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Transactions of the American Mathematical Society

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Injectivity, projectivity, and the axiom of choice

Author: Andreas Blass
Journal: Trans. Amer. Math. Soc. 255 (1979), 31-59
MSC: Primary 04A25; Secondary 03E35, 20K99
MathSciNet review: 542870
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Abstract: We study the connection between the axiom of choice and the principles of existence of enough projective and injective abelian groups. We also introduce a weak choice principle that says, roughly, that the axiom of choice is violated in only a set of different ways. This principle holds in all ordinary Fraenkel-Mostowski-Specker and Cohen models where choice fails, and it implies, among other things, that there are enough injective abelian groups. However, we construct an inner model of an Easton extension with no nontrivial injective abelian groups. In the presence of our weak choice principle, the existence of enough projective sets is as strong as the full axiom of choice, and the existence of enough free projective abelian groups is nearly as strong. We also prove that the axiom of choice is equivalent to “all free abelian groups are projective” and to “all divisible abelian groups are injective."

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  • Peter Aczel, The type theoretic interpretation of constructive set theory: choice principles, The L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981) Stud. Logic Found. Math., vol. 110, North-Holland, Amsterdam, 1982, pp. 1–40. MR 717236, DOI
  • An algebraic equivalent of a multiple choice axiom, Fund. Math. 74 (1972), no. 2, 145–146. MR 290958
  • A. Blass, Two algebraic equivalents of the axiom of choice, Notices Amer. Math. Soc. 22 (1975), A-524.
  • Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
  • Paul J. Cohen, Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0232676
  • Keith I. Devlin and R. B. Jensen, Marginalia to a theorem of Silver, $\vDash $ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974) Springer, Berlin, 1975, pp. 115–142. Lecture Notes in Math., Vol. 499. MR 0480036
  • William B. Easton, Powers of regular cardinals, Ann. Math. Logic 1 (1970), 139–178. MR 269497, DOI
  • Serge Grigorieff, Intermediate submodels and generic extensions in set theory, Ann. of Math. (2) 101 (1975), 447–490. MR 373889, DOI
  • W. Hodges, private communication.
  • Thomas J. Jech, The axiom of choice, North-Holland Publishing Co., Amsterdam-London; Amercan Elsevier Publishing Co., Inc., New York, 1973. Studies in Logic and the Foundations of Mathematics, Vol. 75. MR 0396271
  • Ronald Björn Jensen, Modelle der Mengenlehre. Widerspruchsfreiheit und Unabhängigkeit der Kontinuum-Hypothese und des Auswahlaxioms, Lecture Notes in Mathematics, No. 37, Springer-Verlag, Berlin-New York, 1967 (German). Ausgearbeitet von Franz Josef Leven. MR 0221930
  • John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. MR 0070144
  • Azriel Lévy, Definability in axiomatic set theory. I, Logic, Methodology and Philos. Sci. (Proc. 1964 Internat. Congr.), North-Holland, Amsterdam, 1965, pp. 127–151. MR 0205827
  • Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
  • D. Morris, A model of ZF which cannot be extended to a model of ZFC without adding ordinals, Notices Amer. Math. Soc. 17 (1970), 577.
  • D. G. Northcott, An introduction to homological algebra, Cambridge University Press, New York, 1960. MR 0118752
  • Richard A. Platek, Eliminating the continuum hypothesis, J. Symbolic Logic 34 (1969), 219–225. MR 256872, DOI
  • Dock Sang Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700–712. MR 104721, DOI
  • J. R. Shoenfield, Unramified forcing, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 357–381. MR 0280359
  • B. L. van der Waerden, Algebra. II, 5th ed., Springer-Verlag, Berlin, 1967.

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Article copyright: © Copyright 1979 American Mathematical Society