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14 pages, Republished in: Reprints in Theory and Applications of Categories, No. 15 (2006), pp 1-13

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Originally published in: Diagonal arguments and cartesian closed categories, Lecture Notes in Mathematics, 92 (1969), 134-145, used by permission. 1. Exponentiation, surjectivity, and a fixed-point theorem. 2. Russell' s Paradox is a case of Cantor' s theorem; natural functionals in recursive function theory and smooth manifold theory 3.3. Presentation-free formulations of satisfaction, truth, and provability according to Goodel and Tarski; representability vs. definability. Cited from: Author's Commentary-- "In May 1967 I had suggested in my Chicago lectures certain applications of category theory to smooth geometry and dynamics, reviving a direct approach to function spaces and therefore to functionals. Making that suggestion more explicit led later to elementary topos theory as well as to the line of research now known as synthetic differential geometry. The fuller development of those subjects turned out to involve a truth value object that classifies subobjects, but in the present paper (presented in the 1968 Battelle conference in Seattle) I refer only to weak properties of such an object; it is the other axiom, cartesian closure, that plays the central role. Daniel Kan had recognized that the function space construction for simplicial sets and other categories is a right adjoint, thus unique. Because this uniqueness property of adjoints implies their main calculational rules, I took the further axiomatic step of defining functor categories as a right adjoint to the finite product construction in my 1963 thesis. In 1965, Eilenberg and Kelly introduced the term closed to mean that there is a hom functor valued in the category itself. Such a hom functor is characterized in a relative way as right adjoint to a given tensor product functor; we concentrate here on the absolute case where the tensor is cartesian. Although the cartesian-closed view of function spaces and functionals was intuitively obvious in all but name to Volterra and Hurewicz (and implicitly to Bernoulli), it has counterexamples within the rigid framework advocated by Dieudonne and others. According to that framework the only acceptable fundamental structure for expressing the cohesiveness of space is a contravariant algebra of open sets or possibly of functions. Even though such algebras are of course extremely important invariants, their nature is better seen as a consequence of the covariant geometry of figures. Specific cases of this determining role of figures were obvious in the work of Kan and in the popularizations of Hurewicz' s k--spaces by Kelley, Brown, Spanier, and Steenrod, but in the present paper. I made this role a matter of principle: the Yoneda embedding was shown to preserve cartesian closure, and naturality of functionals was shown to be equivalent to Bernoulli' s principle. Further, I posed the problem of comparing this principle to practice in the specific cases of smooth and recursive mathematics. Later detailed work on those particular cases justified the classical intuition embodied in my general definition. In their books Froelicher and Kriegl (1988), and Kriegl and Michor (1997), extensively develop smooth analysis; their higher-order use of the adequacy of figures is based in part on a lower-order result of Boman 1967 (implicit in Hadamard) concerning the adequacy of paths. They cite the result of Lawvere-Schanuel-Zame showing that the natural functionals in this case are indeed the distributions of compact support, as practice would suggest. Nilpotent infinitesimals fall far short of even one-dimensionality, but if taken to be non-commutative, are already adequate for holomorphic functions, as was strikingly shown by Steve Schanuel (1982). The recursive example was studied by Phil Mulry (1982) who constructed a topos that does include as full sub-categories both the Banach-Mazur and the Ersov versions of higher recursive functionals. I hope that in the future this adequacy of one-dimensional figures will be explained because it occurs in many different examples. Many kinds of cohesion (algebraic geometry, smooth geometry, continuous geometry) are well-expressed as a subtopos of the classifying topos of a finitary single-sorted algebraic theory. But often that algebraic theory is determined by its monoid M of unary operations via naturality only: for example, the binary operations, instead of being independently specified, are just the maps of right M-sets from M2 to M. If a common explanation can be found (for this adequacy of onedimensional considerations in the determination of n-dimensional and infinite-dimensional functionals, in so many disparate cases) it would further establish that the Eilenberg-Mac Lane notion of naturality is far more powerful than the mere tautology it is sometimes considered to be. The original aim of this article was to demystify the incompleteness theorem of Godel and the truth-definition theory of Tarski by showing that both are consequences of some very simple algebra in the cartesian--closed setting. It was always hard for many to comprehend how Cantor' ™s mathematical theorem could be re-christened as a "paradox" by Russell and how Goodel' ™s theorem could be so often declared to be the most significant result of the 20th century. There was always the suspicion among scientists that such extra-mathematical publicity movements concealed an agenda for re-establishing belief as a substitute for science. Now, one hundred years after Godel' s birth, the organized attempts to harness his great mathematical work to such an agenda have become explicit. " 2000 MSC: 08-10, 02-00 Republished in: Reprints in Theory and Applications of Categories, No. 15 (2006), pp 1-13. http://www.tac.mta.ca/tac/reprints/articles/15/tr15.dvi http://www.tac.mta.ca/tac/reprints/articles/15/tr15.ps http://www.tac.mta.ca/tac/reprints/articles/15/tr15.pdf TAC Reprints Home Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Email: wlawvere@buffalo.edu This article may be accessed from http://www.tac.mta.ca/tac/reprints or by anonymous ftp access from ftp://ftp.tac.mta.ca/pub/tac/html/tac/reprints/articles/15/tr15.{dvi,ps}

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http://www.tac.mta.ca/tac/reprints/articles/15/tr15abs.html Originally published in: Diagonal arguments and cartesian closed categories, Lecture Notes in Mathematics, 92 (1969), 134-145, used by permission.

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