Abstract
The purpose of this paper is to justify the claim that Topos theory and Logic (the latter interpreted in a wide enough sense to include Model theory and Set theory) may interact to the advantage of both fields. Once the necessity of utilizing toposes (other than the topos of Sets) becomes apparent, workers in Topos theory try to make this task as easy as possible by employing a variety of methods which, in the last instance, find their justification in metatheorems from Logic. Some concrete instances of this assertion will be given in the form of simple proofs that certain theorems of Algebra hold in any (Grothendieck) topos, in order to illustrate the various techniques that are used. In the other direction, Topos theory can also be a useful tool in Logic. Examples of this are independence proofs in (classical as well as intuitionistic) Set theory, as well as transfer methods in the presence of a sheaf representation theorem, the latter applied, in particular, to model theoretic properties of certain theories.
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References
Artin, M.: 1962, ‘Grothendieck Topologies’, Lecture Notes, Harvard University.
Bunge, M.: 1974, ‘Topos theory and Souslin's hypothesis’, J. Pure and Applied Algebra 4, 159–187.
Bunge, M.: 1981, ‘Sheaves and Prime model extensions’, J. of Algebra 68, 79–96.
Bunge, M.: 1982, ‘On the transfer of an abstract nullstellensatz’, Comm. in Algebra 10 (17), 1891–1906.
Bunge, M.: 1983, ‘Synthetic aspects of C ∞-mappings’, J. Pure and Applied Algebra 28.
Bunge, M. and G. E. Reyes: 1981, ‘Boolean spectra and model completions’, Fundam. Mathem. CXIII, 165–173.
Bunge, M. and M. Heggie: forthcoming, ‘Synthetic Calculus of Variations’ (to appear in the Proceedings of the Special Session on Applied Category Theory of the 89th Annual meeting of the Amer. Math. Soc., Denver, Jan. 1983).
Boileau, A. and A. Joyal: 1981, ‘La logique des topos’, J. of Symb. Logic 46, 6–16.
Fourman, M. and M. Hyland: 1979, ‘Sheaf models for analysis’, in Applications of Sheaves, Springer Lecture Notes in Math. 753, 280–301.
Fourman, M. and D. Scott: 1979, ‘Sheaves and logic’, in Applications of Sheaves, Springer Lecture Notes in Math. 753, 302–402.
Freyd, P.: 1972, ‘Aspects of Topoi’, Bull. of the Australian Math. Soc. 7, 1–76.
Goldblatt, R.: 1979, Topoi, the Categorical Analysis of Logic, North-Holland, Amsterdam.
Gray, J.: 1971, ‘The meeting of the Midwest Category Seminar in Zürich’, in Reports of the Midwest Category Seminar, V. Springer Lecture Notes in Mathem. 195, 248–255.
Grayson, R.: 1981, ‘Concepts of general topology in constructive mathematics and in sheaves’, Annals of Math. Logic 20, 1–41.
Grothendieck, A. and J. L. Verdier: 1972, ‘Théorie des Topos’ (SGA 4, exposés I–VI) Springer Lecture Notes in Math. 269–270.
Heggie, M.: 1982, An Approach to Synthetic Variational Theory, M.Sc. thesis, McGill University.
Johnstone, P. T.: 1977, Topos Theory, Academic Press, New York.
Johnstone, P. T.: 1977, ‘Rings, fields, and spectra’, J. of Algebra 49, 238–260.
Joyal, A.: 1975, ‘Les théorèmes de Chevalley-Tarski et remarques sur l'Algèbre constructive’, Cahiers de Top. et Géom. Diff. XVI, 256–258.
Kock, A. and G. E. Reyes: 1977, ‘Doctrines in categorical logic’, in J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam.
Kock, A.: 1981, Synthetic Differential Geometry, London Mathem. Soc. Lecture Notes Series 51.
Lambek, J. and P. Scott: 1981, ‘Independence of premisses and the free topos’, Proceedings of Symposium in Constructive Mathematics, Springer Lecture Notes in Math. 873, 191–207.
Lawvere, F. W.: 1979 ‘Categorical dynamics’, in Topos Theoretic Methods in Geometry, Aarhus Math. Institute Var. Pub. Series No. 30.
Lawvere, F. W.: 1976, ‘Variable quantities and variable structures in topoi’, in A. Heller and M. Tierney (eds.), Algebra, Topology and Category Theory: a collection of papers in honor of Samuel Eilenberg, Academic Press, New York, 101–131.
Macintyre, A.: 1973, ‘Model completeness and sheaves of structures’, Fund. Math. 81, 73–89.
Maclane, S.: 1971, Categories for the Working Mathematician, Graduate Texts in Mathematics no. 5, Springer-Verlag.
Makkai, M. and G. E. Reyes: 1977, First Order Categorical Logic, Springer Lecture Notes in Math. 611.
Penon, J.: 1981, ‘Infinitésimeaux et intuitionisme’, Cahiers de Top. et Géom. Diff. 22, 67–72.
Reyes, G. E.: 1974, ‘From Sheaves to Logic’, in A. Daigneault (ed.), Studies in Algebraic Logic, Studies in Math. 9, The Mathem. Assoc. of America.
Scedrov, A.: 1981, ‘Sheaves and forcing and their mathematical applications’, Ph.D. Thesis, SUNY at Buffalo, to appear in Memoirs Amer. Math. Soc.
Tierney, M.: 1973, ‘Axiomatic sheaf theory: Some constructions and applications’, Proc. CIME Conference on Categories and Commutative Algebra, Varenna, 1971, Edizioni Cremonese, 249–326.
Wraith, G.: 1975, ‘Lectures on Elementary Topoi’, in Model Theory and Topoi, Springer Lecture Notes in Math 445, 114–206.
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Bunge, M. Toposes in Logic and Logic in Toposes. Topoi 3, 13–22 (1984). https://doi.org/10.1007/BF00136116
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DOI: https://doi.org/10.1007/BF00136116