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Feferman on Foundations pp 3–22Cite as

From Choosing Elements to Choosing Concepts: The Evolution of Feferman’s Work in Model Theory

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Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 13))

Abstract

When Solomon Feferman began his research with Alfred Tarski in the early 1950s, model theory was still in process of becoming a distinct part of mathematical logic. Although Feferman’s doctoral thesis was not in model theory, his interests included model theory from the start, and he published a paper in the field roughly once every six years throughout his career. His earliest work in model theory is recognised in the name ‘Feferman-Vaught theorem’, which stems from some very detailed bare-hands work on sums and products of structures. During the 1960s and 1970s he worked on applications of many-sorted interpolation theorems, in particular to derive results relating implicit and explicit definability in various contexts. In the 1980s he edited with Jon Barwise a monumental collection of essays on ‘Model-theoretic logics’. In more recent papers he reflected on the conceptual basis of model theory from a historical point of view.

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References

  1. Barwise, J., Feferman, S. (eds.): Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer, New York (1985)

    Google Scholar 

  2. Beth, Evert W.: On Padoa’s method in the theory of definition. Nederl. Akad. Wetensch. Proc. Ser. A (= Indagationes Mathematicae 15) 56, 330–339 (1953)

    Google Scholar 

  3. Craig, W.: Review of [2]. J. Symb. Log. 21(2), 194–195 (1956)

    Article  Google Scholar 

  4. Craig, W.: Linear reasoning: a new form of the Herbrand-Gentzen theorem. J. Symb. Log. 22(3), 250–268 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  5. Craig, W.: Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory. J. Symb. Log. 22(3), 269–285 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  6. Derakhshan, J., Macintyre, A.: Some supplements to Feferman-Vaught related to the model theory of adeles. Ann. Pure Appl. Log. 165(11), 1639–1679 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Doner, J.E., Mostowski, A., Tarski, A.: The elementary theory of well-ordering – a metamathematical study. In: Macintyre, A., Pacholski, L., Paris, J. (eds.) Logic Colloquium 77, pp. 1–54. North-Holland, Amsterdam (1978)

    Google Scholar 

  8. Ehrenfeucht, A.: Application of games to some problems of mathematical logic. Bull. de l’Acad. Pol. Sci. Cl 3(5), 35–37 (1957)

    MathSciNet  MATH  Google Scholar 

  9. Eklof, P.C.: Applications to algebra, in [1] pp. 423–441

    Google Scholar 

  10. Evans, David M., Hodges,W., Hodkinson, Ian M.: Automorphisms of bounded abelian groups. Forum Math. 3, 523–542 (1991)

    Google Scholar 

  11. Feferman, A.B., Feferman, S.: Alfred Tarski: Life and Logic. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  12. Feferman, S.: Some recent work of Ehrenfeucht and Fraisse. In: Proceedings of the Summer Institute of Symbolic Logic, Cornell Ithaca (1957), Communications Research Division, pp. 201–209. Institute for Defense Analyses, Princeton NJ, (1960)

    Google Scholar 

  13. Feferman, S.: Lectures on proof theory. In: Proceedings of the Summer School in Logic, Leeds 1967, Lecture Notes in Mathematics 70. pp. 1–107. Springer, Berlin (1968)

    Google Scholar 

  14. Feferman, S.: Persistent and invariant formulas for outer extensions. Compos. Math. 20, 29–52 (1968)

    MathSciNet  MATH  Google Scholar 

  15. Feferman, S.: Ordinals and functionals in proof theory. Actes du Congrès Internationale des Mathématiciens 1, 229–233. Nice (1970)

    Google Scholar 

  16. Feferman, S.: Infinitary properties, local functors, and systems of ordinal functions. In: Hodges, w. (ed.) Conference in Mathematical Logic: London 1970, Lecture Notes in Mathematics 255, pp. 63–97. Springer, Berlin (1972)

    Google Scholar 

  17. Feferman, S.: Applications of many-sorted interpolation theorems. In: Henkin, L. et al., (eds.) Proceedings of the Tarski Symposium. pp. 205–223. American Mathematical Society, Providence RI (1974)

    Google Scholar 

  18. Feferman, S.: ‘Two notes on abstract model theory. I. Properties invariant on the range of definable relations between structures. Fundam. Math. 82, 153–165 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Feferman, S.: Two notes on abstract model theory. II. Languages for which the set of valid sentences is semi-invariantly implicitly definable. Fundam. Math. 89, 111–130 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  20. Feferman, S.: Generalizing set-theoretical model theory and an analogue theory on admissible sets. In: Hintikka, J., Niiniluoto, I., Saarinen, E. (eds.) Essays on Mathematical and Philosophical Logic: Proceedings of the Fourth Scandinavian Logic Symposium and of the First Soviet-Finnish Logic Conference, Jyväskylä, Finland, June 29–July 6, 1976, pp. 171–195. Reidel, Dordrecht (1979)

    Google Scholar 

  21. Feferman, S.: Three conceptual problems that bug me. Draft lecture notes for 7th Scandinavian Logic Symposium, Uppsala, August 18–20. (1996). math.stanford.edu/\(\sim \)feferman/papers/conceptualprobs.pdf

    Google Scholar 

  22. Feferman, S.: Logic, logics, and logicism. Notre Dame J. Form. Log. 40(1), 31–54 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Feferman, S.: Ah, Chu!. In: JFAK, Essays Dedicated to Johan Van Benthem on the Occasion of his 50th Birthday (1999) (compact disc; the paper is available at math.stanford.edu/\(\sim \)feferman/papers)

    Google Scholar 

  24. Feferman, S.: Tarski’s conception of logic. Ann. Pure Appl. Log. 126(1), 5–13 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Feferman, S.: What kind of logic is Independence-Friendly logic?. In: Randall E. Auxier., Hahn, L.E. (eds.) The Philosophy of Jaakko Hintikka, vol. XXX, pp. 453–469. Library of Living Philosophers, Open Court, Chicago IL (2006)

    Google Scholar 

  26. Feferman, S.: Tarski’s conceptual analysis of semantical notions. In: Patterson, D. (ed.) New Essays on Tarski and Philosophy, pp. 72–93. Oxford University Press, Oxford (2008)

    Chapter  Google Scholar 

  27. Feferman, S.: Harmonious logic: Craig’s interpolation theorem and its descendants. Synthese 164(3), 341–357 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Feferman, S.: Andrzej Mostowski: An Appreciation. In: Ehrenfeucht, A., Marek, V.W., Srebrny, M. (eds.) Andrzej Mostowski and Foundational Studies, pp. 388–392. IOS Press, Amsterdam (2008)

    Google Scholar 

  29. Feferman, S.: Set-theoretical invariance criteria for logicality. Notre Dame J. Form. Log. 51(1), 3–20 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Feferman, S.: Which quantifiers are logical? A combined semantical and inferential criterion. Preprint 2013, math.stanford.edu/\(\sim \)feferman/papers/ WhichQsLogical(text).pdf

    Google Scholar 

  31. Feferman, S.: A fortuitous year with Leon Henkin. In: Manzano, M., Sain, I., Alonso, E. (eds.) The Life and Work of Leon Henkin, pp. 35–40. Birkhäuser, Heidelberg (2014)

    Google Scholar 

  32. Feferman, S., Kreisel, G.: Persistent and invariant formulas relative to theories of higher order. Bull. Am. Math. Soc. 72(3), 480–485 (1966)

    Google Scholar 

  33. Feferman, S., Vaught, R.: The first order properties of products of algebraic systems. Fundam. Math. 47, 57–103 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  34. Felscher, I.: Model composition in model-checking, Doctoral Dissertation, Hochschulbibliothek der Rheinisch-Westfälischen Technischen Hochschule Aachen, (2014)

    Google Scholar 

  35. Fraïssé, R.: Sur quelques classifications des relations, basées sur des isomorphismes restreints. II. Applications aux relations d’ordre, et construction d’exemples montrant que ces classifications sont distinctes. In: Publ. Sci. de l’Université d’Alger, Série A (Mathematiques), vol. 2, pp. 273–295. (1957)

    Google Scholar 

  36. Gaifman, H.: Operations on relational structures, functors and classes. I. In: Henkin, L. et al., (eds.) Proceedings of the Tarski Symposium. pp. 21–39. American Mathematical Society, Providence RI (1974)

    Google Scholar 

  37. Gurevich, Y.: Monadic Second-Order Theories, in [1] pp. 479–506

    Google Scholar 

  38. Hintikka, J.: Intellectual autobiography. In: Randall E. Auxier., Hahn, L.E. (eds.) The Philosophy of Jaakko Hintikka. Library of Living Philosophers vol. XXX, pp. 3–84. Open Court, Chicago IL (2006)

    Google Scholar 

  39. Hintikka, J.: Distributive Normal Forms in the Calculus of Predicates, Acta Philosophica Fennica 6. Philosophical Society of Finland, Helsinki (1953)

    Google Scholar 

  40. Hodges, W.: Tarski on Padoa’s Method: a test case for understanding logicians of other traditions. In: Chakraborti, Mihir K. et al., (eds.) Logic, Navya-Nyāya and Applications: Homage to Bimal Krishna Matilal, pp. 155–169. College Publications, London (2008)

    Google Scholar 

  41. Krajewski, S., Srebrny, M.: On the life and work of Andrzej Mostowski (1913–1975). In: Ehrenfeucht, A., Marek, V.W., Srebrny, M. (eds.) Andrzej Mostowski and Foundational Studies, pp. 3–14. IOS Press, Amsterdam (2008)

    Google Scholar 

  42. Kreisel, G.: Set theoretic problems suggested by the notion of potential totality, in Infinitistic Methods. Pergamon Press, Warsaw (1959)

    Google Scholar 

  43. Łoś, Jerzy: On the extending of models (I). Fundam. Math. 42, 38–54 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  44. Lyndon, Roger C.: An interpolation theorem in the predicate calculus. Pac. J. Math. 9, 129–142 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lyndon, Roger C.: Properties preserved under homomorphism. Pac. J. Math. 9, 143–154 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  46. Makkai, M.: An application of a method of Smullyan to logics on admissible sets. Bull. Pol. Acad. Sci. 17, 341–346 (1969)

    MathSciNet  MATH  Google Scholar 

  47. Makowsky, J.A.: Algorithmic uses of the Feferman-Vaught Theorem. Ann. Pure Appl. Log. 126(1), 159–213 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. Mal’tsev, Anatoliĭ I.: Model’niye sootvestvija, Izvestiya Rossiiskoi Akademii Nauk Seriya Matematicheskaya 23(3), pp. 313–336; trans. as ‘Model correspondences.: In Mal’cev, A.I. (ed.) The Metamathematics of Algebraic Systems: Collected Papers 1936–1967, ed. and trans. Benjamin Franklin Wells III, pp. 66–94. North-Holland, Amsterdam (1971)

    Google Scholar 

  49. Marker, D.: A model theoretic proof of Feferman’s preservation theorem. Notre Dame J. Form. Log. 25(3), 213–216 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  50. Mostowski, A.: On direct products of theories. J. Symb. Log. 17(1), 1–31 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  51. Mostowski, A., Tarski, A.: Arithmetical classes and types of well ordered systems. Preliminary report. Bull. Am. Math. Soc. 55, 65 (1949)

    Google Scholar 

  52. Otto, M.: An interpolation theorem. Bull. Symb. Log. 6(4), 447–462 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  53. Padoa, A.: Essai d’une théorie algébrique des nombres entiers, précédé d’une introduction logique à une théorie déductive quelconque. In: Bibliothèque du Congrès international de philosophie, Paris, 1900, vol. 3, Armand Colin, Paris 1902, pp. 309–365; translated in part as ‘Logical introduction to any deductive theory’ in From Frege to Gödel, ed. Jean van Heijenoort, pp. 118–123. Harvard University Press, Cambridge Mass. (1967)

    Google Scholar 

  54. Stern, J.: A new look at the interpolation problem. J. Symb. Log. 40(1), 1–13 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  55. Tarski, A.: Some notions and methods on the borderline of algebra and metamathematics. In: Proceedings of the International Congress of Mathematicians: Cambridge, Massachusetts. vol. 1, pp. 705–720. American Mathematical Society, Providence RI (1952)

    Google Scholar 

  56. Ulsen, Paul van.:E. W. Beth als logicus, Doctoral Dissertation, Institute for Logic, Language and Computation, Amsterdam (2000); ILLC Dissertation Series DS–2000–04

    Google Scholar 

  57. Väänänen, J.: Set-Theoretic Definability of Logics, in [1] pp. 599–643

    Google Scholar 

  58. Zucker, J.I.: The adequacy problem for classical logic. J. Philos. Log. 7(1), 517–535 (1978)

    MathSciNet  MATH  Google Scholar 

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  1. Herons Brook, Sticklepath, Okehampton, EX20 2PY, England

    Wilfrid Hodges

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  1. Wilfrid Hodges
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Correspondence to Wilfrid Hodges .

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  1. Institut für Informatik, Universität Bern, Bern, Switzerland

    Gerhard Jäger

  2. Department of Philosophy, Carnegie Mellon University Dept. Philosophy, Pittsburgh, Pennsylvania, USA

    Wilfried Sieg

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Hodges, W. (2017). From Choosing Elements to Choosing Concepts: The Evolution of Feferman’s Work in Model Theory. In: Jäger, G., Sieg, W. (eds) Feferman on Foundations. Outstanding Contributions to Logic, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-63334-3_1

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