Logic Colloquium '77 A. Macintyre, L. Pacholski, J. P a r k led8.l Q iYorth-Holland Publishing Canpany, 1978

THE TYPE THEORETIC INTERPRETATION OF CONSTRUCTIVE SET THEORY"

Peter Aczel Manchester University

England

By adding to Martin-LSf's intuitionistic theory of types a 'type of sets' we give a constructive interpretation of constructive set theory. constructive version of the classical conception of the cumulative hierarchy of sets.

This interpretation is a

INTRODUCTION Intuitionistic mathematics can be structured into two levels. The first

level arises directly out of Brouwer's criticism of certain methods and notions of classical mathematics. the law of excluded middle was rejected and instead the meaning of mathematical statements was to be based on the notion of 'proof'. level of intuitionism was a theory of meaning quite different from the classical one, it was nevertheless the case that the body of mathematics that could be developed within this level remained a part of classical mathematics. Brouwer felt that Mathematical analysis could not be developed adequately on this basis he was led to formulate his own conception of the continuum. conception involved the mathematical treatment of incompletely specified objects such as free choice sequences. first level but also includes these radical ideas that turn out to be incompatible with classical mathematics. years to make these ideas more transparent they have remained rather obscure to w s t mathematicians and the mathematics based on them has had a very limited following.

In particular the notion of 'truth' that gives rise to

While implicit in this first

Because

This

The second level of intuitionism builds on the

Although considerable effort has been made over the

Since Bishop's book 131 appeared, it has become clear that, in spite of Brouwer's views, constructive analysis can be developed perfectly adequately while staying within the first level of intuitionism. This fact has led to a renewed

*The ideas for this paper were germinating during a visit to Caltech, supported by NSF grant Utrecht University. Oxford Logic Collguiun.

MPS 75-07562 , and came to fruition while visiting The results were first announced in an abstract for the 1976

56 P. ACZEL

interest in this part of intuitionism.

for Bishop's constructive mathematics, of the generally accepted formal system ZF for classical mathematics. Several approaches have been tried and there has been

some controversy over their relative merits.

'Constructive Set Theory' (see [71 and 1131 and 'Intuitionistic Type theory' (see

[ll]). are not necessarily in conflict with each other.

suppresses all explicit constructive notions in order to be as familiar as possible to the classical mathematician.

give a rigorous foundation for the primitive notions of constructive mathematics.

The appeal of Bishop's book is that it wastes little time on the analysis of

The desire has been to find an analogue,

Two such approaches are

In this paper we take the view that these, and perhaps other approaches, Constructive set theory

On the other hand type theory aims to

constructive concepts but instead developes analysis in a language mostly familiar

to the classical mathematician.

further. axiom system that is a subsystem of ZF using intuitionistic logic, and including the axiom of extensionality.

Bishop's notions has been lost and there arises the crucial question:

constructive meaning of the notion of set used in constructive set theory?

to answer that question in this paper.

which the primitive notions of constructive mathematics are directly displayed, together with a natural interpretation of constructive set theory in that frame-

work. We shall give such a framework based on the intuitionistic type theory of [ H I . We could have taken instead a system of 'Explicit Mathematics' (see 151 and

[ 2 1 ) .

whereas type theory is a logic. free theory of constructions within which the

logical notions can be defined.

more fundamental.

In constructive set theory this is taken a stage

As in ZF there are just the notions of set and set membership with an

In such a system the direct constructive content of What is the

We aim

What is needed is a rigorous framework in

But systems of Explicit mathematics leave the logical notions unanalysed,

For this reason we consider type theory to be

The axiom system CZF (Constructive ZF) is set out in 51 and some elementary

properties are given in 02.

considered by Myhill and Friedman in their papers. theoretic notions of [ll] and introduce the type of sets. CZF is set up in 04 and its correctness is proved in 05 and 46.

consider a new principle of choice for constructive set theory.

The system is closely related to the systems

In 13 we sumarise the type The interpretation of

Finally in 17 we

Formally, the type of sets was first introduced by Leversha in 1101 where it

was used to give a complicated type theoretic interpretation of Myhill's original

formulation of constructive set theory in [13].

at that time that the type itself gave a constructive notion of set. was only used as an appropriate constructive notion of ordinal to use in indexing the stages of Leversha's construction.

Unfortunately it was not realised Instead it

THE TYPE THEORETIC INTERPRETATION OF CONSTRUCTIVE SET THEORY 57

1. THE AXIOM SYSTEM CZF

We formulate CZF in a first order languagex having the logical primitives

A . v ,-, +, vx. 3x, the restricted quantifiers (blxcy), (3xcy) and the binary relation symbols E and = . We assume a standard axiomatisation of intuitionistic logic. The remaining axioms of CZF are divided into two groups.

Structural axioms Defining schemes for the restricted quantifiers.

(bkEY)$(X) - vX(XSy -t $(X))

(3XEY)P1(3 - 3 x(xcy 15 O(X))

x = y - Vz(zcx c-) zey)

Equality axioms.

x = y h ycz * xc-z

Set Induction scheme.

V Y t (Vxcy) 4;x) - o(y) 1 -* VX$(X)

Set existence axioms Pairing. ~Z(XEZ A yez)

Restricted Separation. For restricted O(x) Union. 3Z(VYEX) (VUUEY) (UEZ)

3z[(&4 (yex A $4~)) (VYEX) ($(y) + ycz) 1 If $(x,y) is a formula let 4"a.b) denote

M E a ) (3ycb) btx,y) A NyEb) ( 3 x 4 +(x,y).

Strong Collection

(OIxea)3yt&y) --+ 3bQ'(a,b)

3c\dUr(Vxea) (lyebb) O(x.y) - (jdcc)4'(a,d) 1 Subset Collection

where u may occur free in $(x,y). Infinity. 3zNat(z)

where Nat(z) is the conjunction of Mxcz) (Zero(x)V(~yoz)Succ(y,x)), (3x~z)Zero(x)

and &'$Ez)(~xEz)SUCC(~,X). Here Zero(x) is (vyex)I and Succ(y,x) is (&~)(zEx) - ycx .. (&EX) (zcy v z = y) , Remarks. interpretation as smoth as possible. m r e standard way.

existence of a set w such that

Our formulation has been designed to make the correctness proof for our The axioms could have been written in a

Our formlation of the axiom of infinity expresses the

ncu -> n = 0 v &w) (n = mdm)). The mathematical induction scheme can be proved in CZF using set induction.

Recall that in the usual way strong collection implies replacement.

collection is not good enough in the presence of only restricted separation.

Ordinary

The significance of subset collection will become clear in the next section.

58 P. ACZEL

2. ELEMENTARY PROPERTIES OF CZF

We give some simple results that spell out the relationship of CZF to ZF. In particular we reformulate the subset collection scheme as a single axiom and bring out its relationship to the power set axiom and Myhill's exponentiation

axiom. We show that ZF results from CZF by adding classical logic. We use standard set theoretic notation and definitions. So ordered pairs

Let R: A --C B if RGAx B such that (%EA)(~EB)<x.Y>cR, <xly>, Cartesian products A x B and the notion of a function f: A +B etc... are

all defined as usual.

and let R: A >--.< B if in addition &EB)(~xEA)<x,Y>ER.

is A - full if R: A --.c B implies that R: A % D for some DEC.

A set C of subsets of B

Results 2.1-2.6 will be proved informally inside the system CZF- of CZF

without subset collection.

2.1. PROPOSITION. The subset collection scheme is equivalent to the axiom: bAbb3C(C is an A - full set of subsets of B)

PROOF. subset collection where 6(x,y) expresses <X,Y>EU. For the converse, let C be an

A - full set of subsets of B and suppose that (&A) ~ Y E B ) 9(x.y).

+'(A,D) for some DEC. (bk~A)jzJl(x,z) so that by strong collection there is a set R such that (VXEA) (~ZER)$(X.Z) A ~zER)((INW(~EA)J~(X,~). As C is an A - full set of subsets of B we can find DEC such that R: A - D. It

follows that @'(A,D). 2.2. PROPOSITION. implies the exponentiation axiom:

The.above axiom is an immediate consequence of the special instance of -

We show that

Let Jl(x.z) denote (jy~B)(+(~,y) - <x,y> - z). Then

Hence R: A 4 B andvx~(<x,y>E~(x,y)).

The power set axiom implies subset collection, which in turn

VAb$%(C is the set of functions from A to B).

PROOF. If C is the powerset of B then it is trivially an A - full set of subsets of B. For the second one, let C be an A - full set of subsets of A x B. If f: A + B then f': A + A x B where f'(x) - <x.f(x)> for XEA.

Then f': A -<A X B and hence there is a set DEC such that f': A H D . As f' is

a function, D = (f'(x)lx~A) = f, so that fcC. So. using restricted separation we

can define the set of functions from A to B as (fcClf is a function from A to B).

Remarks. be interpreted in weak subsystems of analysis while simple type theory can be interpreted in CZF with the power set axiom.

is a consequence of the exponentiation axiom (although it is easily seen to be, in

the presence of the presentation axiom of 67.) 2.3. PROPOSITION. couhined with the axiom: ((31 has a power set.

PROOF. For any set A let C = {{x~AI@~f(x))lfc%} where %I is the set of functions from A to B. This is a set by the exponentiation axiom, restricted separation and

- Hence the first implication.

The power set axiom is nuch stronger than subset collectiollras CZF can

I do not know if subset collection

It seem unlikely. The power set axiom is equivalent to the exponentiation axiom

One direction is clear. For the other, let B be the power set of (01. -

THE TYPE THEORETIC INTERPRETATION OF CONSTRUCTIVE SET THEORY 59

replacement.

f o r XEA.

2 . 4 . PROPOSITION. f o r a l l r e s t r i c t e d Q , toge ther imply the powerset axiom.

PROOF. By 2.3 i t su f f i ces t o show t h a t

{0,{611 is i t s power set. So le t xC{@1.

x = f b l . Remark.

i n the presence of r e s t r i c t e d excluded middle.

2.5. PROPOSITION.

Clear ly C is a set of subse ts of A.

Then fEAB and hence z = {xeAl@~f(x)}EC. I f z b A l e t f (x ) = i yE{d l lx~z1

Hence C i s t h e power s e t of A.

The exponentiation axiom and r e s t r i c t e d excluded middle,(Ovi 0)

has a power set. I n f a c t w e show t h a t - By hypothesis @ E x V Q ~ X . I f @EX then

I f 01. then x - @. In e i t h e r case xs{Q.{Q)1. It follows t h a t the implications of 2 . 2 . can be replaced by equivalences

Res t r ic ted excluded middle is equiva len t t o the axiom:

b 4 C C X v Ofx).

PROOF. One d i r e c t i o n i s t r i v i a l . For the o the r le t 0 be a r e s t r i c t e d fornula.

Then w e may def ine t h e s e t x = {ye{@11@1 where y i s not f r e e i n a. @EX<->+, so t h a t by the axiom Q w - 0 .

2 . 6 . PROPOSITION.

~X(&->@EX) where x is not f r e e i n 6. PROOF.

f r ee i n 9. Then @EX+>+. Conversely le t A be a set and $(y) a formula. By

assumption the re is a set x such t h a t +(~)<->QEx, f o r each YEA.

t ha t x s{ f l1 i n which case x i s uniquely determined by YEA. there i s a func t ion f defined on A such t h a t ( ~ ~ A ) ( 9 ( y ) < - > g ~ f ( y ) ) . r e s t r i c t e d separa t ion we can form t h e set { y ~ A l @ ~ f ( y ) ) .

l y ~ A I + ( y ) ) so t h a t w e have proved the f u l l separa t ion scheme.

2 . 7 . PROPOSITION. The following systems have t h e same theorems.

Then

The f u l l separa t ion scheme is equivalent t o the scheme:

Given f u l l separa t ion and a fornula 4 le t x = {ys{a l l+} where y i s not

We may assume By s t rong co l l ec t ion

By But t h i s set is

( i ) CZF with c l a s s i c a l logic.

( i i ) ( i i i ) ZF.

CZF with r e s t r i c t e d excluded middle and the f u l l separa t ion scheme.

PROOF. Clear ly ( i i i ) includes both ( i ) and ( i i ) . By 2 . 4 t he powerset axiom is a theorem of both ( i ) and ( i i ) .

f o r any formula 0 w e have O n $ .

e i t h e r case .$<->@EX and we ge t f u l l separa t ion by 2 . 6 .

classical log ic holds i n ( i i ) .

such t h a t +->@EX.

Remark.

t he r e s t r i c t e d excluded middle and le t IZF be CZF-+ POW + SEP.

following diagram of systems, increas ing i n s t r eng th from l e f c t o r i g h t .

Fu l l separa t ion i s a l s o a theorem of ( i ) because

So choose x = {@I i f 41 and x = 0 i f -14. In F ina l ly we observe t h a t

For i f Q is a formula then by 2.6 t.iere i s an x

By r e s t r i c t e d excluded middle ~ ~ E x v I I ) ~ so t h a t $ W T $ .

Let POW denote the powerset axiom, SEP the f u l l separa t ion scheme, REM Gi, have the

60 P. ACZEL

CZF + REM

IZF + REM- ZF

CZF + SEP

IZF (Intuitionistic ZF) arises naturally as the system that is modelled when the classical topological interpretation of intuitionistic logic is extended to set theory (see [81).' IZF would seem to be the impredicative version of CZF. Friedman (in [61) has shown that ZF has a finitist reduction to IZF. (An elegant alternative proof of this appears in 181 where ZF is given a Boolean valued model in IZF) CZF + REM is a natural formalisation of the Liberal intuitionist conception. over sets, but only intuitionistic logic holds when quantifying over the universe (See [14] 4nd also the recent [l?]. 3. THE TYPE THEORETIC FRAM?IWORK

This is the viewpoint that classical logic holds when quantifying

We shall assume that the reader is familiar with the informal notions of For w s t of the purposes of intuitionistic type theory as developed in ill].

this paper it is sufficient that he has read 81 of [ll]. Here we shall only summarise these notions. In type theory the basic forms

of statement are 'A is a type' and 'aEA' ('a is an object of type A'). addition there are definitional equalities between types and between objects of each type. We shall just use ' = I for definitional equality rather than the notation used in [ll]. The basic forms of type are N (the type of natural

numbers), N (the k-element type for k = O,l, ... ), A + B (binary disjoint union) (IIxEA)B(x) (Cartesian product) and (ZxEA)B(x) disjoint union). As special cases of the last two forms, when B(x) is a constant type B , we have the types A + B

(function type) and A xB (binary Cartesian product).

In

k

In addition t o these basic forms of type we need the type V of small types. Thus we have rules This is obtained by a reflection on the basic forms of type.

giving NEV, NksV for k = 0,1,... and A + BEV if AEV and BEV, and (IIxEA)B(x)EV, (ZXEAA)B(X)EV if AEV and B(x)EV for XEA. AEV and BEV. We shall use the notation of A-abstraction. Thus if ... x... is an

expression such that ... a..dB(a) for aEA then by the introduction rule for (IIxEA)B(x) we introduce a function f = Xx...x... ~(lIxsA)B(x) such that f(a) = ... a... for asA.

It follows that A-CBEV and A X E E V if

The key insight in the type theoretic approach to constructive mathematics is the identification of a mathematical proposition with the type of its proofs.

THE TYPE THEORETIC INTERPRETATION OF CONSTRUCTIVE SET THEORY 61

The log ica l cons tan ts are then iden t i f i ed with appropr ia te notions of type theory according t o the following tab le :

Form of propos i t ion I

Form of type

NO A + B A + B

A - B A x B

With these iden t i f i ca t ions , t h e na tu ra l deduction schemes f o r i n t u i t i o n i s t i c pred ica te ca lcu lus reduce t o t h e in t roduct ion and e l imina t ion r u l e s of type theory. Moreover t h i s can be extended t o Heyting a r i thmet ic because the mathematical induction scheme reduces t o the e l imina t ion r u l e f o r N, giving de f in i t i on by pr imi t ive recursion.

I n order t o give our i n t e r p r e t a t i o n of CZF w e need t o introduce a fu r the r type; t h e type U of s e t s . A s usual U w i l l be spec i f ied by in t roduct ion and e l imina t ion ru l e s . The in t roduct ion r u l e f o r U is a type theo re t i c reforrmlatjon of t h e c l a s s i c a l conception of t h e cumulative h ie rarchy of sets. family of previously introduced sets determines a new set, i f it is indexed by a small type. Below we give t h e r u l e s f o r U i n the s t y l e of 8 1 of [ l l ] .

An indexed

U i s a type, namely t h e type of e. I f f ( a ) is a s e t f o r each acA, where A is a small type, then {f (x) IxeA] is a set.

I f B(a) i s a type f o r each set a, and b(A.Xxf(x), Xxg(x)) E B({f (x ) Ix~Al ) whenever A is a small type, f ( a ) i s a set and g(a)EB(f (a)) f o r acA, then w e may define a function F such t h a t F(a)EB(a) f o r asU , by t h e set recurs ion scheme.

F({f(x) IXEA~) = b(A, Axf(x). XxF(f(x))). I f B(a) represents a proposit ion f o r each set a then F i s t h e proof of t he

universal p ropos i t ion ( ~ a s u ) B ( a ) which we ge t by applying set induction t o the

proof b of

(VA EV)(vfoA +U) [(vxEA)B(f(x)) + B ( ~ f ( x ) ( x c A l ) ] 4. THE INTERPRETATION

Before going fu r the r it may help the reader ' s i n t u i t i o n i f we show how some fami l ia r sets a r e b u i l t up using the in t roduct ion r u l e f o r U . le t {q, ..., akl be t h e set I f ( x ) l x ~ N ~ ) where f (1 ) = al, ..., f ( k ) = %. i f k - 0 w e ob ta in t h e empsy set $.

f i n i t e sets. define au 8 t o be t h e set {h(z) IZEA + B l where h ( i ( a ) ) = f ( a ) f o r acA and

h ( j (b ) ) - g(b) f o r bcB. B e t {g(a,y) lyoB(a)] f o r acA then we can de f ine u a t o b e Ih(z) IZE(ZXEA)B(X)] where h ( ( a ,b ) ) - g(a,b) f o r aoA and bcB(a).

Given s e t s a , , ...\ I n pa r t i cu la r ,

In t h i s way we ge t a l l t h e h e r e d i t a r i l y

I f a is a set { f ( x ) IxEA] and 8 i s a set {g(y) ~YEB} then w e can

More genera l ly , i f a i s a set {f(x)lxoAl where f ( a ) i s a

A s an example of an i n f i n i t e s e t we

62 P. ACZEL

can define = {A(n)lnENj where A ( 0 ) = 6 and A(s(n)) = A(n)y{A(n)} for nEN. As a first application of set recursion we define ~ E V and %'Ea -f U for each

set a by the schemes i-1 = A and be used to justify m r e elaborate forms of recursion. type 1 1 a = 6 (I, for sets a,6, that represents the proposition that a is extensionally equal to B .

* Xxf(x). Set recursion can We shall define a small

It is introduced by the following double recursion: II{ f(x) IxEAl = {g(y) ly~BjI1 =

(BxEA) (EyEB)llf(x) = g(y)llx(nyEB)(CxEA)lIf(x) = g(y)ll. To justify this definition we use ordinary set recursion to define a function F such that F({f(x)Ix~Aj) =

X6[(nxd\) (ZyEi)F(f(x))(i(y)) x (fly€:) (Cy~i) (ZxEA)F(f(x)) (8(y)) 1, Now let I la = 611 be F(a) (6) . for AEV and fEA -+ u . The general form of double

set recursion can also be justified in this way, as can triple set recursion etc..

is a member of the set 6, to be (CyE8)lla = 'k(y)II.

We use the same symbol to denote the set and its name in a U .

defined I / a = 611 and 1 1 as611 for sets u.6. We let 111 I( = No, l ld,+Jill = II+II + l l * l l . 116 - Jill = I10 I1 X I I J i li. Il$vJiil = IIoll+ IIJ, 1 1 9

I I (VxEa)+(x) II = (nx~a) II g(g(x)) 1 1 , II&Ea)+(x) I I = (EXE~) II 0(8(x))Il, II Vx$(x) I I =

(flaEu) 11 11 and 11 3x$(x) 11 = (CaEU) I lO(a)ll.

Next we define the small type llaE6(I that represents the proposition that a

Let $U be obtained from the languaged by adding a constant for each set.

We assign a type 11 $ 11 to each sentence $ of dU as follows. We have already

By a trivial induction on restricted sentences 4 we have 4.1. LEMMA. 1 1 011 is a small type for each restricted sentence $ of J U .

A formula $(x,,...,x ) of LU, in the displayed free variables, is valid if n there is an expression a(xl,. . .,xn) such that a(al, .. .,an) E 11 $(al , . . .,an) 11 for sets al,...,a n'

Our aim in this paper is to prove: 4.2. THEOREM. Every theorem of CZF is valid.

As a first step we have: 4 . 3 . LEM. $l,...,$n are valid then so is $.

this lemma it only remains to show that every non-logical axiom of CZF is valid. We do this in the next two sections. 4.4. REMARK.

to argue as follows. As our constructions will always be uniform in the parameters a l , .... a

write a as a(al,,..,an) for some expression a(xl, ..., xn) and hence $(x,, ..., x ) is valid.

If 6 is an intuitionistic logical consequence of 4,. ...,$, and

This is a consequence of the type theoretic representation of logic. With

In proving the validity of a formula $(xl, ..., xn) it is convenient Given sets a , , ..., an we find an object aE\l +(al, ..., an) 11 .

we can

n

THE TYPE THEORETIC INTERPRETATION OF CONSTRUCTIVE SET THEORY 63

5. VALIDITY OF THE STRUCTURAL AXIOMS

We s t a r t with se t induction. Let +(x) be a formula of$U with a t mst x free. each s e t a by the se t recursion: h( ( f (x) lx~A)) - Abb((f(x) Ix~Al)(Axh(f(x))(b)) for AEV and feA + U . valid for the formula +(XI. forrmlae with extra f ree variables.

Let B be the type (I)(~((y((tlxry)O(x) + +(y))ll. Define h(a)EB +ll+(a)ll for

Then XbAah(a)(b)EB +llvx+(x) 11 showing that s e t induction is By remark 4.4. the above is suff ic ient also for

The proof that the remaining s t ructural axioms are valid is not d i f f icu l t ,

but i s a l i t t l e tedious to carry through i n de ta i l . a sequence of lemnas. 5.1. LEMMA.

W e out l ine the main steps i n

The following are valid. ( i ) x - x ( i i ) x = y * y - x ( i i i ) x = y - y - z + x = z .

These require single, double and t r i p l e se t recuVsions to define ro(a)EIl a - all,

cO(a ,6)~I l a = BII + I I s = all and t 0 ( u , 6 , ~ ) ~ l l a * 611 x I ls = rll * I I a = Y I I for se t s

~ , B , Y . 5.2. LEMMA. The fol1owir.g are valid

( i ) u - v <--> &cv) (x - y) A (VyEv) (3xou) (x - y) ( i i ) u E v <--> (jyEv)(u - y)

These follow from the def ini t ions of 1 1 . = .I!. and (I. E. 11. Define a formula +(x) t o be invariant i n x i f x = y + (+(x) + +(y)) i s valid.

5.3. LEMMA. (#EY)+(x) and ~ x E Y ) + ( x ) are val id . 5.4. LEMMA. hence the equality axioms are valid. 5.5. LEMMA.

I f $(x) i s invariant i n x then the s t ructural defining axioms for

The f o m l a e (x = y) and (xey) a re invariant i n both x and y, and

Each fornula of %U i s invariant i n every variable. This i s proved by induction on the way formulae a re b u i l t up. 5.3 and 5 .5

combine t o give the va l id i ty of a l l the defining axioms for the rest r ic ted quantifiers. 6 . VALIDITY OF THE SET EXISTENCE AXIOMS

A s (x - x) is valid there is a function r o ( a ) ~ I I a - all for each se t a . Let N

a* be Ax(x,ro(a(x)) for each se t a. We now consider each s e t existence axiom i n turn. Pairing. Given sets a,6 l e t y be the s e t { g ( u ) l u ~ N ~ l where gE%+U sa t i s f ies

g(1) - a and g(2) = 6.

(y,h)e/l3z(aoz - BEZ) 1 1 . Keeping i n m i n d remark 4.4. w e see that the pairing

Union. Given the se t a l e t y be the se t {g(a) laEA1 where A is (Exca)g(x) and

gcA + U

Then a*(a)E11?(a)EaII for each set a and a€:.

Then hcllaoy - B ~ y l l where h i s (y*(l), y*(2)), and hence

axiom i s valid. - P -

s a t i s f i e s g((x,y)) = a(x) (y) for xsa and y&(x). Then he

64 P. ACZEL

11 ( V X E ~ ) &$EX) (yc$I , where h i s AyXxy*((x,y)) , and hence

(y,h)~113~(&a) (&EX) (YEZ) 1 1 . Restricted Separation. free.

4.1.) type 11 (3xEu)+(x) 1 1 and gEA + u s a t i s f i e s i ( (x ,v)) = h ( x ) for xca and

V E I ~ $($(x)) 11 . Then hlEll (#cy)(uEa - $(u)) 11 and h2cll (Kca)($(x) + XEY) 11 , where hl((x,v)) = (oL*(x),v) and h2(x)(v) = y*((x,v)) for X E ~ and vel!+(k(x)) l l . Hence (yp(hl,h2)) E I I ~ z [ ( % E z ) ( Y E x - $(Y)) - (VYEX)(+(Y) + Y E Z ) ] ~ ~ .

Let $(x) be a restr ic ted formula of du with a t m a t x Given the se t a l e t y be the s e t {g(u)lucAA) where A i s the small (by lermvl

In considering the collection schemes we shal l need the following construction. a,@ be se t s such that a = and f(a)Ell $&(a), b(a)) 1 1 for a€;. Then

K(f)Ell +'(a,@) 1 1 where K(f) i s (Ax(x,f(x)), Ax(x,f(x))). Strong Collection.

aEll (vx~a)3y+(x.y) 11 . be Axq(a(x)) where p and q denote projection functions. {b(x) IxcUJ then K(c)eII + ' ( a , B ) 11 and hence d(a)EIUz$'(a,z) 11 where d(a) i s (B,K(c)) and w e have made expl ic i t the dependence of (B,K(c)) on a .

Aad(a)Ell (vx~afy+(x,y) +3zO'(a,z) 1 1 . Subset Collection.

Az{?i(z(x))lx~al. Now l e t +,(x,y) be a formula of free, and l e t eU(a,6) denote w x ~ a ) ( d y ~ P ) $ ~ ( x , y ) . aEll e ~ ( a 9 6 ) 11 . Then K ( c ) E ~ ~ $b(a,G(b)) 11 where be; +

C E ( ~ X E ~ ) 11 +&@(x),$(b(x))) 11 i s Axq(a(x)). d(6,a) i s (b,K(c)) and we have made expl ic i t the dependence of (b,K(c)) on 6 and

a. Then (~.XaA6d(6,a)) e 113v%[eU(a,6) + (]zov)$:(a,z) 1 1 . Inf ini ty . N o + No respectively. each set a l e t S(a) be the set {h(a)(y)lyea + N 1 l where h(a)E(a + Nr) + U sa t i s f ies h(a) ( i (x) ) = h(x) for xca and h ( a ) ( j ( l ) ) = a.

Also g,(a)Ell (blxca)(xrS(a))ll where g (a). i s AxS(p)*ii(x)), and g2(a)EII ( % E S ( ~ ) ) ( U E ~ ~ u=a)II whereAg2(a\(j(l)) - j ( r o ( a ) ) . It follows tha t g(a)Ell Succ (a,S(a)) ][where g(a) i s ((S(a)*( j ( l ) ) ,gI(a)) , g2(a)). L e t w be the se t {A(n) (nENl where A(o) - @ and A(s(n)) - S(A(n)) for nEN. fsll t ~ x ~ w ~ ( Z e r o ( x ) v ~ y ~ x ) S u c c ( y , x ) ) 11 where f(o) - i(g,) and f(s(n)) = j((n,g(d(n))) for nrN. Also hEll (VyeW)(3xEw)Succ(y,x) 11 where h is Ay(s(y), g(A(y)). In

conclusion (w , ( f , ( (0, go) ,h) 1 ) E 113zNat (z) 11 .

Let $(x,y) be a formula of L u with a t most x and y free. Let

Let +(x,y) be as above. Let a be a s e t and l e t Let bE(l + U be Xxp(a(x)) and let cE(lIxoa) 11 $&x),b(x)) 11

I f 6 is the se t

So

Given se ts a,@, l e t y be the set {G(z) l z ~ a + where G i s u. containing a t mat u,x,y

Finally l e t 6 be a se t and let is Xxp(a(x)) and

Hence d(6 .a )~ l l ( 3 ~ ~ y > $ b ( a , z ) 11 where

Let f o and go be the canonical undefined functions i n No + U and

Then gOEll Zero (@)I1 where 0 is the se t {fo(x) I x E N ~ ~ . For

Then S(a)*(jO))EI(aES(a)II.

gl( L) (S u)) = CX)) m e 5 a d

Then

7. THE PRESENTATION AXIOM

An important aspect of any formalisation of constructive mathematics is the

I n the case of constructive correct treatment of possible principles of choice. set theory Myhill has argued for the correctness of the scheme of dependent choices DC. We can support h i s informal argument with the following resul t .

THE TYPE THEORETIC INTERPRETATION OF CONSTRUCTIVE SET THEORY 65

7.1. THEOREM. Each instance of DC is valid.

This is proved in a straightforward but tedious way as in 96. An analysis of the informal constructive justification for DC leads to the

following notions. can always be found.

base. simple argument in CZF shows us that if a set A has a p sentation then DC holds

for A, i.e. if R: A 4 A then there is an f: w + A such that/f(n)Rf(n+l) for all

new. This suggests consideration of the following axiom

Presentation axiom (PA).

Let us call A a if choice functions defined on the set A For example the countable axiom of choice states that w is a

A surjection of a base onto a set we call a presentation of that set. A

and aeA € f ~ ) - a and /

Every set has a presentation. This axiom has also been considered by A. Blass (see [4]) in a classical

Among other things he has shown that relative to ZF, PA is strictly context. stronger than DC.

The constructive intuition for this axiom is that a presentation is intended

to represent a particular way the set is given to us. are given to us in the form {f(x)lxeAl where A is a small type.

choice functions always exist in type theory because of the type theoretic representation of the quantifiers.

suitable representation as a set in U of each small type A. Unfortunately such a representation does not seem to be available. Nevertheless, if CZF if modified

to CZFI by allowing for a class of individuals, and the type U

type U1 by introducing an individual a E U I for each small type A and each aeA then all the theorems of CZF

sense. fact a strengthened version of PA is valid which requires the base of a

presentation to be a set of individuals. interpretation we need to use the notion of identity discussed in [ll] but not

needed here so far. In addition to what is contained in [ll] we also need to

assum that if A and B are small types then so is I(A,B), the type which represents the proposition that A and B are identical.)

to justifying PA is to take the realisability model of type theory.

induces a realisability model of CZF + PA. realised which requires that the base of a presentation is a subset of 0.

A formalised version of the realisability interpretation of type theory may be found in [l].

References

[l]

The sets in the type U As shown in 1111

So to justify PA it would suffice to have a

is modified to a

A

Each small type A has a representation as the set laAlacA} in U . I

+ PA can be shown to be valid in the appropriate

I n I

(In order to carry out the

An alternative approach This model

In fact a strengthened form of PA is

We hope to discuss these topics in more detail in a future paper.

P. ACZEL, The strength of Martin-Lof's intuitionistic type theory with one

universe.

Report No.2 of Dept. Philosophy, University of Helsinki (1977) pp.1-32.

M. BEESON, Principles of continuous choice and continuity of functions in

Proceedings of the symposium on Mathematical Logic, in Oulu 1974

[21

66 P. ACZEL

formal systems for constructive mathematics, to appear in Annals of Math. Logic.

[3] E. BISHOP, Foundations of Constructive Analysis, McGraw-Hill, (1967). 141 A. BLASS, Injectivity, Projectivity and the axiom of choice, to appear. [5] S. FEFERMAN, A language and axioms for explicit mathematics, in Algebra and

Logic, Springer Lecture Notes No. 450 (1975) pp.87-139.

H. FRIEDMAN, The consistency of set theory relative to a theory with intuitionistic logic, J.S.L. Vo1.38(1973) pp.315-319.

- , Set-theoretic foundations of constructive analysis, Annals of R. GRAYSON, A sheaf approach to models of set theory, M.Sc. thesis Oxford (1975).

H. JERVELL, Constructive Universes I, preprint (1977).

[61

[71 Maths., 105(1977) pp.1-28.

[81

191 [lo] G. LEVERSHA, Formal systems for constructive mathematics. Ph.D. thesis,

[ll] P. MARTIN-LgF, An intuitionistic theory of types: Predicative part, in Logic Manchester University (1976) .

Colloquium'73, eds. H.E. Rose and J.C. Shepherdson, North Holland (1975) pp .73-118.

[12] J. MAYBERRY, The Consistency Problem for Set Theory: An essay on the Cantorian Foundations of Mathematics, Brit. J. Phil. Sci., 28(1977)pp.1-34

(part I), pp.137-170 (part 11).

[131 J. MYHILL, Constructive Set Theory, J.S.L. 40(1975) pp.347-382. 1141 L. POZSGAY, Liberal intuitionism as a basis for set theory, in Proc. Symp.

in Pure Maths. Vol.XII1, Proc. of AMS 1967, Axiomatic Set Theory Conference,

Part I, ed. D. Scott.