- Email: [email protected]
Reflection principles for synthetic theories of smooth manifolds
Nonlinear Analysis, Theory, Methods & Applicorions, Vol. 30, No. 8, pp. 5 135-5 146, 1997 Proc. 2nd World Congress of NonlinearAnalysts 8 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
Pergamon
PII: SO362-546X(%)00153-8
REFLECTION PRINCIPLES FOR SYNTHETIC THEORIES OF SMOOTH MANIFOLDS SHINICHI Department
YAMADA
of Information Noda
and YOSHIHARU
Sciences,
KATO
Science University
of Tokyo,
City, Chiba 278, JAPAN
Key word3 and phrases: algebraic geometry, automated deduction, C”-algebras, categorical logics, calculus of variations, continumn physics, differential geometry, G6del’s incompleteness theorem,
Godel numbering, Grothendieck topology, Gmthendieck topos, HOL, infmitesimals, Isabelle, local set theories, locus, topcs logics, ML, non-standard analysis, operational calculus, proof-checker, rational physics, reflection principles, sieve, sheaf, site, smooth infinitesimalcalculus, smooth marifold, synthetic reasoning, theorem prover, theory of distributions, topos, topos-theoretic models, the Yoneda embedding theorem, the Yoneda Lemma, Whitney’s embedding theorem Dedicated to our colleague the late Nobuo Yoneda in honor of the Yoneda Embedding
Theorem
1. INTRODUCTION Continuous efforts have been made to realize computer-aided reasoning systems for various theories involving continua in the fields of analysis, differential geometry and continuum physics [20, 211. Th e early experimental theorem provers were implemented in Prolog for Mikusiriski’s operational calculus [33] and Schwartz’s theory of distributions [45] with applications to physics [46]. Now a unified method based on a consistent model theory[25, 351 seems to be established for the automated deduction in these theories[52]. Our method employs the synthetic approach began by F. W. Lawvere[29], A. Kock [25], and others[34, 35, 41, 421 by making use of nilpotent infinitesimals. So our idea is in fact a modern version of Leibniz’s idea of machina ratiocinatrix and applicable to a variety of synthetic calculi in the fields of analysis, geometry, and physics. For instance, a formal system of synthetic calculus of variations can be formulated by making use of the axiom system given by M. Bunge and M. Heggie [S]. And a new axiomatic theory may be formalized for operational calculus based on K. Yosida [54]. The idea is also applicable to the fields of continuum physics such as rational mechanics and thermodynamics which underwent theoretical studies by W. No11 [36,37], W. 0. Williams[51], R. D. Coleman et al[8]. Since many problems are left open yet, an informal and intuitive exposition is given here to motivate the further studies and developments. Here we concentrate upon the problems underlying our topos-theoretic model construction by illustrating them with smooth manifolds. We shall see that the Yoneda embedding[2, and in fact provides us with 5, 31, 531 plays a fundamental role in our model construction reflection principles[48] between the given analytic theories and their topos-theoretic models. The problem here is that the Yoneda embedding is somehow weak and loose compared with Gijdel’s arithmetization[l7] in metamathematics. This is a big open problem left for talents. As a whole, our method are roughly divided into two parts: The first part, which is just described, is the topos-theoretic construction of models[25,27,35,41] to yield the algebraic axiom
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systems for given analytic theories. Given an object theory. say, the category of smooth manfolds, we look for its full and faithful embedding into an appropriate topos by making use of the Yoneda embedding. Moreover. to obtain better models. weaker Grothendieck topologies[31,35] are introduced into the categories of models, thus making them into sites[l2,31,35, $91. So we obtain the Grothendieck topos[31, 35, 221. Thus axiom systems for the said theories describe the Grothendieck topoi [2, 5, 22, 31, 351 with smooth differential structures, or smooth topoi. The second part is the formalization of these axiom systems based on constructive categorical logics for analysis, in particular, topos logics or local set theories [2, 4, 5, 1-l. 22, 26, 351. Their calculi are generally given by variants of second order typed X[jn calculus[l6] or by refined variants of Gentzen-style intuitionistic deduction calculi[2, l-1, 15, 16, 311. These formal systems enable us to design and implement their automated deduction systems: We derive logical calculi in the form suitable for interactive proof checking, and then formalize the corresponding reduction calculi based on BHK_interpretation[3, 16, 131 to facilitate their system implementations. We are now planning to implement some of their experimental proof checkBut we ers in standard ML[39], HOL[lB], and/or Isabelle[40] t o study their feasibility[52]. shall not go into the second part here. Our approach is category-theoretic and the underlying log& are intuitionistic or constructive as we shall see in the next section. But we try to incorporate into our theory some crucial counterparts of set-theoretic ideas and results in the non-standard analysis due to A. Robinson[44]. For instance, the notions of hyper-number systems including invertible infinitesimals are inevitable to practical reasoning systems. 2. SYNTHETIC
REASONING
AND INFINITESIMALS
Synthetic reasoning compared with analytic reasoning may be well understood by Euclidean geometry contrasted with analytic geometry. So we review only the two notions of infinitesimals. From this review we shall see that the axioms for synthetic reasoning systems are inconsistent with classical Eogic and no set-theoretic models exist for synthetic theories. On the contrary, topos-theoretic models can be constructed, compatible with constructive or intuitionistic logics. 2.1. INVERTIBLE INFINITESIMALS. Set-theoretic study of infinitesimals have been initiated by A. Robinson [44], which gives rise to non-standard analysis. Its simple axiomatization was given by J. H. Keisler [23]. Let R and R’ be real number field and a hyperreal number field, respectively. By dehnition, R is a complete Archimedean ordered field and R is an ordered field which is a proper extension of R. For each z E R*, .2: is infinitesimal iff Vy E R[ y > 0 -+ 1x1 < y 1. z is finite iff 3:~ E R[ I4 < Y 3. z is infinite iff Vy E R[ llcl > y 1. 2, y E R’ are said to be infinitely close and written z N y, if 2-v is infinitesimal. So, z is infinitesimal iff 2 N-0. 2 is infinite iff z-r is infinitesimal. Let monad(s) = {,y E R’ : z 21 y}. galmy = {y E R’ : x - y is finite). The set of infinitesimals monad(O) and the set of finite hyperreal numbers gadazy(0) are subrings of R’ and nzonad(0) is a maximal ideal of galaxy(O). 2: is an equivalence relation over RI: z N y 9 monad(z) = monad(y). F or each finite hyperreal z E R’, monad(z) contains a unique real T E R. We say this real r the standard part of 2 and write T = st(o). st(~) is not defined for infinite hyperreal 2. For finite z,y E R*, z N y % st(z) = st(y), .st(~) N_Z. and Vr E R. d(r) = T. st : g&my(O) -+ R is a surjective monotone ring homomorphism. Since R* is a proper extension of R, there exist positive and negative infinitesimals in R*. Hence R* also has positive and negative infinite hyperreals. Since R* is a field, 2 E R” - (0)
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has the inverse x-l E R* - (0). In particular, Vx E monad(O) - (0) has the inverse2-l. Thus, in the non-standard analysis a la Robinson, all nonzero inflnitesimak are invertible. On R’ - { 0)) define an equivalence relation M by 2 z y e z/y N 1. If 2 is finite and st(x) # 0, then x z y ($ x N y. EXAMPLE
2.1. To find out the product formula for the Napier number e, we can go in R* az+h_,,= = ax x 9, we find a such that 1 N v for nonzero infinitesimal Since 7 h. Let h = l/n for an infinite hypernatural number n,
asfollows:
n x a’ln -IL a 11%- 1
elln a Hence e = st(1 + 1/7~)~ = lint,,,(l
N
1.
zz
l/n,,
=z 1+1/n. N
(1 + l/n)”
+ l/n)“.
2.2. NILPOTENT INFINITESIMALS. Next we review the notion of nilpotent infinitesimals and derive basic axioms for smooth inl?nitesimal calculus (SIC for short). Geometric line L can be made into a commutative ring R with unit if the two distinct points 0 and 1 are selected on it. So the first basic axiom for SIC is: AXIOM 2.1. There is a non-trivial ring object R in our topos. R is a commutative ring with unit 1, and R is non-trivial in the sense that R has two distinct points 0.1 : 1 -+ R. Geometrically any curve is a piece of a unique straight line in the infinitely small. So, let the curve be the graph of f : R R, then there is a unique pair (a,b) E R x R such that wh ere D is the infinitely small interval. Let 1) be the object V~~I)[f(ll:0+d)=w+d.h]. of first-order infinitesimals defined by D := [x E R : x2 = 01 5 R. The infinitesimals are nilpotent, since Vd E D[d” = 01. Define a map d : R x R x D R by o(u. h. d) = (I + d. h. the map o : R x R -+ R” 1s ’ given by cr(a. h) = Ad : D(a + d. 6) by adjunction. Hence we have the second basic axiom for SIC, called the Kock-Lawvere axiom: AXIOM
2.2.
The arrow o : R x R ---+ R* is an isomorphism.
This axiom asserts the unique existence of (u, h) for Vf E R”, i.e., V’f : R” 3!u. h : R [f = (~(a. h)]. In other words, for V’f E R”, 3!a. b : R Vd : D [f(d) = u + d. 61. Its generalizations to higher order and partial derivatives will be straightforward (see A.Kock[25]). REMARK 2.1. 1) # (0). Suppose that Vd : D[d = 01. Then Vd : D[d = d. 0 A d = d. 11, which implies 0 = 1 by Axiom 2.2. This contradicts Axiom 2.1. REMARK 2.2. Axiom 2.2 is not compatible with the Law of Excluded Middle. For. assume that the law of excluded middle holds. Then we can define a function f : [I --+ R by 1 ifd#O. 0 ifd=O. { Again by the law of excluded middle, Vd E D[d = 0 V d # 01. But since D # (0)) 3do E D with do # 0. By Axiom 2.2, Vd E D[f(d) = f(0) + d . h]. Hence, substituting do for d, 1 = f(do) = 0 -I do . h. Sq uaring bothsides, we have 1 = 0. A contradiction. Therefore Axiom
f(d) =
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Second World Congress ot Nonlinear Analyst\
2.2 is incompatible
with the law of excluded middle.
Hence, The logic underlying SIC is
not classical but constructive or intuitionistic. REMARK
2.3. SIC does not satisfy the Axiom of Choice.
The logic of SIC is not classical. Hence the tbpos which is a model of the logic is not Boolean. But the following theorem due to R. Diaconescu[qj holds.
THEOREM
2.1 (Diaconescu).
If a topos satisfies the Axiom of Choice, then it is Boolean.
EXAMPLE
2.2 (Derivatives).
Now define j”(zrg) = h, then we have usual Taylor’s formula
- d . .9(z~~) x f’(g(zo)). Similarly, f(g(zro + d)) = f(g(xo) + d . .9’(zgj) .:: f(g(zo)) (f o 9)’ = 9’ x (,J’o 9). Other formulae can be derived easily as shown in A. Kock[25].
Hence
EXAMPLE
l-form
I
2.3 (Stokes’ Formula[7]). Consider the integration
along a closed circuit
(C)
on a manifold
of a differential
h1, where d is a symbol
Let (S) be a surface on A! bounded with the boundary
((Y).
of differentiation.
And we introduce
in (S) two
symbols of differentiations dl.dz which are commutative. and then divide (8Y) into the infinitely small parallelograms. If l’ is the vertex of one of these parallelograms and 1’1 and ,132 are vertices given by the operations
Then J;y’III = us. dl x tr(d2).
rl, and dg, respectively.
J;y’ III = Al*
Similarly, Ji’ ‘~1’ = up(d))
and Jlf 71% = J;y u,(,t+dl) + d2 x ?r(dl).
= Jju” v,+dl
xJ,?
~1 = vc(dzj+
Hence the integral of 11’ along the boundary
is equal to m:(dl) + (ul(d2) + dl uv(d2)) - (u*(dl) + d2 . l/l(dl)) - uj(d2) = d, . Ir(dz) - dz . u,(d,). If ‘11’ = Pd;c, then dl . (P&x) - d2 . (Pd,:r) = dl’ A dn;. So we obtain the Stokes’ formula /
F’dx + Qd:u + Rdz
3. TOPOSTHEORETLC
=
s.l
dP A dz -t dQ A dy t dR A dz. MODEL
CONSTRUCTION
From now on. we concentrate upon the model construction.
The construction is illustrated
here with the category of smooth real manifolds.We look into the problem of finding ” bijective” embedding of the category of smooth manifolds and smooth maps into an appropriate topos. The category of C” manifolds and C:” maps is not Cartesian closed. In particular, the space of C” maps is not necessarily a manifold, and pullbacks of manifolds are generally not manifolds. A. Grothendieck[l, 191, and A. Weil[50] tried to construct the categories of spaces, now called smooth topoi, in which the category of (J” manifolds is fully and faithfully embedded. Moreover, function spaces. inverse limits of spaces, and infinitesimal spaces like the space I) of first-order infinitesimals can be constructed in these smooth topoi. A. Weil[50] introduced local algebras, now called the Weil algebras to deal with nilpotent infinitesimals and Ehresmann jets and connections. A. Grothendieck tried to use nilpotent infinitesimals in his theory of
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schemes[l2,17,49] to treat infinitesimal structures in algebraic geometry[l, 12, 191. He drew a general notion, called the Grothendieck topologies, out of the notion of topologies on manifolds and defined a particular site called the Grothendieck topos[2, 5, 22, 311. Lawvere[28] found that the Grothendieck topos can be viewed as a model of topos theory and also a set-theoretic language can be directly interpreted in a topos ( [2, 5, 22, 31, 381). So, he proposed to use the theory of topos as a foundation for synthetic reasoning. This proposal was part of a big project whose objective is to establish a direct, intrinsic axiomatizaton for continuum mechanics. Among these studies, an axiomatic foundation for synthetic differential geometry is given in A. Kock[25]. A well adapted model for the axiom system, C”-schemes, was found by E. J. Dubuc[lO, 111. Obviously, axiomatizations varies according to different theories and no fixed proper axiom system exists for SIC. However, Axiom 2.1,2.2, and Axiom of compatible order relation would be the minimum set of non-logical axioms for SIC. Some further axioms were pursued by A. Kock[25], C. McLarty[32], M. Bunge and M. Heggie[G], I. Moerdijk and G. E. Reys[35] et al.. However many problems remain open. 4. SMOOTH
MANIFOLDS
For simplicity, we shall consider only smooth manifolds. DEFINITION 4.1. Let M be a topological space and II an open subset of 44. If there exists a homeomorphic mapping cp from II to an open set V C R”, then the pair ((J. cp) is called a n-dimensional coordinate neighbourhood. Such a homeomorphism cp : II -+ V is called a chart for fi1. The ntuple (zi o cp, . . . . x,, o p) of the functions defined on IJ by ~iop:[J%,R”%R
i=
l,...
.r1
is called a system of local coordinates in ((1, cp). Here zi is the projection to i th coordinate. we usually write xi for zi o rp. For p E Ii, ((I, ‘p) is called a coordinate neighbourhood of p. DEFINITION 4.2 (Smooth Manifold). Let M be a Hausdorff space with a countable base. Then M is called a smooth n-dimensional manifold (a smooth n-manifold), if M has n-dimensional coordinate neighbourhoods { (rJ$. vi) ( i E I} such that (1) {(Ji] %E I} is an open covering of M, i.e., lJiel [ii = M, (2) andforVi.jEZ,iflJ,nIJ~#lO,then~~o(Pil: cpj(CJj n ffi) + ipi(Clj n Cli) is C”. set { cpz: lJ, + Vi ) i E 1) is called an atlas for M. REMARK
The
4.1. By this definition, a smooth manifold is paracompact.
NOTATION phisms. DEFINITION
4.1. M denotes the category of smooth manifolds with smooth maps as mor4.3 (Sheaf).
Let M be a topological space.
(1) A Preheaf F on M assigns to each open set IJ in M, a set denoted F(U), and every pair of nested open sets U c V c M a restriction map re.sv,u : F(V) + F(U) satisfying the basic properties that resu,u = idu and VIJ c V c W c M, resv,uoresw,v = rt’sw,u. (2) A presheaf is called a sheaf if it satisfies one further condition, called the sheaf axiom. Sheaf Axiom: For each open covering II = lJaEA U, of an open set U c bf, and for each collection of elements fa E ,3T(Ua) Va E A having the property that for all a, b E A the restrictions of fa and fb to U,, II l/b are equal, there is a unique element f E .F(rJ) whose restriction to lJ, is fa for all a.
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Thus we can give an alternative definition of a differential manifold; it is a topological space M with a (sub)sheaf CM(M) such that the pair (111.CF’( M)) is locally isomorphic to an open subset of R” with its sheaf of C” functions.
REMARK
4.2.. The category Sh(M) 0 f s h eaves over a topological space hil is a topos. 5. (‘Y-ALGEBRAS
DEFINITION
5.1. (1) An algebraic theory C” is a category with finite products whose objects are R-modules R”, n = 1,2,3, . . and whose morphisms are smooth mappings in CW(R”, R) and projections $ : R” --+ R, 1 5 i 5 ‘n for each n. (2) A G-algebra A is a covariant finite product preserving functor A : (I” -+ Set, and C03-homomorphism is a natural transformation.
In write map h =
other words, given such a functor A : C” -+ Set, its underlying set A(R)(we will also as A) is a commutative ring with unit since all ring operations on R are smooth. Every f E C”(Rn,R) has an interpretation in A, A(f) : A” -+ A, in such a way that if g o (fi.. ,fm) then A(h) = A(g) 0 (A(fl). ,A(f,,,)), where fi E C”(R”,R):g E Co3(Rm,R) and h E C”(R”,R). In particular, all projections pz, composition and identity maps are preserved: A(@) = py, A(f o g) = A(f) o A(g), A(id) = id. Here are some examples of C” -algebras[27, 35, 41, 421.
EXAMPLE composition C”(f)
5.1.
C”(R”,R) : Cw(Rm,R)n
is a C”
algebra.
+ G”(R*. R)
Here f E C”(R”,R)
(h I.... .hyJ++jO(h*
Coo(R”‘,R) is the free Co”-algebra on m generators, pr. This algebra is also denoted by C” (R”).
is interpreted as the ,... .h,).
these generators being the projections
EXAMPLE
5.2. For each subset X E R”, a function f : X --+ R is smooth if there is an open U 2 X and a smooth g : II --+ R which extends f. The algebra C”( X. R) of smooth functions X -+ R is a (?-algebra. the Coo-structure is given by composition as in example 5.1. This algebra is also denoted by C”“(X).
EXAMPLE 5.3. If M is a smooth manifold, C” (M, R) is a CM-algebra. Here f E G” (R”, R) is interpreted as G”(f) : G” (M, R)” -+ C” (M, R) by composition as in example 5.1. EXAMPLE 5.4. Let A is a CM-algebra and I an ideal in A. Then the quotient algebra A/l is also a C” -algebra. To see this it sufhces to show that if ai = bi mod I for i = 1,. , ‘n and f : R” -+ R is smooth, then A(f)(al, , a,) E A(f)(bl.. , b,) mod I. But, by Hadamard’s lemma, there are smooth functions gi, . g,, :Rn x R” -+RsuchthatforallZ=(z1.... ,zn) and<=(yl,...
,?/n)eRn,
Since A must preserve this equation,
Nf)(al.. . , a,) - A(f)(bl.. . . , bn) =
@z- bi)A(gi)(al,. .
,an,bl,..
.b,,) E 1
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Next we consider some examples of C”Qlgebras of finite type[ll, 27, 351. We say that a CM--algebra is finitely generated if it is of the form CW(R)/Z. Moreover, we say that generated Cm-algebra is finitely presented if it is isomorphic to one of the form a finitely Cm(R”)/I for some n and some finitely generated ideal I. We shall see that if M is a smooth manifold, then C”-algebra C-(M) is finitely presented.
EXAMPLE 5.5. Not all the Cm-algebras C”“(X) with X C R” are quotients of C”(R”). A smooth function on X extends to a function on R” iff X is closed. So for X closed in R”, we have CO”(X,R) g C”“(R”)/m$, where rn$ is the ideal of smooth functions vanishing on X (i.e., taking value 0 on X). Consequently. the following facts hold: Cl C R” is diffeomorphic to a closed subspace [J = {(ST.y) /y&f(z) = since from analysis we know that there is a function f E CO”(R”) such that f(z) 2 0 iff z 4 Cr. So C”(U, R) is finitely generated. Moreover, C”(lJ? R) s F’(R”+‘)/I whre I is the ideal generated by one function y . f(x) - 1. So C,(U, R) is finitely presented. (2) Every paracompact (smooth) manifold M can be embedded into a closed set in some R”. Thus C” (M, R) is finitely generated. But every paracompact manifold is a retract of an open subset in some R”. So P(M, R) is finitely presented.(Recall Whitney’s embedding theorem and the fact that any closed submanifold M of Euclidean space has an open neighborhood IJ and a projection IJ --) M)[43].
(1) $nr;nzbspace
REMARK 5.1 (Finitely Generated Cm-Algebras). C”--homomorphisms between algebras of the form C”(R”)/Z can be treated explicitly. If @ : C”(R”)/Z --) C”(R”)/J is a C” -homomorphism. Then GM), . . . .@(p;) give a smooth map cp : R” + Rn, and @(f mod I) = f o ‘p mod J. Thus C”“-homomorphisms C""(R")/l + Cm(R”‘)/J are in l-1 correspondence with equivalence classes of smooth maps cp : R” -+ R” such that I C v,(J) = J (i=l.... .n). {f E Co3(Rn)l f o cp E J}, whil ecpisequivalenttocp’ifpTocp--p:ocp’E The condition 1 C p,(J) is equivalent to p*(l) C .I, where p*(Z) is the ideal generated by {f 0 cplf E 0. NOTATION 5.1. (1) P denotes the category of finitely presented C”-algebras with smooth maps as morphisms. (2) G denotes the category of finitely generated 6’“~ algebras with smooth maps as morphism% Now any M E M cm be considered as a C”-algebra M : G” + Set: Let M(R”) = F’(M,R”) and, for ‘p E Coo(Rn,Rm), define Al(p) : M(R”) -+ M(R”) by M(p)(f) = cpo f (Vf E M(R“)). If g : M + IV in M, we have a map for each R”, N(R”) + n/r(R”) detied by composition f c-) f o g. This is a natural transformation between the C” --algebras defined by M and N. Conversely, any natural transformation between the C*-‘-algebras M and _V defines a morphism of C”-algebras *V(R) = c~(:V, R) --+ M(R) = C” (M, R) defined by composition J’ H f o g for some g : n/l --+ N in Ml. Thus we have the contravariant embedding of the category M into the category P ( or G). It is also proved [35] that this contravariant functor: M + P sends transversal pullbacks to pushouts: PROPOSITION 5.1. The contravariant functor C* from M to IF’( or G), mapping M to G”(M.R), and f : MI -+ M2 to CM(f) : CW(Mz) + C”“( Ml) (defined by composition ), is full and faithful. This functor c* sends transversal pullbacks to pushouts.
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REMARK 5.2. In other words, we have a correct, i.e., covariant full and faithful embedding c* of the category 1w[into the dual lV’ ( or Gq) of the category P ( or G). It is also proved [35] that this covariant fi,mctor ct preserves transversal pullbacks. 6. THE
CATEGORY
IL OF LOCI
Here we follow Moerdijk and Reyes(34, 351 and introduce the category lL.of loci which is the opposite category of 6. That is, IL = Gq. Then we have the covatiant functor M 2 IL. To distinguish the objects in IL from the objects in 6, we write l(A) for the object A in IL, i.e., for the locus corresponding to an object A in 6. Morphisms CB -+ lA in IL are C” -homomorphisms A -+ B. The explicit description of IL is given as follows: Since a.n object A in G is finitely generated, Cm~-algebra A is isomorphic to an algebra of the form Cw(Rn)/I. An object of IL is explicitely written as 1(C”(R”)/I) for some n and some ideal I. Morphisms of IL from one such dual are equivalence classes of smooth functions ‘p : R” -+ .!(P’(R”)/I) to another e(Cm(Rm)/J) R” with the property that f E J + S o cp E I, while cp is equivalent to (p’if for each >m), p: o cp - py o (d E J. The Cm-algebra A can be recovered projection p:, (i = 1.. from its locus CA by A z Harn~(IlA,K’O”(R)). W e write R for the object K”(R) of IL. So, R” 2 KF’(R”). If LA is an object of lL with A = C”(R”)/Z, then [A is the zero set Z(I) of I in Rn:
t?A = Z(l) := (z E R” 1b’j E I f(z) = 0). PROPOSITION 6.1. The functor c* : M -+ lL defined by M H K?‘(M) faithful covariant embedding, and preserves transversal pullbacks. 7. THE
YONEDA
EMBEDDING
AND SOME
COARSE
gives full and
MODELS
Now we review the Yoneda embedding and introduce some coarse models. Let C be a category. Then for each object A of C, the horn-functor HA : C --) Set is defined by HA(X) = C(A,X) and for each arrow f:X -+ Y, HA(~) : C(A,X) -+ C(A,Y) : g w fog. Dually, for each object A of C, the contravariant horn-jbzctor HA : CT --+Set is defined by IfFandGare gwgof. HA = C(X,A) and for f : X -+ Y, HA(f) : C(Y,A) +C(X,A): set-valued functors on C, we denote by Nat (F. G) the collection of natural transformations from F to G. Then the following fundamental lemma[53, 2, 5, 22, 311 holds: LEMMA 7.1 (The Yoneda Lemma). For each object A of C, the map 0 : Nat(HA, F) -+ F(A) defined by 0 : 17H aA( 1~) is a bijection. For any small category C,consider
the functor category SetCop whose objects
are aJl con-
bavariant set-valued functors on C (such functors are called presheaves on C). Now define a = fog c”’ by y(A) = HA = C(-, A) for every object A of C and y&(g) mapy:C-+Set for C-arrows A s R and C -% A. Then y is a covariant functor and injective on objects. Moreover, by Yoneda Lemma, Nat(HA, HE) g C?(B, A) z C(A, B) for any objects A and Lz. Hence y is full and faithfull. So,we have proved the THEOREM
7.1 (The Yoneda Embedding).
For any small category C, the functor y is a full and faithful covariant embedding of C into Set cop . Dually, we have
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THEOREM
7.2 (The Contravariant Yoneda Embedding).
the functor y’ is a full and faithful contravariant
For any small category C.
embedding of C into SetC.
Recall that Remark 4.2 and [35, 27, 51. (1) If C is a small category, then the category Set c”” . is a topos. (2) If C is a small category, then the category Set
REMARK
7.1.
C. IS a topos.
Then from propositions 5.1 and 6.1 we have the following topos models of MI: PROPOSITION
7.1.
(1) The composition y* o r* of contravariant
functors y’ and r* :
Ml 5 G z Set” gives a covariant embedding of M into !3etG which is full and faithful and preserves transversal pullbacks. (2) The composition y o c+ of covariant functors y and c, : M 3 IL 3 SetLoPgives a covariant embedding of M into SetEO’ which is full and faithful and preserves transversal pullbacks. Also we have the following results: PROPOSITION
7.2. In SetG,
(1) the forgetful functor fi : G --+ Set : A ++ {zc(x E A) is represented by R(A) 2 G(f?‘(R). A). Thi s is a commutative ring with a unit. (2) the functor D = G(CI”“(R)/(x’). -) = {z E RJr2 = 0). So we have Kock-Lawvere axiom. PROPOSITION
7.3. In Sel?“‘“, R = &Z’“(R). and 11 = &(C;*“(R)/(z2)) = {z E R/z2 = 0).
Hence the topoi Set’
and Se?”
are the models of MI.
8. REFLECTION
PRINCIPLES
Now it will be necessary for us to review metamathematidy the topos models and their embeddings obtained in the previous sections by comparing them with G6del’s incompleteness results[l7]. Throughout this section, we consider only formal theories and describe crucial features of reflection principles used by K. G6del[l7] in the proof of his hrst incompleteness theorem (17, 481. To study an object theory 0, we select an arbitrary, but fixed, consistent meta theory M and construct an embedding r1 which maps every terms and relations upof 0 into a closed term r(pl of M, called the Giidel number (or code) of cp. Such an encoding is called as a Giidel numbering ( numbering for short). In order to perform a Godel numbering of a given object theory 0, it is sufficient that all primitive recursive functions are definable in the metatheory M. Now 6x a Godel numbering and let provable0 (,X1, ‘Y I) be the predicate in M corrwtly representing the provability relation X ko Y of 0, where X and Y are a finite sequence of sentences of c) and a sentence of 0, respectively. In addition, all necessary meta concepts “about” 0, such as formula, inference, and provability, should be encoded by this numbering. Then the provability relation t-0 of the object theory 0 is weakZy representable in the meta theory M by the predicate provableo. X F_s Y
if and only if
I-,Q provableB(‘X1,
rY1).
Second
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Let AXIOII4 be the set of axioms of the theory 0 and define the predicate theorem0 in M by theoremn( Y) ++ provableo(’ AXZOW. r Y’). Then th e fo ll owing symmetric rules are derived, which is a meta meta rule commonly called reflection rules: t-+, theorem0 ( r X’)
to -_----x .t,q theoremo(‘X-‘)
k‘7 X
This meta meta rule is usually called reflection principle. We usually assume that M contains 0, but no problem arises if we let M = 0. Then we have the self-reflection of theory (3( or M). Yaw, by Cantor’s diagonal argument, we have: THEOREM 8.1 (Diagonalization Lemma). sentence A such that t-0 A - p(‘A1).
Sow let cp(z) = +heorem(‘?) ness theorem of Godel:
Given any unary formula V(Z), there is a
in the diagonalization
THEOREM 8.2 (Incompleteness nor 7A is provable in 0.
of 0).
Let to
lemma, we have the first incomplete-
A ++ +heorem(‘A’).
Then neither A
Whereas the embeddings defined in Proposition 7.1 gives the reflections between the theories, the resulting models are somehow too big and too coarse. Moreover, the encoding of meta concepts such as R and n are not properly made in Set”. To eliminate the defects, we shall henceforth consider only the topos SetLoP and introduce a Grothendieck topology into it. 9. A BETTER
MODEL
a
We consider here only the category IL. DEFINITION
9.1.
(1) A sieve on an object e E !L is a class S of morphisms in IL with
codomain C such that if A -f, (’ E S then 13 ‘3 (: E S’for any objects A. H and any morphism H 4 A. (2) A Grothendieck topology on IL is a function J assigning to each (J E IL a class .I(C:) of sieves on (J, such that (a) the maximal sieve {flcodon~irt(f) = C> is in J(C). (b) (stability) If S E .I((;‘) and { I1 f, C) is any morphism in IL, then f*(S) n/f 0 g E S} E J(
:= { ki
(c) (transitivity) if ,E E .I(<:) and and 1’ IS a sieve on (J such that for each 0 1 S. f’(7’) E J(c,“), then 7’ E J(C). If S E J(C) , S is said to be a .I -cover. (3) The pair (IL, J) is called a site. Following Moerdijk and Reyes[35], define the site lI%with a weaker Grothendieck on IL, generated by covering families of the form:
( ,’E
topology
(1) (finiteopencovers) {l((?(lf,)/(Zltl,) ‘-)C(Gm(R)/f}r~l,where{I1,.... .lI,}isafinite open cover of R”, and (2) (projections along non-trivial loci) all singleton covers of the form 1(A) x 1(H) -+ t(A), where L(H) # 0: if t(A) # 0. and let B be the topos of sheaves on l5(Recall Remark 4.2). Then:
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PROPOSITION 9.1. (Moerdijk and Reyes[35]) The Grothendieck topology on B is sub canonical. Hence the Yoneda embedding IL it !%d’“’ ( = I5 - SetBop) factors through B c Set”“. In particular, there is a full and faithful embedding of manifolds h4 Y B which preserves transversal pullbacks. The model B shows good properties in metamuthematicd sense, although its construction is technically rather complicated. The best model is not known to date from the logician’s viewpoint. Further scheme-theoretic study of B is necessary but an open problem. REFERENCES 1. Artin, M., Grothendieck: A. & Verdier, J. L., Thiorie des Topos et Cohomologie l&ale des schemas,(SGA-l), Lecture Notes in Mathematics 269 and 270, Springer-Verlag, Berlin (1972). 2. Bell, J. L., Toposes and Local Set Theories, Oxford University Press (1988). 3. Beeson, M. J., Foundations of Constructive Mathematics, Springer-Verlag, Berlin (1985). 4. Boileau, A. et Joyal, A., La Logique des Topos, J. Symbolic Logic, Vol. 46, No. 1, (1981) 616. 5. Borceux, F., Handbook of Categorical Algebm, 3 VO~UXXS,Cambridge University Press (1994). 6. Bunge, M. & Heggie, M., Synthetic Calculus of Variations, in Mathematical Applications of Category Theory (Edited by J.W. Gray), Contemporary Mathematics 30, Amer. Math. Sot. (1984), 30-62. 7. Cartan, E., Lecon sur la gdom&ie des espaces de Riemann, 2’m” edition, Gauthier-Villars, Paris (1945). 8. Coleman, B. D. & Owen, D. R., Recent Research on the Foundations of Thermodynamics, in [30], 83-99. 9. Diaconescu, R., Axiom of Choice and Complementation, Proc. Amer. Math. Sot., 51, (1975), 176178. 10. Dubuc, E. J., Sur les mod&e de la geometric diff&entielle, Cahiers de Topologie et G&ometrie Di&+entielle, 20, (1979) 231-279. 11. -: C” schemes, Amer. J. Math. 103, (1981), 683-690. 12. Eisenbud, D. & Harris, J., Schemes: The Language of Modern Algebraic Geometry, Wadsworth, Belmont, CA (1992). 13. Feferman, S., Constructive Theories of Functions and Classes, in Logic Colloquium ‘78(Edited by M. Boffa et al), North-Holland, Amsterdam (1979) 159-224. 1-l. A Theory of Variable Types, Rev&a Colombiana de Matemdticas Vol. XIX (1985), 95-105. 15. Gentzen, G., Untersuchungeniiber das logische Schliessen, Mathematische Zeitschtift 39 (1935) 176-210, 405431 : English transl.in The Collected Papers of Gerhard Gentzen (Edited by M. E. Szabo), NorthHolland, Amsterdam, London (1969) 68-131. 16. Girard, J.-Y_ et al., Proofs and Qpes, Cambridge University Press (1989). 17. Godel, K., Uber formal unentsheidbare S.&seder Principia mathematics und verwandter Systeme I, Monatshefte fir Mathematik und Physik, 38, (1931), 173-198 : English transl. in tirn F+cge To Gijdel (Edited by J. van Heijenoort), pp. 144-195. Harvard University Press (1967), and Kurt Giidel Collected Works Vol.1 (Edited by S. Feferman et al), pp. 145-195. Oxford University Press (1986) 18. Gordon, M. J. C. and Melham, T. F., eds., Introduction to HOL, Cambridge University Press (1993). 19. Grothendieck, A.; The cohomology theory of abstract algebraic varieties, Proc. Znt. Congr. A4ath., Edinburgh, 1958. Cambridge University Press (1960), 103-118. 20. Hirose, K., Kikyo, H., Kakehi, K., Yamada, S., & Doi, S., Use of Meta-Knowledge on Mathematical Structures in Theorem Proving, The 3rd Asian Conf. in h4athematical Logic, Beijing, China (1987). 21. Hirose, K.; Kikyo, H. & Yamada, S., An Experimental Theorem Prover using Metaknowledge on Mathematical Structures (Japanese), Proc. The 97th National Conf. of Information Processing Society of Japan (1988). 22. Johnstone, P. T., Topos Theory, Academic Press, London (1977). 23. Keisler, H. J., Foundations of Infinitesimal Calculus, Prindle, Weber and Schmidt, Boston (1976). 24. Kock, A., ed., Topos Theoretic n4ethods in Geometry, Matematsk Institut, Aarhus Universitet, Var. Publ. Ser. 30 (1979). 25. __ , Synthetic Differential Geometry, London Mathematical Society Lecture Note Series 51, Cambridge University Press (1981). 26. Lambek, J. & Scott,P. J., Introduction to higher order categoricd logic, Cambridge Univerdity Press (1986).
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Second World Congress of Nonlinear Analysts
27. Lavendhomme, R.. Basic Concepts of Synthetzr Diflerentinl Geometry, Kluwer Academic Publishers, Dordrecht, Boston, London (1996). 28. Lawvere. F. W., Variable Sets Etendu and Variable Structure in Topoi, Notes by S. Landsburg, Lecture Notes in Mathematics, The University of Chicago (1975); pp.17. 29 -pi Categorical Dynamics, in [X], pp. 1 -28. 30. Lawvere, F. W. & Schanuel, S. H., eds., Categories in Continuum PhysZca, Lecture 4otes in Mathematics 1174, Springer-Verlag, Berlin Heidelberg New York Tokyo (1986). 31. ?IIac Lane: S. & I. l\Ioerdijk, I., Sh eaves in Geometry and Logic, Springer-Verlag, New York (1992). 32. UcLarty, C., Elementaq Categories, Elementary Toposes, Oxford University Press (1992). 33. Mikusinski. J., Operational Calculus, Pergamon Press, London (1957). 34. hIoerdijk, I. & Reyes, G. E.. A Smooth Version of the Zariski Topos, Advances in hlathematics 65, (1987); 229-253. 35. ___. Models for Smooth Znfinitesimd Analysis, Springer-Verlag, Sew York (1991). 36. Noll, N., The Foundations of Ilfechanics and Thermodynamics, Springer-Verlag, Berlin (1974). 37. -Continuum Mechanics and Geometric Integration Theory, in [30], 17-29. 38. Osius, G., Logical and Set Theoretical Tools in Elementary Topoi, Model Theory and Topoi, (Edited by F.W. Lawvere et al), Lecture Notes in Mathematics 445, Springer-Verlag, Berlin (1975), 297-354. 39. L.C. Paulson, L. C., ML for the Working Programmer, Cambridge University Press (1992). Isabel/e, Lecture Notes in Computer Science 828, Springer-Verlag, Berlin (1994). 10. -1 41. van &UC, N. & Reyes, G. E., Smooth Functom and Synthetic Calculus, in The L.E.J.Brouwer Centenary Symposinm (Edited by A.S.Troelstra and D. van Dalen), North-Holland, Amsterdam (1982), 377-395. 52. Reyes, G. E., Synthetic reasoning and Variable sets, in [30], 69-82. 43. de Rham, G., Vari&s difl&entiables, Hermann, Paris (1955): English transl., Differentiable Manifolds, Springer-Verlag, Berlin Heidelberg New York Tokyo (1984). -44. Robinson, A., Non-Standad Analysis, Revised edition, North-Holland, Amsterdam (1971). 15. Schwartz, L.: The’orie des distributions, Hermann, Paris, I (1950); II (1951). 46. M&thodes Math6matique.s pour les Sciences Physiques, Hermann, Paris (1961). 47. Shafarevich, Igor R., Basic Algebraic Geometry 2: Schemes and Complex Manifolds, Second Edition, Springer-Verlag, Berlin, Heidelberg (1994). 48. Smorynski, C., The incompleteness theorems, in Handbook of Mathematical Logic (Edited by J. Barwise),North-Holland, Amsterdam, New York, Oxford (1977), 821-865. 49. Tennison, B. R., Sheaf Theory, London Math. Sot. Lecture Note Series 20, Cambridge University Press (1975). 50. Weil, A., Tht%rie des points proches sur les vari&?s diff&entiables, Colloque de Ge’ometrie DifJ&ntielle, C.N.R.S.(1953), Ill-117 : also in We& A; Collected Papers: Oeuores Scientijiques, Vol. 2, 103-109, Springer-Verlag, Berlin (1979). 51. Williams, W. O., Structure of Continuum Physics, in [30], 30-37. 52. Yamada, S., Computer-Aided Synthetic Reasoning in Smooth Infinitesimal Calculus, Proc. 5th Znternational Colloquium on Differential Equations (Vol.Z), (1995), 258-267. 53. Yoneda, N., On the homology theory of modules, J. Fat. Sci. Tokyo, Sec.f. 7 (1954), 193-227. 54. Yosida, K., Operational Calculus: Viewed as a thwq of diatributions(Japane), UP Applied Mathematics Series 5, 3rd Printing, The University of Tokyo Press (1993).
Pergamon
PII: SO362-546X(%)00153-8
REFLECTION PRINCIPLES FOR SYNTHETIC THEORIES OF SMOOTH MANIFOLDS SHINICHI Department
YAMADA
of Information Noda
and YOSHIHARU
Sciences,
KATO
Science University
of Tokyo,
City, Chiba 278, JAPAN
Key word3 and phrases: algebraic geometry, automated deduction, C”-algebras, categorical logics, calculus of variations, continumn physics, differential geometry, G6del’s incompleteness theorem,
Godel numbering, Grothendieck topology, Gmthendieck topos, HOL, infmitesimals, Isabelle, local set theories, locus, topcs logics, ML, non-standard analysis, operational calculus, proof-checker, rational physics, reflection principles, sieve, sheaf, site, smooth infinitesimalcalculus, smooth marifold, synthetic reasoning, theorem prover, theory of distributions, topos, topos-theoretic models, the Yoneda embedding theorem, the Yoneda Lemma, Whitney’s embedding theorem Dedicated to our colleague the late Nobuo Yoneda in honor of the Yoneda Embedding
Theorem
1. INTRODUCTION Continuous efforts have been made to realize computer-aided reasoning systems for various theories involving continua in the fields of analysis, differential geometry and continuum physics [20, 211. Th e early experimental theorem provers were implemented in Prolog for Mikusiriski’s operational calculus [33] and Schwartz’s theory of distributions [45] with applications to physics [46]. Now a unified method based on a consistent model theory[25, 351 seems to be established for the automated deduction in these theories[52]. Our method employs the synthetic approach began by F. W. Lawvere[29], A. Kock [25], and others[34, 35, 41, 421 by making use of nilpotent infinitesimals. So our idea is in fact a modern version of Leibniz’s idea of machina ratiocinatrix and applicable to a variety of synthetic calculi in the fields of analysis, geometry, and physics. For instance, a formal system of synthetic calculus of variations can be formulated by making use of the axiom system given by M. Bunge and M. Heggie [S]. And a new axiomatic theory may be formalized for operational calculus based on K. Yosida [54]. The idea is also applicable to the fields of continuum physics such as rational mechanics and thermodynamics which underwent theoretical studies by W. No11 [36,37], W. 0. Williams[51], R. D. Coleman et al[8]. Since many problems are left open yet, an informal and intuitive exposition is given here to motivate the further studies and developments. Here we concentrate upon the problems underlying our topos-theoretic model construction by illustrating them with smooth manifolds. We shall see that the Yoneda embedding[2, and in fact provides us with 5, 31, 531 plays a fundamental role in our model construction reflection principles[48] between the given analytic theories and their topos-theoretic models. The problem here is that the Yoneda embedding is somehow weak and loose compared with Gijdel’s arithmetization[l7] in metamathematics. This is a big open problem left for talents. As a whole, our method are roughly divided into two parts: The first part, which is just described, is the topos-theoretic construction of models[25,27,35,41] to yield the algebraic axiom
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Second World Congress of Nonlinear Analysts
systems for given analytic theories. Given an object theory. say, the category of smooth manfolds, we look for its full and faithful embedding into an appropriate topos by making use of the Yoneda embedding. Moreover. to obtain better models. weaker Grothendieck topologies[31,35] are introduced into the categories of models, thus making them into sites[l2,31,35, $91. So we obtain the Grothendieck topos[31, 35, 221. Thus axiom systems for the said theories describe the Grothendieck topoi [2, 5, 22, 31, 351 with smooth differential structures, or smooth topoi. The second part is the formalization of these axiom systems based on constructive categorical logics for analysis, in particular, topos logics or local set theories [2, 4, 5, 1-l. 22, 26, 351. Their calculi are generally given by variants of second order typed X[jn calculus[l6] or by refined variants of Gentzen-style intuitionistic deduction calculi[2, l-1, 15, 16, 311. These formal systems enable us to design and implement their automated deduction systems: We derive logical calculi in the form suitable for interactive proof checking, and then formalize the corresponding reduction calculi based on BHK_interpretation[3, 16, 131 to facilitate their system implementations. We are now planning to implement some of their experimental proof checkBut we ers in standard ML[39], HOL[lB], and/or Isabelle[40] t o study their feasibility[52]. shall not go into the second part here. Our approach is category-theoretic and the underlying log& are intuitionistic or constructive as we shall see in the next section. But we try to incorporate into our theory some crucial counterparts of set-theoretic ideas and results in the non-standard analysis due to A. Robinson[44]. For instance, the notions of hyper-number systems including invertible infinitesimals are inevitable to practical reasoning systems. 2. SYNTHETIC
REASONING
AND INFINITESIMALS
Synthetic reasoning compared with analytic reasoning may be well understood by Euclidean geometry contrasted with analytic geometry. So we review only the two notions of infinitesimals. From this review we shall see that the axioms for synthetic reasoning systems are inconsistent with classical Eogic and no set-theoretic models exist for synthetic theories. On the contrary, topos-theoretic models can be constructed, compatible with constructive or intuitionistic logics. 2.1. INVERTIBLE INFINITESIMALS. Set-theoretic study of infinitesimals have been initiated by A. Robinson [44], which gives rise to non-standard analysis. Its simple axiomatization was given by J. H. Keisler [23]. Let R and R’ be real number field and a hyperreal number field, respectively. By dehnition, R is a complete Archimedean ordered field and R is an ordered field which is a proper extension of R. For each z E R*, .2: is infinitesimal iff Vy E R[ y > 0 -+ 1x1 < y 1. z is finite iff 3:~ E R[ I4 < Y 3. z is infinite iff Vy E R[ llcl > y 1. 2, y E R’ are said to be infinitely close and written z N y, if 2-v is infinitesimal. So, z is infinitesimal iff 2 N-0. 2 is infinite iff z-r is infinitesimal. Let monad(s) = {,y E R’ : z 21 y}. galmy = {y E R’ : x - y is finite). The set of infinitesimals monad(O) and the set of finite hyperreal numbers gadazy(0) are subrings of R’ and nzonad(0) is a maximal ideal of galaxy(O). 2: is an equivalence relation over RI: z N y 9 monad(z) = monad(y). F or each finite hyperreal z E R’, monad(z) contains a unique real T E R. We say this real r the standard part of 2 and write T = st(o). st(~) is not defined for infinite hyperreal 2. For finite z,y E R*, z N y % st(z) = st(y), .st(~) N_Z. and Vr E R. d(r) = T. st : g&my(O) -+ R is a surjective monotone ring homomorphism. Since R* is a proper extension of R, there exist positive and negative infinitesimals in R*. Hence R* also has positive and negative infinite hyperreals. Since R* is a field, 2 E R” - (0)
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has the inverse x-l E R* - (0). In particular, Vx E monad(O) - (0) has the inverse2-l. Thus, in the non-standard analysis a la Robinson, all nonzero inflnitesimak are invertible. On R’ - { 0)) define an equivalence relation M by 2 z y e z/y N 1. If 2 is finite and st(x) # 0, then x z y ($ x N y. EXAMPLE
2.1. To find out the product formula for the Napier number e, we can go in R* az+h_,,= = ax x 9, we find a such that 1 N v for nonzero infinitesimal Since 7 h. Let h = l/n for an infinite hypernatural number n,
asfollows:
n x a’ln -IL a 11%- 1
elln a Hence e = st(1 + 1/7~)~ = lint,,,(l
N
1.
zz
l/n,,
=z 1+1/n. N
(1 + l/n)”
+ l/n)“.
2.2. NILPOTENT INFINITESIMALS. Next we review the notion of nilpotent infinitesimals and derive basic axioms for smooth inl?nitesimal calculus (SIC for short). Geometric line L can be made into a commutative ring R with unit if the two distinct points 0 and 1 are selected on it. So the first basic axiom for SIC is: AXIOM 2.1. There is a non-trivial ring object R in our topos. R is a commutative ring with unit 1, and R is non-trivial in the sense that R has two distinct points 0.1 : 1 -+ R. Geometrically any curve is a piece of a unique straight line in the infinitely small. So, let the curve be the graph of f : R R, then there is a unique pair (a,b) E R x R such that wh ere D is the infinitely small interval. Let 1) be the object V~~I)[f(ll:0+d)=w+d.h]. of first-order infinitesimals defined by D := [x E R : x2 = 01 5 R. The infinitesimals are nilpotent, since Vd E D[d” = 01. Define a map d : R x R x D R by o(u. h. d) = (I + d. h. the map o : R x R -+ R” 1s ’ given by cr(a. h) = Ad : D(a + d. 6) by adjunction. Hence we have the second basic axiom for SIC, called the Kock-Lawvere axiom: AXIOM
2.2.
The arrow o : R x R ---+ R* is an isomorphism.
This axiom asserts the unique existence of (u, h) for Vf E R”, i.e., V’f : R” 3!u. h : R [f = (~(a. h)]. In other words, for V’f E R”, 3!a. b : R Vd : D [f(d) = u + d. 61. Its generalizations to higher order and partial derivatives will be straightforward (see A.Kock[25]). REMARK 2.1. 1) # (0). Suppose that Vd : D[d = 01. Then Vd : D[d = d. 0 A d = d. 11, which implies 0 = 1 by Axiom 2.2. This contradicts Axiom 2.1. REMARK 2.2. Axiom 2.2 is not compatible with the Law of Excluded Middle. For. assume that the law of excluded middle holds. Then we can define a function f : [I --+ R by 1 ifd#O. 0 ifd=O. { Again by the law of excluded middle, Vd E D[d = 0 V d # 01. But since D # (0)) 3do E D with do # 0. By Axiom 2.2, Vd E D[f(d) = f(0) + d . h]. Hence, substituting do for d, 1 = f(do) = 0 -I do . h. Sq uaring bothsides, we have 1 = 0. A contradiction. Therefore Axiom
f(d) =
513x
Second World Congress ot Nonlinear Analyst\
2.2 is incompatible
with the law of excluded middle.
Hence, The logic underlying SIC is
not classical but constructive or intuitionistic. REMARK
2.3. SIC does not satisfy the Axiom of Choice.
The logic of SIC is not classical. Hence the tbpos which is a model of the logic is not Boolean. But the following theorem due to R. Diaconescu[qj holds.
THEOREM
2.1 (Diaconescu).
If a topos satisfies the Axiom of Choice, then it is Boolean.
EXAMPLE
2.2 (Derivatives).
Now define j”(zrg) = h, then we have usual Taylor’s formula
- d . .9(z~~) x f’(g(zo)). Similarly, f(g(zro + d)) = f(g(xo) + d . .9’(zgj) .:: f(g(zo)) (f o 9)’ = 9’ x (,J’o 9). Other formulae can be derived easily as shown in A. Kock[25].
Hence
EXAMPLE
l-form
I
2.3 (Stokes’ Formula[7]). Consider the integration
along a closed circuit
(C)
on a manifold
of a differential
h1, where d is a symbol
Let (S) be a surface on A! bounded with the boundary
((Y).
of differentiation.
And we introduce
in (S) two
symbols of differentiations dl.dz which are commutative. and then divide (8Y) into the infinitely small parallelograms. If l’ is the vertex of one of these parallelograms and 1’1 and ,132 are vertices given by the operations
Then J;y’III = us. dl x tr(d2).
rl, and dg, respectively.
J;y’ III = Al*
Similarly, Ji’ ‘~1’ = up(d))
and Jlf 71% = J;y u,(,t+dl) + d2 x ?r(dl).
= Jju” v,+dl
xJ,?
~1 = vc(dzj+
Hence the integral of 11’ along the boundary
is equal to m:(dl) + (ul(d2) + dl uv(d2)) - (u*(dl) + d2 . l/l(dl)) - uj(d2) = d, . Ir(dz) - dz . u,(d,). If ‘11’ = Pd;c, then dl . (P&x) - d2 . (Pd,:r) = dl’ A dn;. So we obtain the Stokes’ formula /
F’dx + Qd:u + Rdz
3. TOPOSTHEORETLC
=
s.l
dP A dz -t dQ A dy t dR A dz. MODEL
CONSTRUCTION
From now on. we concentrate upon the model construction.
The construction is illustrated
here with the category of smooth real manifolds.We look into the problem of finding ” bijective” embedding of the category of smooth manifolds and smooth maps into an appropriate topos. The category of C” manifolds and C:” maps is not Cartesian closed. In particular, the space of C” maps is not necessarily a manifold, and pullbacks of manifolds are generally not manifolds. A. Grothendieck[l, 191, and A. Weil[50] tried to construct the categories of spaces, now called smooth topoi, in which the category of (J” manifolds is fully and faithfully embedded. Moreover, function spaces. inverse limits of spaces, and infinitesimal spaces like the space I) of first-order infinitesimals can be constructed in these smooth topoi. A. Weil[50] introduced local algebras, now called the Weil algebras to deal with nilpotent infinitesimals and Ehresmann jets and connections. A. Grothendieck tried to use nilpotent infinitesimals in his theory of
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schemes[l2,17,49] to treat infinitesimal structures in algebraic geometry[l, 12, 191. He drew a general notion, called the Grothendieck topologies, out of the notion of topologies on manifolds and defined a particular site called the Grothendieck topos[2, 5, 22, 311. Lawvere[28] found that the Grothendieck topos can be viewed as a model of topos theory and also a set-theoretic language can be directly interpreted in a topos ( [2, 5, 22, 31, 381). So, he proposed to use the theory of topos as a foundation for synthetic reasoning. This proposal was part of a big project whose objective is to establish a direct, intrinsic axiomatizaton for continuum mechanics. Among these studies, an axiomatic foundation for synthetic differential geometry is given in A. Kock[25]. A well adapted model for the axiom system, C”-schemes, was found by E. J. Dubuc[lO, 111. Obviously, axiomatizations varies according to different theories and no fixed proper axiom system exists for SIC. However, Axiom 2.1,2.2, and Axiom of compatible order relation would be the minimum set of non-logical axioms for SIC. Some further axioms were pursued by A. Kock[25], C. McLarty[32], M. Bunge and M. Heggie[G], I. Moerdijk and G. E. Reys[35] et al.. However many problems remain open. 4. SMOOTH
MANIFOLDS
For simplicity, we shall consider only smooth manifolds. DEFINITION 4.1. Let M be a topological space and II an open subset of 44. If there exists a homeomorphic mapping cp from II to an open set V C R”, then the pair ((J. cp) is called a n-dimensional coordinate neighbourhood. Such a homeomorphism cp : II -+ V is called a chart for fi1. The ntuple (zi o cp, . . . . x,, o p) of the functions defined on IJ by ~iop:[J%,R”%R
i=
l,...
.r1
is called a system of local coordinates in ((1, cp). Here zi is the projection to i th coordinate. we usually write xi for zi o rp. For p E Ii, ((I, ‘p) is called a coordinate neighbourhood of p. DEFINITION 4.2 (Smooth Manifold). Let M be a Hausdorff space with a countable base. Then M is called a smooth n-dimensional manifold (a smooth n-manifold), if M has n-dimensional coordinate neighbourhoods { (rJ$. vi) ( i E I} such that (1) {(Ji] %E I} is an open covering of M, i.e., lJiel [ii = M, (2) andforVi.jEZ,iflJ,nIJ~#lO,then~~o(Pil: cpj(CJj n ffi) + ipi(Clj n Cli) is C”. set { cpz: lJ, + Vi ) i E 1) is called an atlas for M. REMARK
The
4.1. By this definition, a smooth manifold is paracompact.
NOTATION phisms. DEFINITION
4.1. M denotes the category of smooth manifolds with smooth maps as mor4.3 (Sheaf).
Let M be a topological space.
(1) A Preheaf F on M assigns to each open set IJ in M, a set denoted F(U), and every pair of nested open sets U c V c M a restriction map re.sv,u : F(V) + F(U) satisfying the basic properties that resu,u = idu and VIJ c V c W c M, resv,uoresw,v = rt’sw,u. (2) A presheaf is called a sheaf if it satisfies one further condition, called the sheaf axiom. Sheaf Axiom: For each open covering II = lJaEA U, of an open set U c bf, and for each collection of elements fa E ,3T(Ua) Va E A having the property that for all a, b E A the restrictions of fa and fb to U,, II l/b are equal, there is a unique element f E .F(rJ) whose restriction to lJ, is fa for all a.
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Thus we can give an alternative definition of a differential manifold; it is a topological space M with a (sub)sheaf CM(M) such that the pair (111.CF’( M)) is locally isomorphic to an open subset of R” with its sheaf of C” functions.
REMARK
4.2.. The category Sh(M) 0 f s h eaves over a topological space hil is a topos. 5. (‘Y-ALGEBRAS
DEFINITION
5.1. (1) An algebraic theory C” is a category with finite products whose objects are R-modules R”, n = 1,2,3, . . and whose morphisms are smooth mappings in CW(R”, R) and projections $ : R” --+ R, 1 5 i 5 ‘n for each n. (2) A G-algebra A is a covariant finite product preserving functor A : (I” -+ Set, and C03-homomorphism is a natural transformation.
In write map h =
other words, given such a functor A : C” -+ Set, its underlying set A(R)(we will also as A) is a commutative ring with unit since all ring operations on R are smooth. Every f E C”(Rn,R) has an interpretation in A, A(f) : A” -+ A, in such a way that if g o (fi.. ,fm) then A(h) = A(g) 0 (A(fl). ,A(f,,,)), where fi E C”(R”,R):g E Co3(Rm,R) and h E C”(R”,R). In particular, all projections pz, composition and identity maps are preserved: A(@) = py, A(f o g) = A(f) o A(g), A(id) = id. Here are some examples of C” -algebras[27, 35, 41, 421.
EXAMPLE composition C”(f)
5.1.
C”(R”,R) : Cw(Rm,R)n
is a C”
algebra.
+ G”(R*. R)
Here f E C”(R”,R)
(h I.... .hyJ++jO(h*
Coo(R”‘,R) is the free Co”-algebra on m generators, pr. This algebra is also denoted by C” (R”).
is interpreted as the ,... .h,).
these generators being the projections
EXAMPLE
5.2. For each subset X E R”, a function f : X --+ R is smooth if there is an open U 2 X and a smooth g : II --+ R which extends f. The algebra C”( X. R) of smooth functions X -+ R is a (?-algebra. the Coo-structure is given by composition as in example 5.1. This algebra is also denoted by C”“(X).
EXAMPLE 5.3. If M is a smooth manifold, C” (M, R) is a CM-algebra. Here f E G” (R”, R) is interpreted as G”(f) : G” (M, R)” -+ C” (M, R) by composition as in example 5.1. EXAMPLE 5.4. Let A is a CM-algebra and I an ideal in A. Then the quotient algebra A/l is also a C” -algebra. To see this it sufhces to show that if ai = bi mod I for i = 1,. , ‘n and f : R” -+ R is smooth, then A(f)(al, , a,) E A(f)(bl.. , b,) mod I. But, by Hadamard’s lemma, there are smooth functions gi, . g,, :Rn x R” -+RsuchthatforallZ=(z1.... ,zn) and<=(yl,...
,?/n)eRn,
Since A must preserve this equation,
Nf)(al.. . , a,) - A(f)(bl.. . . , bn) =
@z- bi)A(gi)(al,. .
,an,bl,..
.b,,) E 1
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Next we consider some examples of C”Qlgebras of finite type[ll, 27, 351. We say that a CM--algebra is finitely generated if it is of the form CW(R)/Z. Moreover, we say that generated Cm-algebra is finitely presented if it is isomorphic to one of the form a finitely Cm(R”)/I for some n and some finitely generated ideal I. We shall see that if M is a smooth manifold, then C”-algebra C-(M) is finitely presented.
EXAMPLE 5.5. Not all the Cm-algebras C”“(X) with X C R” are quotients of C”(R”). A smooth function on X extends to a function on R” iff X is closed. So for X closed in R”, we have CO”(X,R) g C”“(R”)/m$, where rn$ is the ideal of smooth functions vanishing on X (i.e., taking value 0 on X). Consequently. the following facts hold: Cl C R” is diffeomorphic to a closed subspace [J = {(ST.y) /y&f(z) = since from analysis we know that there is a function f E CO”(R”) such that f(z) 2 0 iff z 4 Cr. So C”(U, R) is finitely generated. Moreover, C”(lJ? R) s F’(R”+‘)/I whre I is the ideal generated by one function y . f(x) - 1. So C,(U, R) is finitely presented. (2) Every paracompact (smooth) manifold M can be embedded into a closed set in some R”. Thus C” (M, R) is finitely generated. But every paracompact manifold is a retract of an open subset in some R”. So P(M, R) is finitely presented.(Recall Whitney’s embedding theorem and the fact that any closed submanifold M of Euclidean space has an open neighborhood IJ and a projection IJ --) M)[43].
(1) $nr;nzbspace
REMARK 5.1 (Finitely Generated Cm-Algebras). C”--homomorphisms between algebras of the form C”(R”)/Z can be treated explicitly. If @ : C”(R”)/Z --) C”(R”)/J is a C” -homomorphism. Then GM), . . . .@(p;) give a smooth map cp : R” + Rn, and @(f mod I) = f o ‘p mod J. Thus C”“-homomorphisms C""(R")/l + Cm(R”‘)/J are in l-1 correspondence with equivalence classes of smooth maps cp : R” -+ R” such that I C v,(J) = J (i=l.... .n). {f E Co3(Rn)l f o cp E J}, whil ecpisequivalenttocp’ifpTocp--p:ocp’E The condition 1 C p,(J) is equivalent to p*(l) C .I, where p*(Z) is the ideal generated by {f 0 cplf E 0. NOTATION 5.1. (1) P denotes the category of finitely presented C”-algebras with smooth maps as morphisms. (2) G denotes the category of finitely generated 6’“~ algebras with smooth maps as morphism% Now any M E M cm be considered as a C”-algebra M : G” + Set: Let M(R”) = F’(M,R”) and, for ‘p E Coo(Rn,Rm), define Al(p) : M(R”) -+ M(R”) by M(p)(f) = cpo f (Vf E M(R“)). If g : M + IV in M, we have a map for each R”, N(R”) + n/r(R”) detied by composition f c-) f o g. This is a natural transformation between the C” --algebras defined by M and N. Conversely, any natural transformation between the C*-‘-algebras M and _V defines a morphism of C”-algebras *V(R) = c~(:V, R) --+ M(R) = C” (M, R) defined by composition J’ H f o g for some g : n/l --+ N in Ml. Thus we have the contravariant embedding of the category M into the category P ( or G). It is also proved [35] that this contravariant functor: M + P sends transversal pullbacks to pushouts: PROPOSITION 5.1. The contravariant functor C* from M to IF’( or G), mapping M to G”(M.R), and f : MI -+ M2 to CM(f) : CW(Mz) + C”“( Ml) (defined by composition ), is full and faithful. This functor c* sends transversal pullbacks to pushouts.
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REMARK 5.2. In other words, we have a correct, i.e., covariant full and faithful embedding c* of the category 1w[into the dual lV’ ( or Gq) of the category P ( or G). It is also proved [35] that this covariant fi,mctor ct preserves transversal pullbacks. 6. THE
CATEGORY
IL OF LOCI
Here we follow Moerdijk and Reyes(34, 351 and introduce the category lL.of loci which is the opposite category of 6. That is, IL = Gq. Then we have the covatiant functor M 2 IL. To distinguish the objects in IL from the objects in 6, we write l(A) for the object A in IL, i.e., for the locus corresponding to an object A in 6. Morphisms CB -+ lA in IL are C” -homomorphisms A -+ B. The explicit description of IL is given as follows: Since a.n object A in G is finitely generated, Cm~-algebra A is isomorphic to an algebra of the form Cw(Rn)/I. An object of IL is explicitely written as 1(C”(R”)/I) for some n and some ideal I. Morphisms of IL from one such dual are equivalence classes of smooth functions ‘p : R” -+ .!(P’(R”)/I) to another e(Cm(Rm)/J) R” with the property that f E J + S o cp E I, while cp is equivalent to (p’if for each >m), p: o cp - py o (d E J. The Cm-algebra A can be recovered projection p:, (i = 1.. from its locus CA by A z Harn~(IlA,K’O”(R)). W e write R for the object K”(R) of IL. So, R” 2 KF’(R”). If LA is an object of lL with A = C”(R”)/Z, then [A is the zero set Z(I) of I in Rn:
t?A = Z(l) := (z E R” 1b’j E I f(z) = 0). PROPOSITION 6.1. The functor c* : M -+ lL defined by M H K?‘(M) faithful covariant embedding, and preserves transversal pullbacks. 7. THE
YONEDA
EMBEDDING
AND SOME
COARSE
gives full and
MODELS
Now we review the Yoneda embedding and introduce some coarse models. Let C be a category. Then for each object A of C, the horn-functor HA : C --) Set is defined by HA(X) = C(A,X) and for each arrow f:X -+ Y, HA(~) : C(A,X) -+ C(A,Y) : g w fog. Dually, for each object A of C, the contravariant horn-jbzctor HA : CT --+Set is defined by IfFandGare gwgof. HA = C(X,A) and for f : X -+ Y, HA(f) : C(Y,A) +C(X,A): set-valued functors on C, we denote by Nat (F. G) the collection of natural transformations from F to G. Then the following fundamental lemma[53, 2, 5, 22, 311 holds: LEMMA 7.1 (The Yoneda Lemma). For each object A of C, the map 0 : Nat(HA, F) -+ F(A) defined by 0 : 17H aA( 1~) is a bijection. For any small category C,consider
the functor category SetCop whose objects
are aJl con-
bavariant set-valued functors on C (such functors are called presheaves on C). Now define a = fog c”’ by y(A) = HA = C(-, A) for every object A of C and y&(g) mapy:C-+Set for C-arrows A s R and C -% A. Then y is a covariant functor and injective on objects. Moreover, by Yoneda Lemma, Nat(HA, HE) g C?(B, A) z C(A, B) for any objects A and Lz. Hence y is full and faithfull. So,we have proved the THEOREM
7.1 (The Yoneda Embedding).
For any small category C, the functor y is a full and faithful covariant embedding of C into Set cop . Dually, we have
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THEOREM
7.2 (The Contravariant Yoneda Embedding).
the functor y’ is a full and faithful contravariant
For any small category C.
embedding of C into SetC.
Recall that Remark 4.2 and [35, 27, 51. (1) If C is a small category, then the category Set c”” . is a topos. (2) If C is a small category, then the category Set
REMARK
7.1.
C. IS a topos.
Then from propositions 5.1 and 6.1 we have the following topos models of MI: PROPOSITION
7.1.
(1) The composition y* o r* of contravariant
functors y’ and r* :
Ml 5 G z Set” gives a covariant embedding of M into !3etG which is full and faithful and preserves transversal pullbacks. (2) The composition y o c+ of covariant functors y and c, : M 3 IL 3 SetLoPgives a covariant embedding of M into SetEO’ which is full and faithful and preserves transversal pullbacks. Also we have the following results: PROPOSITION
7.2. In SetG,
(1) the forgetful functor fi : G --+ Set : A ++ {zc(x E A) is represented by R(A) 2 G(f?‘(R). A). Thi s is a commutative ring with a unit. (2) the functor D = G(CI”“(R)/(x’). -) = {z E RJr2 = 0). So we have Kock-Lawvere axiom. PROPOSITION
7.3. In Sel?“‘“, R = &Z’“(R). and 11 = &(C;*“(R)/(z2)) = {z E R/z2 = 0).
Hence the topoi Set’
and Se?”
are the models of MI.
8. REFLECTION
PRINCIPLES
Now it will be necessary for us to review metamathematidy the topos models and their embeddings obtained in the previous sections by comparing them with G6del’s incompleteness results[l7]. Throughout this section, we consider only formal theories and describe crucial features of reflection principles used by K. G6del[l7] in the proof of his hrst incompleteness theorem (17, 481. To study an object theory 0, we select an arbitrary, but fixed, consistent meta theory M and construct an embedding r1 which maps every terms and relations upof 0 into a closed term r(pl of M, called the Giidel number (or code) of cp. Such an encoding is called as a Giidel numbering ( numbering for short). In order to perform a Godel numbering of a given object theory 0, it is sufficient that all primitive recursive functions are definable in the metatheory M. Now 6x a Godel numbering and let provable0 (,X1, ‘Y I) be the predicate in M corrwtly representing the provability relation X ko Y of 0, where X and Y are a finite sequence of sentences of c) and a sentence of 0, respectively. In addition, all necessary meta concepts “about” 0, such as formula, inference, and provability, should be encoded by this numbering. Then the provability relation t-0 of the object theory 0 is weakZy representable in the meta theory M by the predicate provableo. X F_s Y
if and only if
I-,Q provableB(‘X1,
rY1).
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Let AXIOII4 be the set of axioms of the theory 0 and define the predicate theorem0 in M by theoremn( Y) ++ provableo(’ AXZOW. r Y’). Then th e fo ll owing symmetric rules are derived, which is a meta meta rule commonly called reflection rules: t-+, theorem0 ( r X’)
to -_----x .t,q theoremo(‘X-‘)
k‘7 X
This meta meta rule is usually called reflection principle. We usually assume that M contains 0, but no problem arises if we let M = 0. Then we have the self-reflection of theory (3( or M). Yaw, by Cantor’s diagonal argument, we have: THEOREM 8.1 (Diagonalization Lemma). sentence A such that t-0 A - p(‘A1).
Sow let cp(z) = +heorem(‘?) ness theorem of Godel:
Given any unary formula V(Z), there is a
in the diagonalization
THEOREM 8.2 (Incompleteness nor 7A is provable in 0.
of 0).
Let to
lemma, we have the first incomplete-
A ++ +heorem(‘A’).
Then neither A
Whereas the embeddings defined in Proposition 7.1 gives the reflections between the theories, the resulting models are somehow too big and too coarse. Moreover, the encoding of meta concepts such as R and n are not properly made in Set”. To eliminate the defects, we shall henceforth consider only the topos SetLoP and introduce a Grothendieck topology into it. 9. A BETTER
MODEL
a
We consider here only the category IL. DEFINITION
9.1.
(1) A sieve on an object e E !L is a class S of morphisms in IL with
codomain C such that if A -f, (’ E S then 13 ‘3 (: E S’for any objects A. H and any morphism H 4 A. (2) A Grothendieck topology on IL is a function J assigning to each (J E IL a class .I(C:) of sieves on (J, such that (a) the maximal sieve {flcodon~irt(f) = C> is in J(C). (b) (stability) If S E .I((;‘) and { I1 f, C) is any morphism in IL, then f*(S) n/f 0 g E S} E J(
:= { ki
(c) (transitivity) if ,E E .I(<:) and and 1’ IS a sieve on (J such that for each 0 1 S. f’(7’) E J(c,“), then 7’ E J(C). If S E J(C) , S is said to be a .I -cover. (3) The pair (IL, J) is called a site. Following Moerdijk and Reyes[35], define the site lI%with a weaker Grothendieck on IL, generated by covering families of the form:
( ,’E
topology
(1) (finiteopencovers) {l((?(lf,)/(Zltl,) ‘-)C(Gm(R)/f}r~l,where{I1,.... .lI,}isafinite open cover of R”, and (2) (projections along non-trivial loci) all singleton covers of the form 1(A) x 1(H) -+ t(A), where L(H) # 0: if t(A) # 0. and let B be the topos of sheaves on l5(Recall Remark 4.2). Then:
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PROPOSITION 9.1. (Moerdijk and Reyes[35]) The Grothendieck topology on B is sub canonical. Hence the Yoneda embedding IL it !%d’“’ ( = I5 - SetBop) factors through B c Set”“. In particular, there is a full and faithful embedding of manifolds h4 Y B which preserves transversal pullbacks. The model B shows good properties in metamuthematicd sense, although its construction is technically rather complicated. The best model is not known to date from the logician’s viewpoint. Further scheme-theoretic study of B is necessary but an open problem. REFERENCES 1. Artin, M., Grothendieck: A. & Verdier, J. L., Thiorie des Topos et Cohomologie l&ale des schemas,(SGA-l), Lecture Notes in Mathematics 269 and 270, Springer-Verlag, Berlin (1972). 2. Bell, J. L., Toposes and Local Set Theories, Oxford University Press (1988). 3. Beeson, M. J., Foundations of Constructive Mathematics, Springer-Verlag, Berlin (1985). 4. Boileau, A. et Joyal, A., La Logique des Topos, J. Symbolic Logic, Vol. 46, No. 1, (1981) 616. 5. Borceux, F., Handbook of Categorical Algebm, 3 VO~UXXS,Cambridge University Press (1994). 6. Bunge, M. & Heggie, M., Synthetic Calculus of Variations, in Mathematical Applications of Category Theory (Edited by J.W. Gray), Contemporary Mathematics 30, Amer. Math. Sot. (1984), 30-62. 7. Cartan, E., Lecon sur la gdom&ie des espaces de Riemann, 2’m” edition, Gauthier-Villars, Paris (1945). 8. Coleman, B. D. & Owen, D. R., Recent Research on the Foundations of Thermodynamics, in [30], 83-99. 9. Diaconescu, R., Axiom of Choice and Complementation, Proc. Amer. Math. Sot., 51, (1975), 176178. 10. Dubuc, E. J., Sur les mod&e de la geometric diff&entielle, Cahiers de Topologie et G&ometrie Di&+entielle, 20, (1979) 231-279. 11. -: C” schemes, Amer. J. Math. 103, (1981), 683-690. 12. Eisenbud, D. & Harris, J., Schemes: The Language of Modern Algebraic Geometry, Wadsworth, Belmont, CA (1992). 13. Feferman, S., Constructive Theories of Functions and Classes, in Logic Colloquium ‘78(Edited by M. Boffa et al), North-Holland, Amsterdam (1979) 159-224. 1-l. A Theory of Variable Types, Rev&a Colombiana de Matemdticas Vol. XIX (1985), 95-105. 15. Gentzen, G., Untersuchungeniiber das logische Schliessen, Mathematische Zeitschtift 39 (1935) 176-210, 405431 : English transl.in The Collected Papers of Gerhard Gentzen (Edited by M. E. Szabo), NorthHolland, Amsterdam, London (1969) 68-131. 16. Girard, J.-Y_ et al., Proofs and Qpes, Cambridge University Press (1989). 17. Godel, K., Uber formal unentsheidbare S.&seder Principia mathematics und verwandter Systeme I, Monatshefte fir Mathematik und Physik, 38, (1931), 173-198 : English transl. in tirn F+cge To Gijdel (Edited by J. van Heijenoort), pp. 144-195. Harvard University Press (1967), and Kurt Giidel Collected Works Vol.1 (Edited by S. Feferman et al), pp. 145-195. Oxford University Press (1986) 18. Gordon, M. J. C. and Melham, T. F., eds., Introduction to HOL, Cambridge University Press (1993). 19. Grothendieck, A.; The cohomology theory of abstract algebraic varieties, Proc. Znt. Congr. A4ath., Edinburgh, 1958. Cambridge University Press (1960), 103-118. 20. Hirose, K., Kikyo, H., Kakehi, K., Yamada, S., & Doi, S., Use of Meta-Knowledge on Mathematical Structures in Theorem Proving, The 3rd Asian Conf. in h4athematical Logic, Beijing, China (1987). 21. Hirose, K.; Kikyo, H. & Yamada, S., An Experimental Theorem Prover using Metaknowledge on Mathematical Structures (Japanese), Proc. The 97th National Conf. of Information Processing Society of Japan (1988). 22. Johnstone, P. T., Topos Theory, Academic Press, London (1977). 23. Keisler, H. J., Foundations of Infinitesimal Calculus, Prindle, Weber and Schmidt, Boston (1976). 24. Kock, A., ed., Topos Theoretic n4ethods in Geometry, Matematsk Institut, Aarhus Universitet, Var. Publ. Ser. 30 (1979). 25. __ , Synthetic Differential Geometry, London Mathematical Society Lecture Note Series 51, Cambridge University Press (1981). 26. Lambek, J. & Scott,P. J., Introduction to higher order categoricd logic, Cambridge Univerdity Press (1986).
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27. Lavendhomme, R.. Basic Concepts of Synthetzr Diflerentinl Geometry, Kluwer Academic Publishers, Dordrecht, Boston, London (1996). 28. Lawvere. F. W., Variable Sets Etendu and Variable Structure in Topoi, Notes by S. Landsburg, Lecture Notes in Mathematics, The University of Chicago (1975); pp.17. 29 -pi Categorical Dynamics, in [X], pp. 1 -28. 30. Lawvere, F. W. & Schanuel, S. H., eds., Categories in Continuum PhysZca, Lecture 4otes in Mathematics 1174, Springer-Verlag, Berlin Heidelberg New York Tokyo (1986). 31. ?IIac Lane: S. & I. l\Ioerdijk, I., Sh eaves in Geometry and Logic, Springer-Verlag, New York (1992). 32. UcLarty, C., Elementaq Categories, Elementary Toposes, Oxford University Press (1992). 33. Mikusinski. J., Operational Calculus, Pergamon Press, London (1957). 34. hIoerdijk, I. & Reyes, G. E.. A Smooth Version of the Zariski Topos, Advances in hlathematics 65, (1987); 229-253. 35. ___. Models for Smooth Znfinitesimd Analysis, Springer-Verlag, Sew York (1991). 36. Noll, N., The Foundations of Ilfechanics and Thermodynamics, Springer-Verlag, Berlin (1974). 37. -Continuum Mechanics and Geometric Integration Theory, in [30], 17-29. 38. Osius, G., Logical and Set Theoretical Tools in Elementary Topoi, Model Theory and Topoi, (Edited by F.W. Lawvere et al), Lecture Notes in Mathematics 445, Springer-Verlag, Berlin (1975), 297-354. 39. L.C. Paulson, L. C., ML for the Working Programmer, Cambridge University Press (1992). Isabel/e, Lecture Notes in Computer Science 828, Springer-Verlag, Berlin (1994). 10. -1 41. van &UC, N. & Reyes, G. E., Smooth Functom and Synthetic Calculus, in The L.E.J.Brouwer Centenary Symposinm (Edited by A.S.Troelstra and D. van Dalen), North-Holland, Amsterdam (1982), 377-395. 52. Reyes, G. E., Synthetic reasoning and Variable sets, in [30], 69-82. 43. de Rham, G., Vari&s difl&entiables, Hermann, Paris (1955): English transl., Differentiable Manifolds, Springer-Verlag, Berlin Heidelberg New York Tokyo (1984). -44. Robinson, A., Non-Standad Analysis, Revised edition, North-Holland, Amsterdam (1971). 15. Schwartz, L.: The’orie des distributions, Hermann, Paris, I (1950); II (1951). 46. M&thodes Math6matique.s pour les Sciences Physiques, Hermann, Paris (1961). 47. Shafarevich, Igor R., Basic Algebraic Geometry 2: Schemes and Complex Manifolds, Second Edition, Springer-Verlag, Berlin, Heidelberg (1994). 48. Smorynski, C., The incompleteness theorems, in Handbook of Mathematical Logic (Edited by J. Barwise),North-Holland, Amsterdam, New York, Oxford (1977), 821-865. 49. Tennison, B. R., Sheaf Theory, London Math. Sot. Lecture Note Series 20, Cambridge University Press (1975). 50. Weil, A., Tht%rie des points proches sur les vari&?s diff&entiables, Colloque de Ge’ometrie DifJ&ntielle, C.N.R.S.(1953), Ill-117 : also in We& A; Collected Papers: Oeuores Scientijiques, Vol. 2, 103-109, Springer-Verlag, Berlin (1979). 51. Williams, W. O., Structure of Continuum Physics, in [30], 30-37. 52. Yamada, S., Computer-Aided Synthetic Reasoning in Smooth Infinitesimal Calculus, Proc. 5th Znternational Colloquium on Differential Equations (Vol.Z), (1995), 258-267. 53. Yoneda, N., On the homology theory of modules, J. Fat. Sci. Tokyo, Sec.f. 7 (1954), 193-227. 54. Yosida, K., Operational Calculus: Viewed as a thwq of diatributions(Japane), UP Applied Mathematics Series 5, 3rd Printing, The University of Tokyo Press (1993).