ so that f (z) ∈ V(Y∗ ) where
^
f (z) := (β̃(p
0 z))(p1 z).
Now define
Σ(α, β) := (sup z ∈ C)hα̃(p0 z), f (z)i.
Note that
D := Πi∈ᾱ β̃(i)
is in Y∗ . Let z ∈ D. For i ∈ ᾱ we have α̃(i) ∈ V(Y∗ ), β̃(i) ∈ V(Y∗ ) and z(i) ∈ β̃(i) so that
(^
β̃(i))(z(i)) ∈ V(Y∗ ). Hence g(z) ∈ V(Y∗ ) and g(z) = ᾱ where
g(z) := (sup i ∈ ᾱ)(^
β̃(i))(z(i)).
30
The Formulae-as-Classes Interpretation of Constructive Set Theory
Hence S(α, g(z)) ∈ V(Y∗ ). Now define
Π(α, β) := (sup z ∈ D)S(α, g(z)).
Then Π(α, β) ∈ V(Y∗ ).
Lemma 4.28. The partial functions α, β 7→ Σ(α, β) and α, β 7→ Π(α, β) are E℘ -recursive
partial functions. Also, whenever ᾱ = β̄ then Σ(α, β) ↓ and Π(α, β) ↓.
t
u
Proof. This is obvious by their definition.
Proposition 4.29. There are E℘ -recursive partial functions hs , hp such that whenever α is
injectively presented and ᾱ = β̄ then the following hold:
hs (α, β) Σ(α, β) is the disjoint union of the family of sets S(α, β).
hp (α, β) Π(α, β) is the cartesian product of the family of sets S(α, β).
(43)
(44)
Moreover, if β̃(i) is injectively presented for all i ∈ ᾱ then both Σ(α, β) and Π(α, β) are
injectively presented.
Proof. By 4.22,
ha (α, β) S(α, β) is a function with domain α
for a E℘ -recursive partial function ha . Let η ∈ V(Y∗ ). Suppose
e η is in the disjoint union of S(α, β).
There is an E℘ -recursive partial function h0 such that the latter yields
h0 (α, β, η, e) ∃x ∈ α ∃v ∃u ∈ S(α, β) hx, viV = u ∧ ∃y ∈ v OP(x, y) = η
so that
∃v
∃u
∈
S(α,
β)
hα̃(i),
vi
=
u
∧
∃y
∈
v
OP(α̃(i),
y)
=
η
,
V
1
where i = h0 (α, β, η, e) 0 . Hence by 4.22 (ii),
h0 (α, β, η, e)
h1 (α, β, η, e) δ = β̃(i) ∧ ∃y ∈ δ OP(α̃(i), y) = η,
where δ = h0 (α, β, η, e) 10 and h1 is an E℘ -recursive partial function. Thus
h2 (α, β, η, e) ∃y ∈ β̃(i) OP(α̃(i), y) = η,
with another E℘ -recursive partial function h2 . The latter yields
h3 (α, β, η, e) hα̃((z)0 ), f (z)iV = η,
where z ∈ C is defined by z = hi, h2 (α, β, η, e) 0 i and h3 is yet another E℘ -recursive partial
function. Therefore
h4 (α, β, η, e) η ∈ Σ(α, β)
The Formulae-as-Classes Interpretation of Constructive Set Theory
31
for a final E℘ -recursive partial function h4 .
Going in the other direction and starting from
e0 η ∈ Σ(α, β),
we can reverse the above implication so that we get
h0 (α, β, η, e0 ) η is in the disjoint union of S(α, β)
for some E℘ -recursive partial function h0 . As a consequence of the above we then can compose an E℘ -recursive partial function hs such that (43) holds.
For the cartesian product suppose
e η is in the cartesian product of the family of sets S(α, β).
(45)
Then in particular
hc0 (α, β, η) η is a function with domain α
for some E℘ -recursive partial function hc0 , and by 4.22 (ii) we get
`∗1 (α, η, hc0 (α, β, η)) η = S(α, γ)
(46)
with γ = `∗1 (α, η, hc0 (α, β, η)) and γ̄ = ᾱ. Hence from (45) we obtain
hc1 (α, β, η, e) η = S(α, γ) ∧ θ(α, β, γ)
(47)
for some E℘ -recursive partial function hc1 , where θ(α, β, γ) stands for
∀x ∈ α ∃u ∃v ∈ S(α, β) OP(x, u) = v ∧ ∃y ∈ u ∃v ∈ Σ(α, γ) OP(x, u) = v .
Using 4.22 (i) twice, from (47) we obtain that for all i ∈ ᾱ,
hc2 (α, β, η, e, i) ∃y ∈ β̃(i) y = γ̃(i)
so that
hc3 (α, β, η, e, i) (^
β̃(i))(j) = γ̃(i),
(48)
where
j = hc2 (α, β, η, e, i) 0
and hc2 , hc3 are appropiate E℘ -recursive partial functions. Now let z be the function with domain ᾱ and
z(i) = hc2 (α, β, η, e, i) 0
so that z ∈ D. Then (48) yields
hc3 (α, β, η, e, i) (^
β̃(i))(z(i)) = γ̃(i),
(49)
whence, owing to the definition of g(z),
hc4 (α, β, η, e) Σ(α, g(z)) = Σ(α, γ)
(50)
32
The Formulae-as-Classes Interpretation of Constructive Set Theory
for an E℘ -recursive partial function hc4 . It follows from (46) and (50) that
hc6 (α, β, η, e) η ∈ Π(α, β)
(51)
for an E℘ -recursive partial function hc5 .
By inspecting the implications leading from (45) to (51) one realizes that they can be reversed
so that if
e0 η ∈ Π(α, β)
we can construct an E℘ -recursive partial function hc7 such that
hc7 (α, β, η, e0 ) η is in the cartesian product of the family of sets S(α, β).
(52)
As a consequence of the above we can compose an E℘ -recursive partial function hp such that
(44) holds.
Now suppose that β̃(i) is injectively presented for all i ∈ ᾱ. Let γ := Σ(α, β) and let z1 , z2 ∈
C such that e γ̃(z1 ) = γ̃(z2 ). We must show that z1 = z2 . Let xi := p0 zi and yi := p1 zi
for i = 1, 2. As e γ̃(z1 ) = γ̃(z2 ) it follows that
l0 (α, β, z1 , z2 , e) hα̃(x1 ), (^
β̃(x1 ))(y1 )iV = hα̃(x2 ), (^
β̃(x2 ))(y2 )iV
so that
l1 (α, β, z1 , z2 , e) α̃(x1 ) = α̃(x2 )
l2 (α, β, z1 , z2 , e) (^
β̃(x1 ))(y1 ) = (^
β̃(x2 ))(y2 )
for some E℘ -recursive partial functions l0 , l1 , l2 . As α is injectively presented x1 = x2 and
hence
l2 (α, β, z1 , z2 , e) (^
β̃(x1 ))(y1 ) = (^
β̃(x1 ))(y2 ).
Whence as β̃(x1 ) is injectively presented, y1 = y2 , so that z1 = z2 . Thus Σ(α, β) is injectively
presented.
Next let δ := Π(α, β) and let z1 , z2 ∈ D such that e δ̃(z1 ) = δ̃(z2 ). We must show that
z1 = z2 . Note that
^
δ̃(zi ) = S(α, g(zi )) = (sup i ∈ ᾱ)hα̃(i), (g(z
2 ))(i)iV .
Hence, as
l3 (α, β, e, z1 , z2 ) β̃(z1 ) ⊆ β̃(z1 )
for some E℘ -recursive partial function l3 , we conclude that for all i ∈ ᾱ,
^
^
l4 (α, β, z1 , z2 , e, i) hα̃(i), (g(z
(53)
1 ))(i)iV = hα̃(j), (g(z2 ))(j)iV
where j := l3 (α, β, e, z1 , z2 ) (i) 0 and l4 is an E℘ -recursive partial function. But if e0 00
^
^
α̃(i) = α̃(j) for some e0 then i = j, so that if also e00 (g(z
1 ))(i) = (g(z2 ))(j) for some e
then
e00 (^
β̃(i))(z1 (i)) = (^
β̃(i))(z2 (i)),
and as β̃(i) is injectively presented, z1 (i) = z2 (i). Hence it follows from (53) that z1 (i) =
z2 (i) for all i ∈ ᾱ, so that z1 = z2 .
t
u
The Formulae-as-Classes Interpretation of Constructive Set Theory
33
Lemma 4.30. For A ∈ Y∗ and a, b ∈ A, Î(A, a, b) ∈ V(Y∗ ), where
Î(A, a, b) := sup z ∈ I(A, a, b) hω (0),
and
1. Î(A, a, b) is injectively presented,
2. there is an E℘ -recursive partial function `I such that if α ∈ V(Y∗ ) is injectively presented
and ᾱ = A then for all η ∈ V(Y∗ ),
`I (α, η, a, b) η ∈ Î(A, a, b) iff
η = hω (0) ∧ α̃(a) = α̃(b) .
Proof. As I(A, a, b) ∈ Y∗ and hω (0) ∈ V(Y∗ ) it follows that Î(A, a, b) ∈ V(Y∗ ).
(i): Let γ := Î(A, a, b) and let z1 , z2 ∈ I(A, a, b) such that γ̃(z1 ) = γ̃(z2 ). We must show that
z1 = z2 . But, as z1 , z2 ∈ I(A, a, b), z1 = 0 = z2 so that z1 = z2 .
(ii): As α is injectively presented and a, b ∈ A = ᾱ, we have |=∗ α̃(a) = α̃(b) if, and only if
a = b.
Let η ∈ V(Y∗ ). Suppose e η ∈ Î(A, a, b). Then (e)1 η = hω (0) and a = b, yielding
h(e)1 , ids (α̃(a))i η = hω (0) ∧ α̃(a) = α̃(b).
Conversely, if c η = hω (0) ∧ α̃(a) = α̃(b), then (c)0 η = hω (0) and a = b, so that
0 ∈ I(A, a, b) and whence h0, (c)0 i η ∈ Î(A, a, b).
In view of the above it is clear how to construct `I .
t
u
Theorem 4.31. ΠΣI−AC is validated in V(Y∗ ).
Proof. Note that the ΠΣI-generated sets are an inductively defined class I(Φ). By lemma 2.7
there is a formula θ(u, v) of set theory such that I(Φ) = {x | ∃u θ(u, x)} and, moreover,
CZF ` ∀u∀x θ(u, x) iff
5
_
θi (u, x) ,
(54)
i=1
where
θ1 (u, x) := x ∈ ω
θ2 (u, x) := x = ω
θ3 (u, x) := ∃v ∈ u ∃a θ(v, a) ∧ ∃f Fun(f ) ∧ dom(f ) = a ∧ ∀i ∈ a ∃z ∈ u θ(z, f (i))
∧ x is the disjoint union of f
θ4 (u, x) := ∃v ∈ u ∃a θ(v, a) ∧ ∃f Fun(f ) ∧ dom(f ) = a ∧ ∀i ∈ a ∃z ∈ u θ(z, f (i))
∧ x is the cartesian product of f
θ5 (u, x) := ∃v ∈ u ∃a θ(v, a) ∧ ∃w ∈ a ∃w0 ∈ a ∀z [z ∈ x iff (z = 0 ∧ w = w0 )] .
We shall show that there are E℘ -recursive partial functions `& and `# such that whenever
ξ, α ∈ V(Y∗ ) and
e θ(ξ, α)
34
The Formulae-as-Classes Interpretation of Constructive Set Theory
then
`& (e, ξ, α) ∈ V(Y∗ ),
`& (e, ξ, α) is injectively presented,
`# (e, ξ, α) `& (e, ξ, α) = α.
(55)
`& (e, ξ, α) and `# (e, ξ, α) will be defined by recursion on ξ so that, ultimately, the E℘ recursive partial functions `# and `& will have to be defined by invoking the recursion theorem 4.7. Note that (54) and Theorem 4.17 entail that there are E℘ -recursive partial functions
g1 , g2 , g3 , g4 , g5 such that e θ(ξ, α) implies
gi (e, ξ, α) 0 = 0 ∨ gi (e, ξ, α) 0 = 1
(56)
gi (e, ξ, α) 0 = 0 ⇒ gi (e, ξ, α) 1 θi (ξ, α)
for i = 1, 2, 3, 4, 5.
Now assume that α, ξ ∈ V(Y∗ ), e θ(ξ, α) and that for all β ∈ V(Y∗ ), j ∈ ξ¯ and c, if
˜
˜
˜
c θ(ξ(j),
β), then `# (e0 , ξ(j),
β) ↓ and `& (e0 , ξ(j),
β) ↓ and that the properties pertaining
to (55) obtain. Owing to (56) we can proceed by division into cases. Let
gij (e, ξ, α) := gi (e, ξ, α) j .
Case 1: Let g10 (e, ξ, α) = 0. Then g11 (e, ξ, α) α ∈ ω, so that by the proof of 4.17,
Υ1 (g11 (e, ξ, α), α) 1 α = hω (n)
for some E℘ -recursive partial function Υ1 , where
n := Υ1 (g11 (e, ξ, α), α) 0 ∈ ω.
Hence we let `& (e, ξ, α) := hω (n) and
`# (e, ξ, α) := Υ1 (g11 (e, ξ, α), α) 1 .
It follows from Lemma 4.24 that `& (e, ξ, α) is injectively presented.
Case 2: Suppose g10 (e, ξ, α) = 1 and g20 (e, ξ, α) = 0. Then g21 (e, ξ, α) α ∈ ω, so that by
the proof of 4.17,
Υ2 (g21 (e, ξ, α), α) α = ω ∗
for some E℘ -recursive partial function Υ2 . This time let
`& (e, ξ, α) := ω ∗ ,
`# (e, ξ, α) := Υ2 g21 (e, ξ, α), α .
It follows from Lemma 4.24 that `& (e, ξ, α) is injectively presented.
Case 3: Suppose g10 (e, ξ, α) = g20 (e, ξ, α) = 1 and g30 (e, ξ, α) = 0. Then g31 (e, ξ, α) The Formulae-as-Classes Interpretation of Constructive Set Theory
35
θ3 (ξ, α). Unpacking the definition of g31 (e, ξ, α) θ3 (ξ, α), we can E℘ -computably extract
from e, ξ, α sets γ, β ∈ V(Y∗ ) and j ∈ ξ¯ such that
˜
γ) ∧ Fun(β) ∧ dom(β) = γ
Υ∗3 (e, ξ, α) θ(ξ(j),
∧ ∀y ∈ γ ∃z ∈ ξ ∃w ∈ β ∃w0 [OP(y, w0 ) = w ∧ θ(z, w)]
∧ α is the disjoint union of β.
for some E℘ -recursive partial function Υ∗3 . From the first conjunct above we get
˜
Υ∗3 (e, ξ, α) 0 θ(ξ(j),
γ).
(57)
(58)
From (57) we also get
Υ03 (e, ξ, α) 0 ∀w ∈ β ∃z ∈ ξ θ(z, w),
(59)
and hence we can E℘ -computably extract a function l0 with domain β̄ from e, ξ, α such that
for all i ∈ β̄ we have l0 (i) ∈ ξ¯ and
˜ 0 (i)), β̃(i))
Υ13 (e, ξ, α, i) θ(ξ(l
(60)
for some E℘ -recursive partial function Υ13 . Put
˜
ρ := `& Υ∗3 (e, ξ, α) 0 , ξ(j),
γ
˜ 0 (i)), β̃(i)
τi := `& Υ1 (e, ξ, α, i), ξ(l
3
¯ By the inductive hypotheses we get ρ ↓, ρ ∈ V(Y∗ ), ρ is injectively presented,
for i ∈ ξ.
τi ↓, τi ∈ V(Y∗ ), τi is injectively presented and
˜
`# Υ∗3 (e, ξ, α) 0 , ξ(j),
γ ρ=γ
(61)
#
1
0
˜ (i)), β̃(i) τi = β̃(i)
` Υ3 (e, ξ, α), ξ(l
(62)
for all i ∈ β̄. Put
τ := (sup i ∈ β̄)τi
˜
γ
e := `# Υ∗3 (e, ξ, α) 0 , ξ(j),
˜ 0 (x)), β̃(x) .
f # := ab β̄, λx.`# Υ1 (e, ξ, α), ξ(l
#
3
Then τ ∈ V(Y∗ ). From (57), (61) and (62) we can conclude that
#
#
Υ∗∗
3 (e , f , e, ξ, α) Fun(τ ) ∧ dom(τ ) = ρ
∧ α is the disjoint union of τ
(63)
for some E℘ -recursive partial function Υ∗∗
3 . Thus, by 4.22 we can E℘ -computably extract
∗
∗∗ #
#
δ ∈ V(Y ) from Υ3 (e , f , e, ξ, α), ρ, τ such that δ̄ = ρ̄ and
∗∗ #
#
Υ++
Υ
(e
,
f
,
e,
ξ,
α),
ρ,
τ
τ = S(ρ, δ)
(64)
3
3
for some E℘ -recursive partial function Υ++
3 . Thence, from (63), (64) and Proposition 4.29
we get
#
#
Υ$3 Υ∗∗
Σ(ρ, δ) = α
(65)
3 (e , f , e, ξ, α), ρ, τ
36
The Formulae-as-Classes Interpretation of Constructive Set Theory
and also that Σ(ρ, δ) is injectively presented. As a result, we define
`& (e, ξ, α) := Σ(ρ, δ)
#
#
`# (e, ξ, α) := Υ$3 Υ∗∗
3 (e , f , e, ξ, α), ρ, τ .
Case 4: Suppose g10 (e, ξ, α) = g20 (e, ξ, α) = g30 (e, ξ, α) = 1 and g40 (e, ξ, α) = 0. Then
g41 (e, ξ, α) θ4 (ξ, α). Here we proceed in the same vein as in the previous case, crucially
utilizing 4.22 (ii) and 4.29.
Case 5: Suppose g10 (e, ξ, α) = g20 (e, ξ, α) = g30 (e, ξ, α) = g40 (e, ξ, α) = 1 and g50 (e, ξ, α)
= 0. Then g51 (e, ξ, α) θ5 (ξ, α). Unpacking the definition of g51 (e, ξ, α) θ5 (ξ, α), we can
E℘ -computably extract from e, ξ, α sets β ∈ V(Y∗ ), j ∈ ξ¯ and i0 , i1 ∈ β̄ such that
˜
Υ∗5 (e, ξ, α) θ(ξ(j),
β) ∧ ∀z z ∈ α iff z = hω (0) ∧ β̃(i0 ) = β̃(i1 ) .
(66)
for some E℘ -recursive partial function Υ∗5 . From the first conjunct above we get
˜
β).
Υ∗5 (e, ξ, α) 0 θ(ξ(j),
(67)
Put
˜
β .
ρ := `& Υ∗5 (e, ξ, α) 0 , ξ(j),
By the inductive hypotheses we get that ρ ↓, ρ ∈ V(Y∗ ), and that ρ is injectively presented
and
˜
`# Υ∗5 (e, ξ, α) 0 , ξ(j),
γ ρ = β.
(68)
Put
˜
e# := `# Υ∗5 (e, ξ, α) 0 , ξ(j),
β .
From (66) and (68) we can extract j0 , j1 ∈ ρ̄ such that
#
Υ∗∗
5 (e , e, ξ, α) ∀z z ∈ α iff z = hω (0) ∧ ρ̃(j0 ) = ρ̃(j1 ) .
(69)
for some E℘ -recursive partial function Υ∗∗
5 . Thus, by 4.30 we get that
∗∗ #
Υ++
Υ
(e
,
e,
ξ,
α),
ρ
α = Î(ρ̄, j0 , j1 )
5
5
(70)
for some E℘ -recursive partial function Υ++
5 , and, moreover, that Î(ρ̄, j0 , j1 ) is injectively
presented. As a result, we define
`& (e, ξ, α) := Î(ρ̄, j0 , j1 )
#
`# (e, ξ, α) := Υ++
Υ∗∗
5
3 (e , e, ξ, α), ρ .
Finally, by combining (55) and Theorem 4.23, we can design an E℘ -recursive partial function
Ψ such that if
c ∃z θ(z, α)
then
Ψ(α, c) α is a base,
showing that ΠΣI−AC is validated in V(Y∗ ).
t
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The Formulae-as-Classes Interpretation of Constructive Set Theory
37
Remark 4.32. Aczel uses the notion of a strong base to show that ΠΣI−AC is validated
in Martin-Löf type theory. α ∈ V(Y∗ ) is said to be a strong base if |=∗ α = β for some
injectively presented β ∈ V(Y∗ ). The proof of 4.31 does not use this notion. The main
reason for its avoidance is that I could not see how to internally characterize the notion of a
strong base in V(Y∗ ), that is, via a formula ψ(u) such that |=∗ ψ(α) would hold iff α were
a strong base. As Aczel also invokes the axiom of choice in type theory in connection with
propositions involving the notion of a strong base there was a danger that dealing with it
would require some forms of the axiom of choice to hold in the background theory.
4.5
The formulae-as-classes interpretation for CZF + REA
As the reader may expect, the formulae-as-classes interpretation given for CZF above can
be extended to CZF + REA also. The first step is to add the following condition to the
definition of E℘ -recursive functions, giving rise to the E℘w -recursive functions:
{w̄}(x, g) = Wz∈x g(z) if g is a (set-)function with dom(g) = x,
where w̄ is a “fresh” natural number.
.
∗
)-assignment M, a class k θ k M as in
One then defines for every formula θ ∈ L∈ and V(Yw
Definition 4.10, where, however, the definition of the product
Y
f
Bx := {a | ∀x∈A({a}(x) ∈ Bx )}
(71)
x∈A
is to be understood in the sense of E℘w -recursive functions. Correspondingly, we obtain the
following result.
Theorem 4.33. Let θ(u1 , . . . , ur ) be a formula of L∈ all of whose free variables are among
u1 , . . . , ur . If
CZF + REA + RDC + ΠΣW−AC ` θ(u1 , . . . , ur ),
then one can effectively construct an index of a E℘w -recursive partial function g such that
CZFExp + REA ` ∀α1 , . . . , αr ∈ V(Y∗ ) g(α1 , . . . , αr ) ∈ k θ(α1 , . . . , αr ) k.
Proof. The proof builds on the proof of Theorem 4.13. First we have to deal with REA.
Since CZF proves that every set is a subset of a transitive set it suffices to show that there is
a term trea such that if α ∈ V(Y∗ ) and e Tran(α) then
trea (α, e) ∃x α ⊆ x ∧ x is regular .
So suppose e Tran(α). Define a function fα with domain ᾱ by fα (i) := α̃(i) and put
A := Wi∈ᾱ fα (i).
∗
Note that A is E℘w -computable from α and that A ∈ Yw
. By the recursion theorem we can
construct an index c such that
c(x, y, z) ' p fa(x, y), ab fa(x, y), λu.c(x, y, fa(z, u)) .
(72)
38
The Formulae-as-Classes Interpretation of Constructive Set Theory
Note that the elements of A are of the form sup(i, g) where i ∈ ᾱ and g : fα (i) → A. By
∈-induction on the elements sup(i, g) of A one shows that
(73)
c(fα , i, g) ' sup fα (i), ab fα (i), λu.c(fα (i), i, g(u)) ,
∗
so that c(fα , i, g) ↓ and, moreover, c(fα , i, g) ∈ V(Yw
). Now put
hα := ab A, λx.c(fα , p0 x, p1 x)
and let
β := sup(A, hα ).
∗
Then β ∈ V(Yw
) and by the above, β is E℘w -computable from α.
In what follows, we shall frequently suppress the witnessing information and write |=∗ θ(~
α)
rather than e θ(~
α) for a specific e. In all cases the witnessing information could be supplied
and is indeed E℘w -computable from the exhibited parameters. Having progressed thus far in
the paper, we consider it unnecessary and too tedious to make witnesses explicit all of the
time.
First we prove that
|=∗ β is transitive.
(74)
So assume |=∗ γ ∈ β. Then |=∗ γ = hα (j) for some j ∈ A. But j = sup(i, g) for some i ∈ ᾱ
and g : fα (i) → A. Hence, letting δ := sup(fα (i), g 0 ) and g 0 be the function with domain
fα (i) satisfying g 0 (u) = hα (g(u)), we get |=∗ δ ⊆ β and |=∗ γ = δ, whence |=∗ γ ⊆ β.
To demonstrate that β is regular we further need to show that if |=∗ γ ∈ β and |=∗ ∀u ∈ γ ∃v ∈
β θ(u, v), then
|=∗ ∃x ∈ β θ0 (x, γ),
(75)
where θ0 (x, γ) stands for
∀u ∈ γ ∃v ∈ x θ(u, v) ∧ ∀v ∈ x ∃u ∈ γ θ(u, v).
So assume |=∗ γ ∈ β and |=∗ ∀u ∈ γ ∃v ∈ β θ(u, v). Then |=∗ γ = hα (j) for some j ∈ A and
` ∀u ∈ hα (j) ∃v ∈ β θ(u, v)
(76)
for a function with domain hα (j). Then j = sup(i, g) for some i ∈ ᾱ and g : fα (i) → A. Also
hα (j) = fα (i). From (76) we conclude that
∀ν ∈ fα (i) p0 (`(ν)) ∈ A ∧ p1 (`(ν)) θ h]
α (j)(ν), hα (p0 (`(ν))) .
Now let the function σ with domain fα (i) be defined by σ(ν) = p0 (`(ν)). So sup(i, σ) ∈ A
and if
δ := hα (sup(i, σ))
then |=∗ δ ∈ β. Moreover, we have
∀ν ∈ δ̄ p1 (`(ν)) θ h]
α (j)(ν), hα (σ(ν)) .
The Formulae-as-Classes Interpretation of Constructive Set Theory
39
From the latter we obtain |=∗ θ0 (δ, hα (j)) and hence |=∗ θ0 (δ, γ), so that |=∗ ∃x ∈ β θ0 (x, γ).
It remains to prove that
|=∗ α ⊆ β.
(77)
By induction on η ∈ V(Y∗ ) we shall show that |=∗ η ∈ α ⊃ η ∈ β. So as induction hypothesis
we assume that
|=∗ ∀u ∈ η u ∈ α ⊃ u ∈ β .
Now if η ∈ α then |=∗ η = α̃(i) for some i ∈ ᾱ. Hence |=∗ ∀u ∈ α̃(i) u ∈ β. The latter implies
that we have a function ` with domain fα (i) = α̃(i) such that ` : fα (i) → A and for all
j ∈ fα (i)
^
|=∗ (α̃(i))(j)
= hα (`(j)).
Then sup(i, `) ∈ A. Let δ = hα (sup(i, `)). Then |=∗ δ ∈ β and for all j ∈ fα (i) = δ̄,
^
|=∗ (α̃(i))(j)
= δ̃(j).
The latter entails |=∗ α̃(i) = δ, whence |=∗ η = δ, so that |=∗ η ∈ β. This completes the
∗
proof that V(Yw
) validates REA.
Finally we turn to ΠΣWI−AC. The proof of the validity of ΠΣI−AC was carried out in
Theorem 4.31. Here we shall only discuss the additional constructions that are needed to ac∗
∗
)
) and Π(α, β) ∈ V(Yw
commodate the W-operation. In Definition 4.27, Σ(α, β) ∈ V(Yw
∗
) such that ᾱ = β̄, and 4.30 they are related to the disjoint union
are defined for α, β ∈ V(Yw
and cartesian product operations when α is injectively presented and β̃(i) is injectively presented for each i ∈ ᾱ. Moreover, Σ(α, β) and Π(α, β) are shown to be injectively presented
under these conditions. Aczel has carried a similar construction for the operation W.
∗
∗
), where
) such that ᾱ = β̄. First note that A ∈ V(Yw
Let α, β ∈ V(Yw
A = Wi∈ᾱ β̃(i).
∗
Define hα,β : A → V(Yw
) by transfinite recursion on A so that for i ∈ ᾱ and f : β̃(i) → A,
hα,β sup(i, f ) := hα̃(i), S β̃(i), (sup j ∈ β̃(i))hα,β (f (j)) iV .
Finally let
W(α, β) := sup(A, hα,β ).
A is obviously E℘w -computable from α, β. Similarly as in (72) one can show with the aid
of the recursion theorem that α, β 7→ hα,β is E℘w -computable form α, β so that W(α, β) is
E℘w -computable from α, β also.
Under the above conditions, if α is injectively presented and β̃(i) is injectively presented for
all i ∈ ᾱ, it can be shown that W(α, β) is injectively presented, too. The proof consists in
adapting the proof of [3] A2.2 to the present context.
Furthermore, by using W(α, β) and the techniques of the proof of Theorem 4.31 one can
∗
show that V(Yw
) validates ΠΣWI−AC. Due to space limitations a proof with all the
details cannot be included in this paper. However, as I trust that the reader has by now
amassed enough experience from earlier proofs in this paper as to how to do things in an
E℘w -computable way, he or she can glean the details from the proof of [3] A2.2.
t
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40
The Formulae-as-Classes Interpretation of Constructive Set Theory
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