|
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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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]
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
(85g:03085), http://dx.doi.org/10.1016/S0049-237X(09)70120-X
- [2]
An algebraic equivalent of a multiple choice axiom, Fund. Math.
74 (1972), no. 2, 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]
Henri
Cartan and Samuel
Eilenberg, Homological algebra, Princeton University Press,
Princeton, N. J., 1956. MR 0077480
(17,1040e)
- [5]
Paul
J. Cohen, Set theory and the continuum hypothesis, W. A.
Benjamin, Inc., New York-Amsterdam, 1966. MR 0232676
(38 #999)
- [6]
Keith
I. Devlin and R.
B. Jensen, Marginalia to a theorem of Silver, ⊨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
(58 #235)
- [7]
William
B. Easton, Powers of regular cardinals, Ann. Math. Logic
1 (1970), 139–178. MR 0269497
(42 #4392)
- [8]
Serge
Grigorieff, Intermediate submodels and generic extensions in set
theory, Ann. Math. (2) 101 (1975), 447–490. MR 0373889
(51 #10089)
- [9]
W. Hodges, private communication.
- [10]
Thomas
J. Jech, The axiom of choice, North-Holland Publishing Co.,
Amsterdam, 1973. Studies in Logic and the Foundations of Mathematics, Vol.
75. MR
0396271 (53 #139)
- [11]
Ronald
Björn Jensen, Modelle der Mengenlehre. Widerspruchsfreiheit
und Unabhängigkeit der Kontinuum-Hypothese und des Auswahlaxioms,
Ausgearbeitet von Franz Josef Leven. Lecture Notes in Mathematics, No. 37,
Springer-Verlag, Berlin, 1967 (German). MR 0221930
(36 #4982)
- [12]
John
L. Kelley, General topology, D. Van Nostrand Company, Inc.,
Toronto-New York-London, 1955. MR 0070144
(16,1136c)
- [13]
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
(34 #5653)
- [14]
Saunders
MacLane, Categories for the working mathematician,
Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5. 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 University Press, New York, 1960. MR 0118752
(22 #9523)
- [17]
Richard
A. Platek, Eliminating the continuum hypothesis, J. Symbolic
Logic 34 (1969), 219–225. MR 0256872
(41 #1528)
- [18]
Dock
Sang Rim, Modules over finite groups, Ann. of Math. (2)
69 (1959), 700–712. MR 0104721
(21 #3474)
- [19]
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
(43 #6079)
- [20]
B. L. van der Waerden, Algebra. II, 5th ed., Springer-Verlag, Berlin, 1967.
- [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.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
04A25,
03E35,
20K99
Retrieve articles in all journals
with MSC:
04A25,
03E35,
20K99
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1979-0542870-6
PII:
S 0002-9947(1979)0542870-6
Article copyright:
© Copyright 1979 American Mathematical Society
|
