Abstract
The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on the problem whether there are (enough) injective sets in CZF. We show that no two element set is injective unless the law of excluded middle is admitted for negated formulas, and that the axiom of power set is required for proving that “there are strongly enough injective sets”. The latter notion is abstracted from the singleton embedding into the power set, which ensures enough injectives both in every topos and in IZF. We further show that it is consistent with CZF to assume that the only injective sets are the singletons. In particular, assuming the consistency of CZF one cannot prove in CZF that there are enough injective sets. As a complement we revisit the duality between injective and projective sets from the point of view of intuitionistic type theory.
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References
Aczel P.: The type theoretic interpretation of constructive set theory. In: MacIntyre, A., Pacholski, L., Paris, J. (eds.), Logic Colloquium ’77, pp. 55–66. North–Holland, Amsterdam (1978)
Aczel P.: The type theoretic interpretation of constructive set theory: choice principles. In: Troelstra, A.S., van Dalen, D. (eds.), The LEJ Brouwer Centenary Symposium, pp. 1–40. North–Holland, Amsterdam (1982)
Aczel P., Crosilla L., Ishihara H., Palmgren E., Schuster P.: Binary refinement implies discrete exponentiation. Studia Logica 84, 361–368 (2006)
Aczel, P., and M. Rathjen, Notes on Constructive Set Theory, Draft available at the address: http://www.mittag-leffler.se/preprints/meta/AczelMon_Sep_24_09_16_56.rdf.htm.
Banaschewski B., Schuster P.: The shrinking principle and the axiom of choice. Monatshefte für Mathematik 151, 263–270 (2007)
Blass A.: Injectivity, projectivity, and the axiom of choice. Transactions of the American Mathematical Society 255, 31–59 (1979)
van den Berg B., Moerdijk I.: Aspects of predicative algebraic set theory II: realizability. Theoretical Computer Science 412, 1916–1940 (2011)
Crosilla L., Ishihara H., Schuster P.: On constructing completions. The Journal of Symbolic Logic 70, 969–978 (2005)
Devlin K.: Constructibility. Springer, Berlin and Heidelberg (1984)
Friedman H.: Set-theoretic foundations for constructive analysis. Annals of Mathematics 19, 868–870 (1977)
MacLane S.: Homology. Springer, Berlin and Heidelberg (1975)
MacLane S., Moerdijk I.: Sheaves in Geometry and Logic. A First Introduction to Topos Theory. Springer, New York (1992)
Martin–Löf, P., Intuitionistic Type Theory. Notes by G. Sambin of a series of lectures given in Padua, June 1980. Bibliopolis, Napoli. Studies Proof Theory 1, 1984.
Martin-Löf, P., 100 years of Zermelo’s axiom of choice: what was the problem with it?, in S. Lindström, E. Palmgren, K. Segerberg, and V. Stoltenberg-Hansen (eds.), Logicism, Intuitionism, and Formalism—What Has Become of Them? Springer, Dordrecht. Synthese Library 341:209–219, 2009.
McLarty C.: What does it take to prove Fermat’s Last theorem?. The Bulletin of Symbolic Logic 16, 359–377 (2010)
Mines R., Ruitenburg W., Richman F.: A Course in Constructive Algebra. Springer, New York (1987)
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Aczel, P., van den Berg, B., Granström, J. et al. Are There Enough Injective Sets?. Stud Logica 101, 467–482 (2013). https://doi.org/10.1007/s11225-011-9365-8
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DOI: https://doi.org/10.1007/s11225-011-9365-8