Continuous Truth I Non-constructive Objects

LOGIC COLLOQUIUM '82
G. Lalli, G. Long0 and A. 'Marcia (editors)
0 Elsevier Science Publishers B. V. (North-Holland), 1984 161
CONTINUOUS TRUTH I
Non-constructive Objects
Michael P. Fourman
Department of Mathematics
Department of Pure Mathematics
Columbia University
Uni vers i ty of Sydney
New York, N.Y. 10027 N.S.W. 2006
U.S.A.
Australia
We give a general theory of the logic of potentially
infinite objects, derived from a theory of meaning for
statements concerning these objects. The paper has two
main parts which may be read independently but are
intended to complement each other. The first part is
essentially philosophical. In it, we discuss the theory
of meaning. We believe that even the staunchest realist
must view potential infinities operationally. The second
part is formal. In it, we consider the interpretation of
logic in the gros topos of sheaves over the category of
separable locales equipped with the open cover topology.
We show that general principles of continuity, local
choice and local compactness hold for these models. We
conclude with a brief discussion of the philosophical
significance of our formal results. They allow us to
reconc!le our explanation of meaning with the "equivalence
thesis , that 'snow is white is true' iff snow is white.
PROLEGOMENON
Classical mathematics is based on a platonic view of mathematical objects. The
meanings of mathematical statements are determined truth-functionally. This
Fregean explanation of meaning justifies classical logic. The deficiencies of
such a view are amply discussed by Dummett C19781.
A constructive mathematician rejects the completed infinities of classiGa1 mathematics.
For him, the objects of mathematics are essentially finite. The meaning
of quantification over infinite domains is given operationally in terms of a
theory of constructions. The resulting logic includes Heyting's predicate calculus
and other principles (e.g. choice principles).
As Dummett has stressed, one task of any philosophy of mathematics is to explain
the applicability of mathematics. The potential infinities of experience exceed
the finite objects of the strict constructivist. They demanda mathematics of infinite
objects. Naive abstraction leads to the ideal infinite objects of classical
mathematics. This idealisation has enjoyed remarkable success. However,
the meaning of statements .of classical mathematics remains problematic.
Brouwer C19811 introduced to mathematics potentially infinite objects such as freechoice
sequences. Consideration of these justified, for Brouwer, intuitionistic
logic, including various choice and continuity princip2e.s. We shall consider a
general notion of non-constructive object. For us, to present such a notion is to
give a theory of meaning for statements involving non-constructive objects.
Our non-constructive objects are not the platonic ideal objects of classical
mathematics nor the finitary objects of pure constructivism. They are potentially

162 M.P. FOURMAN
infinite objects related to the lawless sequences of Kreisel 119681and to Brouwer's
free-choice sequences (Troelstra 119771). The meanin s of statements about these
objects cannot be given in terms of truth conditions ?as for classical Platonist
mathematics) or in terms of constructions (as for naive constructivism). The
essence of these non-constructive objects lies in their infinite character. They
are not, in general, totally grasped. They are given in terms of partial data
which may later be refined. Meaning for statements about non-constructive objects
is given by saying what data justifies a given assertion.
To describe a particular notion of non-constructive object is to describe the type
of data on which it is based. We consider various such notions. Each conception
of data gives an explanation of meaning which extends the range of meaningful
statements and may be viewed as introducing new objects in that it ascribes meaning
to new forms of quantification. In fact for each type of data we introduce a
concrete representation of the non-constructive objects based on it.
Such a project is not novel: Beth 119471 introduced his models to provide just
such an explanation of meaning for choice sequences. Our models generalise Beth's.
Dumnett 119771 makes a lengthy critique of the view that the intended meanings of
of the logical constants are faithfully represented on Beth trees. Since our
models generalise Beth's they appear prima facie to be susceptible to the same
criticisms. However, Dummett's remarks on the (non)-consonance of the intended
meanings of the connectives with their interpretation in Beth trees are directed
at a different problem from the one we address. Dummett appears to have oierlooked
the possibility of separating the problem of explaining the constructive meaning
of statements concerning lawlike objects from that of explaining the intuitionistic
meaning of statements concerning choice sequences. Although we know of no
satisfactory explanation of constructive truth (in particular, we agree with
Dummett that Beth models do not give one), such a separation appears natural. It
is possible to conceive of constructive truth independently of choice sequences.
Given such a conception, Beth models provide an account of the introduction of
non-lawlike objects. It is this type of account we have generalised. By way of
example we now consider two notions of data closely related to Beth models. They
both arise from the same informal picture.
Imagine receiving from Mars an infinite sequence a of natural numbers. The
picture is of a ticker-tape which produces an indefinitely continued finite
initial segment a of the sequence CL. (We write CL E a to mean that a is an initial
segment of a.) We want to examine the consequences of treating such undetermined
sequences seriously as sequences. (Later we shall introduce more interesting
examples .)
A naive view of this example considers the stages by which information arises:
at any stage, the possible future data is represented by the collection N

Continuous Truth I 163
and
inductive
a* lt $ for all n E N
alk @
Persistence reflects the idea that knowledge, once justified, is secure. The inductive
clause comes from reflection on the infinite character of a. Given a E a,
the collection {a* I nc NI covers all possibilities for future data.
In general, if we stipulate blk $ for b E B 5 N

164 M.P. FOURMAN
forming one such process into another, generally less free: its restriction.
We give a general definition of this transformation as follows:
clk $(sib) iff b*clk $(6)
(Where 6 is a non-constructive object given by stipulating what data justifies
$(6) for various $.)
For example, alb = a' = a; Olb = a'; ylb = y'.
This change of viewpoint amount sformally to a change in our representation of data.
Formerly we considered the partially ordered set or tree N

Continuous Truth I 165
top element T). The poset IP is equipped with a notion of covering family. We
demand that this be
ref Zective
stabZe
ip} covers p
If U covers p and q 5 p then Iq A w I w E Ul covers q
monotone If V 2 U covers p then V covers p.
The notion of a covering family is crucial to our explanation of meaning for incomplete
objects. It formalises the sense in which they are potentially infinite.
We avoid the metaphor of Wright C19811 which represents such a covering family as
embodying the recognition that the state of information is capable of effective
enlargement to one of type a* because it seems to leave open to us the choice
of not performing this enlargement. The idea we have is to introduce consideration
of a particular type of incomplete object by specifying the type of data which
generates it. This specification includes a notion of covering family. Differences
over which is the proper collection of covering families do not affect the
basic conception but merely lead to different types of data.
We are not as mathematicians or logicians interested in the result of a particular
experiment. Rather, we are interested in those properties which would remain invariant
no matter what the outcome or methodology of a particular experiment. It
is not the result but the uses to which the result might be put in defining mathematical
quantities which interest us. Were the temperature scale non-linear,or
the time scale given by the unequal time of the sun, physics would be different
(it was). But mathematics and logic should be immune to such vagaries.
Our solution is similar to that we employed in giving an objective view of. open
data. The possibility we envisage is that of a change of scale which in some
sense refines our possibilities for measurement. The measurements of the old context
should be meaningful in the new one but the new one may afford finer distinctions.
To describe such a change of scale is to say which new observations q E Q
are to be viewed as refining an old observation p E IP. We write this relation
q 5 f*(p) and demand that it be
monotone
p 5 p'
muZtip Zicative q 5 f*(p) q 5 f*(p')
q 5 f*(p A p')
continuous
pi I i E I covers p r 5 f*(p)
{q I q f*(pi) some i E 11 covers r .
The motivation for the first two is clear. Continuity may be viewed as the requirement
that a previous conviction that a certain family covers, cannot be overturned.
The change of viewpoint induced by such a transformation f is given by
Mathematically, our notion of data gives a presentation of a ZocaZe. Change of
scale is represented by a continuous function between locales. Abstractly we
write such a change f: Y -f X.
We now consider a supplement to our notion of justification. Suppose, we consider,
that consideration of a particular type of data would justify 0, then C$ is justified.
This is the reflection on which our whole project is based: that we can
justify talk of incomplete objects by reflecting on hypothetica2 indefinitely
continued processes.

166 M.P. FOURMAN
We shall formulate this by saying that if f: Y + X represents the introduction of
new distinctions independent of those represented by X then
or that such an f is a cover.
Our final problem of formalisation is to characterise the introduction of
independent data. A simple example is, given Pand Q with notions of covering, to
consider P x Q the product poset with coverings
I I i E I1 covers p.q when {pi 1 i E I} covers p
I I i E I} covers p,q when qi 1 i E I} covers q.
The projection given by 5 n*(p') iff p s p' represents the introduction of
data of type Q independently of the data IP under consideration. We shall require
that all such projections be covers.
In general there are two conditions we require to view a change of scale as the
introduction of independent data. The first is obvious: no new covers should be
introduced between existing observations.
qi s f*(ri) {qi 1 i E I} covers each q 5 f*(r)
{ri I i E I} covers r
The second is subtle: no new conditional relationships should be introduced between
existing observations. We explain: if w E Q is such that
r 5 f*(p) r 5 w
r 5 f*(q)
(we view w as establishing a conditional relationship between f*(p) and f*(q)),
we demand that w 5 f*(s) for some s E IP such that
r s p r s s
r s q
(that the relationship be already established in IP). Technically, these requirements
amount to demanding that the continuous map f: Y + X be a surjection and
that it be open. The structure of data we have arrived at may be viewed as the
category of locales equipped with the topology of covering by open maps.
Before turning to a formal examination of the interpretation of logic over this
site, we sum up our intentions.
We introduce non-constructive objects by explaining the meanings of the connectives
for statements concerning them. This is not a matter of characterising a
domain of quantification. We have to explain the connectives anew in terms of
the way such an object is given to us. Moreover, it is not sufficient to merely
paraphrase the new quantifiers Wa and 3u. Such a paraphrase entails a revision
of the interpretations of -+ and v.
Our aim is to show that it is possible to derive rigorously properties of various
domains of incomplete objects by giving a formal representation of the data which
presents them as a site. We consider that the passage from an informal notion of
data to the corresponding site is simple and natural. (Indeed, for us, to have a
clear conception of a type of data is to be able to describe the corresponding
site.) Once this passage is made, the derivation of properties (choice and continuity
principles, for example) is a mathematical matter. Our hope in presenting
these modeis is Leibnitzian: to eliminate further discussion of the justification
of such principles by reducing the matter to calculation.
In our paper "Notions of Choice Sequence" C19821 we presented various notions of

Continuous Truth I 167
choice sequence, including ones satisfying the axioms of LS and CS, with the same
purpose. Unfortunately, as the literature on choice sequences makes clear, clarity
is in the eye of the beholder. Hence the present attempt at a more careful explanation
of our informal notion of non-constructive object and its formalisation.
CONTINUOUS TRUTH
We start with a concrete presentation of the interpretation of higher-order logic
in a Grothendieck topos. This material (561-3) is well-known to cognoscenti
( tautologously), but is otherwise accessible only through a study of scattered
references. We give some of these sources but make no systematic attempt at a
complete list. Many important and historically significant contributions are not
mentioned. Our account is fuller than is logically necessary for the sequel in
order to point out some connections between different approaches. It is not, however,
exhaustive.
51 Frames and Locales
1.1 Definition. A frame is a complete lattice with finite A distributive over
arbitrary V. Frame morphisms, "and-or maps", are maps preserving these operations;
T,A,V.
1.2 Example. The lattice O(X) of open subsets of a topological space is a frame.
If f: Y + X is a continuous map then the inverse image f*: O(X) -r O(Y) is an
A,V-map.
1.3 Definition. The category of Zocalos or generalised spaces is the dual of the
category of frames. We call the morphisms continuous maps f: Y + X and write
f*: U(X) + O(Y) for the corresponding inverse image maps between the frames of
opens of X and Y (as in the topological case).
Example 1.2 gives a functor 6: Top + LOC from topological spaces and continuous
maps to locales.
1.4 Discrete spaces.
Spatially P(A) corresponds to the discrete topology on A.
point space with O( ll) = P( ll) .
An example is the one-
1.5 Definition. A topological space X is sober iff Top[Il,Xl 1 LocCll,pX1.
1.6 Lemma. On the full subcategory of sober spaces 6 is full and faithful.
We tacitly restrict our attention to sober spaces and henceforth omit mention of 8.
We view locales as generalised spaces. (The relationship between LOC and Top is
better expressed in terms of the right adjoint, pt: LOC + Top, to 6.)
, Quotient maps of frames induce congruences: if f*: O(X) + U(Y) is
p a q iff f*p = f*q. Each congruence class !PI has a canonical
representative jp = Vtq I p a q}. The maps j: O(X) + O(X) arising in this way are
monotone
idempotent
muttip licative
P 2 jp
j2 = j
j(p A 9) = jp A jq .
Such maps are called nuctei. The quotient may be identified as the image (or
fixed points) of j. The quotients of O(X) are isomorphic (as posets) with the
nuclei on U(X). Spatially we view these quotients as giving rise to subspaces
of x.

168 M.P. FOURMAN
1.8 Surjections. Dually, we view injective inverse image maps as giving rise
to surjections of spaces.
1.9 Right adjoints.
given by
The map q
p A
where p + r = v!q I p A q 2 r}.
morphisms are different).
Each frame map f* has a right adjoint f,, direct image,
=
V{q 1 f*q 5 pl .
f*p
q has a right adjoint r + p -f r defined by
p ~ q s r iff qsp+r
Thus frames are complete Heyting algebras (but the
1.10 Definition. A map of spaces f: X -f Y is o en if the inverse image map
f*: O(y) + O(X) has a left adjoint 3,: U(X) + O(V7 commuting with A:
3f(f*(Y) A x) = Y A jf(X)
or, equivalently, if f* preserves +.
1.11 Proposition. The category of locales is complete and cocomplete. Open
surjections are stable (under pull-back).
The theory of locales is developed extensively by Joyal and Tierney C19821.
Johnstone 119821 uses locales systematically and has a comprehensive bibliography.
52 Sites and Sheaves
2.1 Definitions. Let 0 be a small categor . A cribte K of A E !El is a suboil
functor of the representable functor A E S' : that is, for each B E 181 a set
K(B) 5 IB,AI, stable under composition; for each f E K(B) and g: C -f B in C, the
composite f 0 g E K(C).
2.2 Lemma. The cribles of A form a frame, P(A).
If f: B + A in B we have an inverse image map f*: P(A) + P(B) given by
f*K = Ig f 0 g E K} for K E P(A). By abuse we write f: B -f A for the corresponding
continuous map. This map is open.
2.3 Definition. (Lawvere-Tierney) A Grothendieck topology j on is a
family of nuclei jA: P(A) -f P(A), natural in A: that is f*o jA= jBof*for f: B + A.
2.4 Lemma. If j is a Grothendieck topology on 0, the quotient frames n(A) have
induced inverse image maps f*: n(A) + n(B) and the corresponding map of locales,
which we write f: Bj +A', is open.
2.5 Definitions.
which is
A pretopotogy J on 0 i s a family J(A) c P(A) for each A E
reflerrive A E J(A)
multiplicative
K E J(A) L E J(A)
K n L E J(A)
B
stable
K E J(A) f: B + A
f*K E J(B)
(For example, let K E J(A) iff j K = T).
A crible K E P(A) is inductively ctosed for J iff

Continuous Truth I 169
f: B + A f*K E J(B) .
f c K
As A is closed and an intersection of closed cribles is closed, each crible
K E P(A) has a cZosure jAK. This gives a topology j on C. We say K inductiveZy
covers A iff j K = A, and write this K E J(A).
___R_
2.6 Exam les. (1) Let U(X) be a frame viewed as a poset viewed as a category.
Let K E J A iff VK = A. (Identifying B + A with its domain.) Then n(X) 1 O(X).
(2) Let be a small category of locales closed under finite limits and open
inclusions. Let K E J(A) iff K contains some family Ifi: Bi + Ali E 11 of open
maps such that V 3fi(Bi) = A.
is closed for this topology. This assignment gives an A;V map r*: U(A) + n(A),
spatially a surjection r: AJ + A. Each closed crible contains a largest open
inclusion. This assignment gives an A V map i*: n(A) + U(A), spatially we have
an inclusion i: A c+JJ. Furthermore, r o i = idA and r*i* s id,(A) so A is an
adjoint retract of AJ (Fourman C19821).
2.7 Definition. A presheaf on C is a functor X: Cop + Sets. If f: B + A E C and
a E X(A) we use the notation alf, "a restricted along f", for X(f)(a) E X(B).
Note that alflg = alfo g and alid = a. The appropriate morpkisms between presheaves
F: Y + X are natural transformations, maps FA: Y(A) + X(A) which commute
with restrictions FB(a,lf) = (FAa)lf.
2.8 Exam les. (1) The representable functor [-,A], (or by abuse, justified by
Yone a s emma, A) is a presheaf. Restrictions are by composition glf = gof.
Yoneda's lemma tells us that for any other presheaf, X, we have X(A) [A,X]. In
-7++
The crible generated by each open inclusion U 4 A
particular the embedding C + Stop is full and faithful. Each crible K E P(A) is a
subpresheaf K H A.
(2) P and n are presheaves with restrictions given by inverse images Klf = f*(K).
2.9 Definition. A presheaf X on C is a sheaf for the (pre) topology J if whenever
K E J(A), each natural transformation x: K + X has a unique extension along
K >+ A. Equivalent1 y, if K E J(A) and we have a family xf E X(B) for f: B -f A E K
such that xgf = xflg for each g: C + 8, there is a unique x E
xf = xlf for each f E K.
Grothendieck topos.
X(A) such that
The cateaorv of sheaves and natural transformations is a
There is, as yet, no staisfactory introductory text on topos theory.
references are SGA4, Wraith C19751, Johnstone C19771, Freyd '19721.
The basic
53 Forcing over a site
Here we describe Joyal's presentation of interpretations in topoi in terms of a
notion of forcing. Let C and a (pre) topology J be fixed. The basic structures
we consider are diagrams
-
of preskeaves on C. Each presheaf A interprets a type
or sort of variable. A morphism f: A1 x ... x An + B interprets an n-ary operation.
A subobject R A1 x ... x An interprets an n-ary relation.
3.1 Definitions. Let L be a first-order language (possibly many-sorted) with
equality. An interpretation of L is given by assigning to each sort A of L a
presheaf A, to each operation F from A1,. . .,An to B a natural transformation
F: A1 x ... x An + B, and to each relation R on A1 x ... x An a subfunctor
R * A1 x .. . x An.
Given such an interpretation, for U E JCCJ we let LU be the

$
170 M.P. FOURMAN
expansion of L obtained by adding constants of the appropriate sorts for the elements
of A(U). If f: V + U then for any term T or formula Q of LU we obtain a
term T l f or formula @If of LV by restricting any new constants which occur.
for U E IC1 we define for each closed term T of sort A of LU an interpretation
[lTnu E A(U) by induction:
Note that UTlfl" = [TDulf.
for 4 a sentence of Lu.
ucnu = c for c E A(U)
[IF( TI,. . . ,Tn) 1,
= F(lT,n,. . . ,iTnj) .
Now we define inductively the relation, U forces Q,
UIk Q
Now
INDUCTIVE DEFINITION OF FORCING
V f l t @If all f: Vf
UIt @
+ K
K E
J(U)
for all f: V + U, if VIE $If then V l t $If
V l t @ +
for all f: V + U, for all c E A(U), VIE ~lfCc/xl
U l t wx.+
We now give some "derived rules" for forcing:
3.2 Lemma. Basic properties of forcing
(PI
U I t 0 f : V + U
V l t +If

Continuous Truth I 171
for each f in some K E
UIt $ y Ji
J(U) either Vflt$4f or V f l t $ l f
Ult- 3x.g
for each f in some K E j(U) we have V f l t $Cc/xl for some c E A(Vf)
V l t 8 * +
for all f: V + U if V l t glf then VIE +If
VIE WX.$
for all f: V -f U and all c E A(U), we have Vlk $lf[c/xl
(Atomic)-
for each f in some K E j(U)'we hav; E R(Vf)
Our presentation here is non-standard in that the definition of forcing is usually
given by stipulating both positive and negative rules for each connective, (I) and
(P) are then derived. The resulting relation is the same.
3.3 Definition. A sequent r - is
pretation iff
uaZid (written r 1 $) in the given inter-
UI~$CS(X)/XI all q E
u I t $CS(X)/Xl
where E is an interpreation of the variables of L by elements of the appropriate
A(U).
3.4 Proposition. If each sort is interpreted by an inhabited presheaf (each
A(U) is inhabited) then the axioms and rules of Heyting's propositional calculus
are valid for k.
(Adaptations for domains which are not inhabited are discussed in Fourman r19771,
Scott 119781, Joyal & Boileau C19811, Makkai & Reyes 119771.)
3.5 Definitions. A presheaf A is separated iff
'Ita =
a = b
A subobject R t+ A is cZosed iff
for a,b
r
E A(U)
U I t R(a) for a E A(U).
FaquJ
A higher-order type-theory is merely a many-sorted first-order theory with some
structure on the collection of sorts and certain distinguished operations and
relations. One of the insights due to Lawvere and Tierney is that topoi have
such higher-order structure. We consider languages where for any two sorts A and
B we can form the product A x B with appropriate pairing and projection operations,
the function space BA with an evaluation operation -( -), and also the power type

172 M.P. FOURMAN
P(A) with a membership reZation E . An interpretation is standard if all this
structure is interpreted by the corresponding structure on Sh((C).
3.6 Proposition. In any standard interpretation the following schemata, which
combine comprehension and extensionality, are valid.
A
x E A 3!y E B.@(x,y) - 3 !f E B WX E A.$(x,f(x))
3!z E P(A) W X E A (X E z ++ @(x)). 0
Thus power-types and function spaces behave as they should. The categorical
characterisation of this higher-order structure in terms of adjoints is very
simple, products are categorical products,
We shall not describe this structure in general here. We shall be dealing primarily
with sorts interpreted by representables. These are particularly simple to
deal with because they have generic elements. A well-known consequence of this is
the Yoneda Lemma:
OP
F(U) [U,FI for F E ISc I and U E /cC1 .
We use this to calculate some examples of the higher-order structure. For this
exercise, we suppose that ci: has finite products and that each representable
functor is a sheaf.
3.7 Lemma.
If F is a sheaf and U,V
are representable
(1) F'(U) F(U x C)
with evaluation for u: V + U and E E F(U x V) given by ~ ( u ) = sI.
(2) (PU)(V) = n(U x V) with Ulka E R iff Rl = V.
U
Proof. F (V)
U
[V,F 1 [UxV,FI 2 F(UxV)
(PU)(V) CV,PUI Sub(UxV) = 6?(UxV). 0
A logical counterpart to Yoneda's lemma is the following.
3.8 Lemma. Generic elements for representables.
If U is representable then VIkWx E U.$ iff V x Ulk $IT~[T~/XI.
Proof. In one direction this is immediate from In the other, suppose
-+ V with a: W + U E U(W) then cf,a>: W + V x U and, by persistence, if
V x U(k $fn1C.rr2/xl then WIk $lf[a/xl.
We give an example of the use of generic elements in the simple case of a category
of presheaves.
3.9 Proposition. Choice holds for representables in categories of presheaves.
Proof.
then
Let U be a representable and suppose
Vlk WX E u.3~ E F.@(x,y)
UX VIk 3~
So by (W)' we have V[k Wx.$ . 0
F.$IT~(T~,Y)
UxVlk $ 1 ~ ~ ( . r r ~ , C for ) some 6 E
F(UxV)

X
U
regarding 5 as an element of F (V) this gives
Since 51 = 5. Thus
and so
Continuous Truth I 173
ux V l t ~l+1,(51n2)(~l)) .
vit wx t U.4(X,E(X)),
vlt- 3f.Vx.+(x,f(x)). 0
From a category-theoretic viewpoint this result is well-known in the form,
"Representables are internally projective".
54 POINTS, LOCAL CHOICE, CONTINUITY
Now we let CC be a category of locales closed under finite limits and open inclusions,
equipped with the open cover topology, J. We write E for the topos Sh(C,J).
For each locale X we define an internal locale X by
O(X)(u); O(XX UJ).
This is generated internally by the basis given by
B()#)(U)E O(Xx U)
or even by the constant basis
Bo(X)(U) = O(X),
with the inclusions go(#) 4 B(X) 4 I )()#) induced by the projections
X x UJ -f
x U -f X. (In the terminology of Joyal & Tierney X = P*(X).)
The internal space of points of X is given by
(ptX)(U) ,z cruj,x1.
This is the space of E-valued models of X. For a: UJ + X and W E O(XxUJ)
iff -'(w) = T.
since i*: o(u~) + U(U) reflects
4.1 Proposition. For any X E BI the internal locale X has enough points
Proof.
We must show for W 6 O(X) that
Ult K covers pt W U l t K is an inductive crible .
UIFW E K
We assume the hypotheses, and let
M = {Wi x Ui I UiItWi E KIUil
Clearly IK is a doubly inductive crible of O(W) x U(U), that is an open of W x U.
We show that IK covers W x U that is that W x U E K, which is evidently sufficient
since then Ult W E K.
By persistence
W
x UlF KAn2 covers nl

174 M.P. FOURMAN
that is
so
W x U(k3V E Kln2.rl E V.
IK" = {Wi x Uilfor some Vi c U(W) we have Wi x Ui It Vi E K1r2 A rl c Vi}
covers W x U. [Because, if p: X + Y is an open surjection and Xlk VEKJPA elp E V
then YlkV E K A e E V: the basic opens of )# are constant and thus descend open
surjections.1 But now we claim IK* IK because, by definition
wi x ui It n1 E vi iff Mi 5 Vi
and, as projections are covers,
Wi x Uilb Vi E KIn2 iff UiIkVi E KIUi. 0
Special cases of this are worthy of mention. When X is Baire space NN, Cantor
N
space 2 , Dedekind reals R, to say that X has enough points is the internal
statement of Bar induction, Fan theorem, Heine Bore1 theorem (respectively). For
these cases it is sufficeint to take the topology on Q generated by covering
families of open inclusions: since each of these spaces X has a point the proections
X x U + X are covers for this topology. We call this topology the open
inclusion topology.
We introduce some more general spaces. Let f: X + U in LOC. We consider the
internal locale X/f defined at U by the basis U(V) with all its standard covers.
More properly for 9: W + U we define
W/f)lg = 0(9*X)
given by pulling back f along g. Any commuting triangle
x
F Y
induces an internal map of locales
5: X/f + Y/h
defined at U. Given by 5-l on basis elements, this clearly takes basic covers to
covers. Furthermore, if 5: X + Y is open (and surjective) then 5: X/f + Y/h is
open, since it suffices to define 4 comnuting with A on basis elements, (and
surjective since if 5: X + Y is an open surjection then so are all its pullbacks,
so internally 5-l reflects basic open covers). These spaces include the spaces )K
we introduced earlier as
U1l-X (X x U)/v .
We now specialise to the case where the objects of b are TI.
an isomorphism
(ptX)(U) CCUj,Xl CCU,Xl
Then U 4 U j induces
so that X represents the functor ptX. This happens in particular for the spaces
N, NN, pN, R and their basic opens (see Fourman 119831.) Furthermore, any element
of pt(X/f) defined at U induces a comnuting triangle

So we have a presentation
Continuous Truth I 175
UJ - X which for TI spaces
\/
X corresponds to a
section of g
U
U
pt(X/f)'g
correspond to
commuting triangles
//dl
with restriction given by composition.
/
W .U
9
We extend our earlier lemma on generic elements:
4.2 Lemma. If objects of (I: are TI then
iff
ulk Vx E
Pt(X/f).$
Xlk ($lf)(id). 0
O f course these generalised representables can be defined internally in any
Grothendieck topos and this result holds.
e-
4.3 Pro osition. - If the objects of C are T, then for any X E ICI and any
Proof.
iff
Wx E
pt(X) .3a c A.$(x,a)
+ 3 open cover p: Z ->> X and a function f: pt Z + A
WX E A.WZ E pt ZCpz = x + $(x,f(z))l.
As pt X is representable,
UIkVx 3a o(x,a)
X x ulk 3a $ln2(nl,a)
iff for some open cover p: Z --> X x U
zlk $ln20~(nlo~,S) for some 5 c A(Z)
iff
Ult-'Jz E z $(P(Z),dZ)). 0
such that
We do not know under what conditions 6 descends to give a function defined on a
cover by open sets. We can ensure this by considering the open inclusion topology
on C in which case we obtain
1 Wx E pt X.3a E A.$(x,a)
+ 3 open cover Ui E U(X) and functions fi: Ui + A such that
Wx E Ui.$(X'fi(X)) .
We now consider continuity.
4.4. Proposition. If X,Y are TI then 1 V f: pt )# + pt W, f is continuous.

176 M.P. FOURMAN
Proof. If Uik f: pt )# + pt W then f is represented by 5: X x U + Y in (c. For
VE()(Y) a basic open of W, w: W + U and x: W + X we have
Wlk (t;lw)(x) E
iff [S 0 l-l(v) = w
-1 -1
iff 5 (V) = w
w It- x E
iff
c5-1(v)lwl
1
regarding 5- V E O(Xx U) as an open of )# defined at U.
1
Thus Ulk 5- (V) is open. 0
v
55 Iteration
We return for a while to consideration of a general Grothendieck topos B = Sh(O,J).
We consider the internal category (I in E given by
(E(U)
with restrictions given by pulling back.
[For those who worry about coherence (one should worry), we remark that a concrete
category in E with an equivalent category of sections over U is given by
considering V/f to be represented as the element S of (PV)(U) determined by
W / ~ V E S iff ~ ~ f o v = g .
So & is an internal small full subcategory of E whose objects are subfunctors of
representables.]
We give C_ a topology by letting
xi
\/
-x
cover
Now for A E IEI we define A, E ShE(C,J) by
with restrictions for g: V
(c/u
X/f in & if Xi + X cover X in c.
UkA_(X/f)
A(X)
+ U given by restriction along f*g
and for 5: Y/h + X/g in a/U, by restriction along 5
Y A X
Any morphism A + B in E induces
U
+,B in ShE($,J).

Continuous Truth I 177
For those who prefer global descriptions, we associate to A E If[ (pseudo)
functors
6/U + E/U
natural in U (i.e. comnuting with g* for g: V + U) as follows:
where
31
U
For Y
'
+
U X
nf ,
E/U
a, B .
X we have nh P npE whence nhS* * nf (as E.* 4 ng)
and nhAy * nPx (as E*Ax Ay) .
This gives the required arrow nhayA + nfAXA.
functor
OP
C+EC *
We shall show that this preserves first order logic.
What we obtain is an (internal)
liere we work concretely for the sake of computations. A simple but more abstract
treatment will appear in Fourman and Kelly C19831. We now consider a first-order
language L with sorts for the objects of E and operations symbols for its
morphisms. In fact to avoid size problems, we consider an arbitrary small fragment
of such a language. We may consider L also as a language in K as a constant
object (via A).
Working in E we consider the interpretation of L given by interpreting the sort A
by A and each operation f: A + B by the corresponding morphism 4 + &.
5.1 Lemma. For f: X + U and g: X + V
~lk x/fk 9 iff vlt- X/gl!- +
Proof. By induction, it suffices to show that if Ulk X/flk *g
v~F X/glk 0 for all g: X + v
then IF is closed under the rules of . As no rule decreases the complexity of
9 we say assume that the result holds for subformulae of 9.
Only (+)+ and (W)' present any difficulties.
result for @ and $.
is defined to mean
Me consider (-+)+, and suppose the
Suppose that for all E: W -+ U and all h: Z + g*X, if W
then WlkZ/(E*f h)Ipd(f*E-h). Then if n: W' + V and h': Z + rr*X are such
that W' Z'/(n*g 0 h') IF @l(g*n 0 h') then by induction hypothesis
Z/(E*f h) Ip*01(f*E 0 h)

178 M.P. FOURMAN
Uk- Z'(f0 g*no h') It * $1(g*no h')
whence (letting 5 = id and h = g*n 0 h') we have
U Z'/( f 0 g*n 0 h' It * $1 (g*n 0 h')
in particularW'IkZ'(n*goh')lkJil(g*qoh').
I
for V is similar. 0
5.2 Theorem.
So VlkX/g/k~-t$. The proof
For Q a formula of L with appropriate parameters
U IF'' X/flk Q" iff xlk Q .
Proof. Firstly, this is well formed: Parameters for Q at X/f are elements of
m) which are given as elements of A(X) and are thus parameters for $ at X.
We proceed by induction.
That is, we show that if we define It* internally by
-
Ulk X/f It * Q iff X l t Q
X\k $) and if we define \I+
then it* is closed under the defining clauses oflk internally, (whence
UIk X/f 1 @
by X It+ $ iff Ulc X/f Ik @ then \kt
is closed under the defining clauses of (whence Xlk Q *VIE X/flt- Q).
As the operations A + B are just those inherited from E,
alike in both contgxts:
under (=)+ and if
so IF* is closed under (=)+
That It + and
terms are interpreted
Thus if [TI = Uo] then UlkU-rl = Dull, so 1' is closed
Xi 11'$Ifi for fi: Xi + X in some cover of X then X I 1 Xi/fi
internally Xik X/idlk $. In the contrary direction, suppose Ulk Xi/g 0 fi IF* $Ifi
for some cover of X as above. Then Xi $Ifi so Xlk Q that is Ulk X/g Q. For
(+)+, first suppose that for all f: V + U if V I--+~lf then V Ik+~lf. Then we
claim UI U/idlk @ + I$, because for all g: W + U and all h: V +W, if
W V/h IF @1g 0 h, then V It+ $lg 0 h so V IF + $19 0 h, that is W v/h Jilg 0 h.
Conversely, if for all g: W + U and all h: Z + g*X, where f: X + U, if
WIE Z/g*f o h \I* $lf*g 0 h, then XIk @ + $, because for h: Z -t X if Z\k $lh then
U l t Z/f 0 h It-* $lh so Ulk Z/f 0 h It-*Jl?h which gives Zlk $Ih, so Ulk X/f I/-* Q + Ji.
The proof for W+
is similar.
Ulk U ~ l l = Uol then UIk T = a,
\I-* are closed under (A)', (v)', (3.)' is trivial. For I, suppose
We view this thorem as asserting that in the topos E the naive notion of truth
given by the equivalence thesis is consonant with the theory of meaning given by
the notion of forcing over the site &. Of course this may seem vacuous as it
appears that B is manufactured with this result in mind. However, in the case of
primary interkt for this paper, the results of 84 allow us to regard (I internally
as a full subcategory of Loc(E) equipped with the open cover topology. In fact,
if Q is the category of separable locales, we may identify (I as a category of
sepgrable locales in E. We shall deal with this, among other things, in a sequel
to this paper.
Given f: X -f U we may view an element a of A(X) as a function:
U It a: X/f + A,.
This allows us to rephrase our theorem.
5.3 Corollary. ulkX/flk $(a) iff Ulk~t E X/f@(a[t)). 0
0
$Ifi and by I
We view this as a general form of the elimination theorem (cf. Troelstra C19771

Continuous Truth I 179
pp.33,79). The appropriate theory of continuous truth CT has an axiom for each
clause in the definition of X/f/k $(a). For example, the clause for 3 gives the
axiom of local choice Yt E X 3y$(a(t),y) iff 3 open cover p: Z ->> X and continuous
f: Z + Y such that Wz$(a(p(z)),f(z)). The translation T$ of a formula $ with-
$.
out free lawless variables is given by T$ : def/k
- CODA
A general notion of non-constructive object is given by interpretations in
Grothendieck topoi. The process of iteration described in 55 shows how we may
view (internal) truth in this interpretation as given by a non-standard theory
of meaning. The clauses defining this give axioms for the corresponding theory
of continuous truth CT and an "elimination" translation. By construction,
CT I$ tf T$ and for formulae in the lawlike part of the language T$ 5 $. The
proof theoretic content of the elimination;
CT $ iff ID T$,
requires formalisation of our treatment in an appropriate theory ID of inductive
definitions. We do not undertake this here.
A final example of an unfinished object is this paper. Some of the results, in
particular continuity principles in sheaves over sites, go back to 1978 and were
much influenced by discussions with Scott and Hyland. Some results are still
being refined. Other persistent influences have been those of Joyal and Lawvere
on the one hand and of Kreisel, Troelstra and Dummett on the other. This
research has been supported at various times by the N.S.F. (U.S.A.), the S.R.C.
(U.K.), the Z.W.O. (Netherlands), and the A.R.G.S. (Australia), and made easier
by the hospitality of many people notably Christine Fox, Irene Scott, Karen Green,
and Imogen Kelly. I am grateful.
REFERENCES
Artin, M., Grothendieck, A., Verdier, J.L., ThGorie des Topos et Cohomologie,
Etale des Sch6mas (SGA4), (Lecture Notes in Math. 269, 270, Springer-Verlag,
Berlin, 1972).
Beth, E.W., Semantical Considerations on Intuitionistic Logic, Indag. Math.,
9(1947), p.572-7.
Boileau, Andr6 & Joyal, Andr6, La logique des topos, J.S.L.
46(1981), p.6-16.
Brouwer, L.E.J., Cambridge Lectures on Intuitionism, D. van Dalen, ed. (Cambridge
University Press, 1981).
Dummett, Michael, Elements of Intuitionism, (Oxford University Press, 1977).
Dummett, Michael, Truth and other enigmas, (Duckworth, London, 1978).
Fourman, Michael P., The logic of Topoi, in Handbook of Math. Logic (ed.
J.), (North-Holland, 1977), p.1053-90.-
Barwise,
Fourman, Michael P., Notions of Choice Sequence, Proc. Brouwer Symposium, (ed.
Troelstra, A. and van Dalen, D.), (North-Holland, 1982).
Fourman, Michael P. & Grayson, Robin J., Formal Spaces, Proc. Brouwer Symposium,
(ed. Troelstra, A. and van Dalen, D.), (North-Holland, 1982).
Fourman, Michael P., T1 spaces over topological sites, JPAA, (to appear), 1983.

180 M.P. FOURMAN
Freyd, Peter, Aspects of Topoi, Bull. Austral. Math. SOC., 7(1972), p.1-76.
Isbell, John, Atomless parts of spaces, Math. Scand., 31(1972), p.5-32.
Johnstone, Peter T., Topos Theory, (Acad. Press, London, 1977).
Johnstone, Peter T., Stone spaces, (Acad. Press, London, 1982).
Joyal, Andre, & Tierney, Myles, An extension of the Galois theory of Grothendieck,
preprint, 1982.
Kreise!, Georg, Lawless sequences of natural numbers.
p .222-48.
Comp. Math. 20(1968),
Makkai , Michael & Reyes, Gonzalo, First-Order Categorical Logic, (Lecture Notes
in Math. 611, Springer-Verlag, 1977).
Moschovakis, Joan R., A topological interpretation of second-order intuitionistic
arithmetic, Comp. Math., (3), 26( 1973), p.261-75.
Scott, Dana S., Extending the topological interpretation to intuitionistic
analysis, Comp. Math. 20(1968), 222-48.
Scott, Dana S., Identity and Existence in Intuitionistic Logic, Proc. Durham
Symposium, (ed. Fourman et al.) (Lecture Notes in Math. 753, Springer-Verlag,
1978) , p. 660-96.
Troelstra, Anne S., Choice Sequences, (Oxford University Press, 1977).
Wraith, Gavin C., Lectures on elementary topoi, Model theory and topoi, (ed.
Lawvere F.W. et al.), (Lecture Notes in Math. 445, Springer-Verlag. Berlin,
1975), p. 114-206.
Wright, Crispin, Wittgenstein's Philosophy of Mathematics, (Duckworth, 1981).