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

DOI:
https://doi.org/10.1090/S0002-9947-1979-0542870-6

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."

**[1]**P. Aczel,*Interpretations of constructive set theory*, (Logic Colloquium, 1977, Wroclaw), North-Holland, Amsterdam. MR**717236 (85g:03085)****[2]**M. K. Armbrust,*An algebraic equivalent of a multiple choice axiom*, Fund. Math.**74**(1972), 145-146. MR**0290958 (45:52)****[3]**A. Blass,*Two algebraic equivalents of the axiom of choice*, Notices Amer. Math. Soc.**22**(1975), A-524.**[4]**H. Cartan and S. Eilenberg,*Homological algebra*, Princeton Univ. Press, Princeton, N. J., 1956. MR**0077480 (17:1040e)****[5]**P. J. Cohen,*Set theory and the continuum hypothesis*, Benjamin, New York, 1966. MR**0232676 (38:999)****[6]**K. Devlin and R. B. Jensen,*Marginalia to a theorem of Silver*, Proc. Logic Conference, Kiel, 1974 (ed. by G. H. Müller, A. Oberschelp, and K. Potthoff), Lecture Notes in Math., vol. 499, Springer-Verlag, Berlin and New York, 1975, pp. 115-142. MR**0480036 (58:235)****[7]**W. B. Easton,*Powers of regular cardinals*, Ann. Math. Logic**1**(1970), 139-178. MR**0269497 (42:4392)****[8]**S. Grigorieff,*Intermediate submodels and generic extensions in set theory*, Ann. of Math. (2)**101**(1975), 447-490. MR**0373889 (51:10089)****[9]**W. Hodges, private communication.**[10]**T. Jech,*The axiom of choice*, North-Holland, Amsterdam, 1973. MR**0396271 (53:139)****[11]**R. B. Jensen,*Modelle der Mengenlehre*, Lecture Notes in Math., vol. 37, Springer-Verlag, Berlin and New York, 1967. MR**0221930 (36:4982)****[12]**J. L. Kelley,*General topology*, Van Nostrand, Princeton, N. J., 1955. MR**0070144 (16:1136c)****[13]**A. Levy,*Definability in axiomatic set theory*. I, Proc. Int. Congress on Logic, Methodology, and Philosophy of Science, 1964 (ed. by Y. Bar-Hillel), North-Holland, Amsterdam, 1965. MR**0205827 (34:5653)****[14]**S. MacLane,*Categories for the working mathematician*, Springer-Verlag, New York, 1971. MR**0354798 (50:7275)****[15]**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.**[16]**D. G. Northcott,*An introduction to homological algebra*, Cambridge Univ. Press, Cambridge, 1960. MR**0118752 (22:9523)****[17]**R. Platek,*Eliminating the continuum hypothesis*, J. Symbolic Logic**34**(1969), 219-225. MR**0256872 (41:1528)****[18]**D. S. Rim,*Modules over finite groups*, Ann. of Math.**69**(1959), 700-712. MR**0104721 (21:3474)****[19]**J. R. Shoenfield,*Unramified forcing*, Axiomatic Set Theory, (ed. by D. Scott), Proc. Sympos. Pure Math., vol. 13, Part I, Amer. Math. Soc., Providence, R. I., 1971, pp. 357-381. MR**0280359 (43:6079)****[20]**B. L. van der Waerden,*Algebra*. II, 5th ed., Springer-Verlag, Berlin, 1967.

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DOI:
https://doi.org/10.1090/S0002-9947-1979-0542870-6

Article copyright:
© Copyright 1979
American Mathematical Society