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Infinite Populations, Choice and Determinacy

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Abstract

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow’s theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted axiom of choice do not meet this criterion. The technically novel part of this paper is a proposal to use a set-theoretic principle known as the axiom of determinacy (\(\mathsf {AD}\)), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of \(\mathsf {AD}\) from the point of view of the research area in question is its game-theoretic character.

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References

  1. Aumann, R. J., Markets with a continuum of traders, Econometrica 32(1–2):39–50, 1964.

    Article  Google Scholar 

  2. Bedrosian, G., A. Palmigiano, and Z. Zhao, Generalized ultraproduct and Kirman-Sondermann correspondence for vote abstention, in Proceedings of LORI 2015, 2015, pp. 27–39.

    Chapter  Google Scholar 

  3. Bell, J. L., The axiom of choice, in E. N. Zalta, (ed.), The Stanford Encyclopedia of Philosophy, summer 2015 edn., http://plato.stanford.edu/archives/sum2015/entries/axiom-choice/ 2015.

  4. Bell, J. L., Toposes and Local Set Theories: an Introduction, vol. 14 in Logic Guides, Oxford University Press, 1988.

  5. Bernays, P., A system of Axiomatic Set Theory: Part III. Infinity and enumerability. Analysis, The Journal of Symbolic Logic 7(2):65–89, 1942.

    Article  Google Scholar 

  6. Boolos, G., The iterative conception of set, The Journal of Philosophy 68(8):215–231, 1971.

    Article  Google Scholar 

  7. Brunner, N., and H. R. Mihara, Arrow’s theorem, Weglorz’ models and the Axiom of Choice, Mathematical Logic Quarterly 46(3):335–359, 2000.

    Article  Google Scholar 

  8. Chichilnisky, G., and G. Heal, Social choice with infinite populations: construction of a rule and impossibility results, Social Choice and Welfare 14(2):303–318, 1997.

    Article  Google Scholar 

  9. Diaconescu, R., Axiom of choice and complementation, Proc. Amer. Math. Soc. 51(1):176–178, 1975.

    Article  Google Scholar 

  10. Fishburn, P. C., Arrow’s impossibility theorem: concise proof and infinite voters, Journal of Economic Theory 2(1):103–106, 1970.

    Article  Google Scholar 

  11. Forster, T., The iterative conception of set, The Review of Symbolic Logic 1:97–110, 2008.

    Article  Google Scholar 

  12. Fraenkel, A. A., Y. Bar-Hillel, and A. Levy, Foundations of Set Theory. Second Revised Edition, vol. 67 of Studies in Logic and the Foundations of Mathematics, Elsevier, 1973.

  13. Goodman, N., and J. Myhill, Choice implies excluded middle, Mathematical Logic Quarterly 24(25–30):461–461, 1978.

    Article  Google Scholar 

  14. Hamkins, J. D., The set-theoretic multiverse, Review of Symbolic Logic 5:416–449, 2012.

    Article  Google Scholar 

  15. Hellman, G., Does category theory provide a framework for mathematical structuralism?, Philosophia Mathematica 11(2):129–157, 2003.

    Article  Google Scholar 

  16. Herzberg, F., and D. Eckert, Impossibility results for infinite-electorate abstract aggregation rules, Journal of Philosophical Logic 41(1):273–286, 2011.

    Article  Google Scholar 

  17. Herzberg, F., and D. Eckert, The model-theoretic approach to aggregation: Impossibility results for finite and infinite electorates, Mathematical Social Sciences, Computational foundations of social choice 64(1):41–47, 2012.

    Google Scholar 

  18. Herzberg, F., L. Lauwers, L. van Liedekerke, and E. S. Fianu, Addendum to L. Lauwers and L. Van Liedekerke Ultraproducts and aggregation [J. Math. Econ. 24 (3) (1995)], Journal of Mathematical Economics 46(2):277–278, 2010.

    Article  Google Scholar 

  19. Hildenbrand, W., On economies with many agents, Journal of Economic Theory 2(2):161–188, 1970.

    Article  Google Scholar 

  20. Jech, T. J., The Axiom of Choice, vol. 75 of Studies in Logic and the Foundations of Mathematics, Elsevier, 1973.

  21. Kanamori, A., The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer Monographs in Mathematics, Springer Berlin Heidelberg, 2008.

    Google Scholar 

  22. Kechris, A. S., The axiom of determinancy implies dependent choices in L(\(\mathbb{R}\)), The Journal of Symbolic Logic 49(1):161–173, 1984.

    Article  Google Scholar 

  23. Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, Springer New York, 1985.

    Google Scholar 

  24. Kirman, A. P., and D. Sondermann, Arrow’s theorem, many agents, and invisible dictators, Journal of Economic Theory 5(2):267–277, 1972.

    Article  Google Scholar 

  25. Kuratowski, K., and A. Mostowski, Set Theory, vol. 53 of Studies in Logic and the Foundations of Mathematics, Elsevier, 1968.

  26. Lambek, J., and P. J. Scott, Introduction to Higher Order Categorical Logic, no. 7 in Cambridge studies in advanced mathematics, Cambridge University Press, 1986.

  27. Lauwers, L., Ordering infinite utility streams comes at the cost of a non-Ramsey set, Journal of Mathematical Economics 46(1):32–37, 2010.

    Article  Google Scholar 

  28. Lauwers, L., Intergenerational equity, efficiency, and constructibility, Economic Theory 49(2):227–242, 2012.

    Article  Google Scholar 

  29. Lauwers, L., and L. Van Liedekerke, Ultraproducts and aggregation, Journal of Mathematical Economics 24(3):217–237, 1995.

    Article  Google Scholar 

  30. Martin-Löf, P., 100 Years of Zermelo’s Axiom of Choice: What was the Problem with It?, Springer, Netherlands, Dordrecht, 2009, pp. 209–219.

    Book  Google Scholar 

  31. Martin-Löf, P., and G. Sambin, Intuitionistic type theory, Studies in proof theory, Bibliopolis, 1984.

    Google Scholar 

  32. McLarty, C., Two constructivist aspects of category theory, Philosophia Scientae (Cahier spécial 6):95–114, 2006.

  33. Mihara, R. H., Arrow’s theorem and Turing computability, Economic Theory 10(2):257–276, 1997.

    Article  Google Scholar 

  34. Mihara, R. H., Arrow’s theorem, countably many agents, and more visible invisible dictators, Journal of Mathematical Economics 32(3):267–287, 1999.

    Article  Google Scholar 

  35. Monjardet, B., Chapter 5 - on the use of ultrafilters in social choice theory, in P. K. Pattanaik, and M. Salles, (eds.), Social Choice and Welfare, vol. 145 of Contributions to Economic Analysis, Elsevier, 1983, pp. 73–78.

  36. Mycielski, J., On the axiom of determinateness, Fundamenta Mathematicae 53(2):205–224, 1964.

    Article  Google Scholar 

  37. Mycielski, J., On the axiom of determinateness (II), Fundamenta Mathematicae 59(2):203–212, 1966.

    Article  Google Scholar 

  38. Mycielski, J., and H. Steinhaus, A mathematical axiom contradicting the axiom of choice, Bulletin de l’Académie Polonaise des Sciences 10:1–3, 1962.

    Google Scholar 

  39. Mycielski, J., and S. Świerczkowski, On the Lebesgue measurability and the axiom of determinateness, Fundamenta Mathematicae 54(1):67–71, 1964.

    Article  Google Scholar 

  40. Ostroy, J. M., and W. R. Zame, Nonatomic economies and the boundaries of perfect competition, Econometrica 62:593–633, 1994.

    Article  Google Scholar 

  41. Parsons, C., What is the iterative conception of set?, in R. E. Butts, and J. Hintikka, (eds.), Logic, Foundations of Mathematics, and Computability Theory, vol. 9 of The University of Western Ontario Series in Philosophy of Science, Springer Netherlands, 1977, pp. 335–367.

  42. Pitts, A., Nominal techniques, ACM SIGLOG News 3(1):57–72, 2016.

    Google Scholar 

  43. Pitts, A. M., Nominal Sets: Names and Symmetry in Computer Science, vol. 57 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2013.

  44. Pivato, M., Additive representation of separable preferences over infinite products, Theory and Decision 77(1):31–83, 2014.

    Article  Google Scholar 

  45. Shelah, S., Can you take Solovay’s inaccessible away?, Israel Journal of Mathematics 48(1):1–47, 1984.

    Article  Google Scholar 

  46. Sierpiński, W., Sur une proposition qui entraîne l’existence des ensembles non mesurables, Fundamenta Mathematicae 34(1):157–162, 1947.

    Article  Google Scholar 

  47. Solovay, R. M., A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics 92(1):1–56, 1970.

    Article  Google Scholar 

  48. Sørensen, M. H., and P. Urzyczyn, Lectures on the Curry-Howard Isomorphism, vol. 149 of Studies in Logic and the Foundations of Mathematics, Elsevier Science Inc., New York, NY, USA, 2006.

    Chapter  Google Scholar 

  49. Steel, J. R., What is ...a Woodin cardinal?, Notices of the AMS 54(9):1146–1147, 2007.

  50. Svensson, L.-G., Equity among generations, Econometrica 48(5):1251–1256, 1980.

    Article  Google Scholar 

  51. Zame, W., Can intergenerational equity be operationalized?, Theoretical Economics 2:2, 2007.

    Google Scholar 

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  1. Informatik 8, FAU Erlangen-Nürnberg, Erlangen, Germany

    Tadeusz Litak

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Special Issue: Logics for Social Behaviour

Edited by Alessandra Palmigiano and Marcus Pivato

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Litak, T. Infinite Populations, Choice and Determinacy. Stud Logica 106, 969–999 (2018). https://doi.org/10.1007/s11225-017-9730-3

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