Abstract
Partially ordered sets are studied, and the techniques of inductive proofs and recursive constructions over partially ordered sets with ascending or descending chain condition is formalized. The axioms of Zermelo–Fraenkel Set Theory with Choice, and the arithmetic of ordinals and cardinals, are reviewed, and Zorn’s Lemma is developed. We end with some thoughts on whether set theory is “real”.
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Numbers in angle brackets at the end of each listing show pages of these notes on which the work is referred to. “MR” refers to the review of the work in Mathematical Reviews, readable online at http://www.ams.org/mathscinet/ .
References
Numbers in angle brackets at the end of each listing show pages of these notes on which the work is referred to. “MR” refers to the review of the work in Mathematical Reviews, readable online at http://www.ams.org/mathscinet/ .
Garrett Birkhoff, Lattice Theory, third edition, AMS Colloq. Publications, v. XXV, 1967. MR 37 #2638.
Paul Halmos, Naive Set Theory, Van Nostrand University Series in Undergraduate Mathematics, 1960; Springer Undergraduate Texts in Mathematics, 1974. MR 22 #5575, MR 56 #11794.
J. Donald Monk, Introduction to Set Theory, McGraw-Hill, 1969. MR 44 #3877.
Robert L. Vaught, Set Theory, an Introduction, Birkhäuser, 1985; second edition 1995. MR 95k :03001.
Thomas W. Hungerford, Algebra, Springer GTM, v. 73, 1974. MR 50 #6693.
Serge Lang, Algebra, Addison-Wesley, third edition, 1993. Reprinted as Springer GTM v. 211, 2002. MR 2003e :00003.
George M. Bergman, Boolean rings of projection maps, J. London Math. Soc. (2) 4 (1972) 593–598. MR 47 #93.
George M. Bergman and P. M. Cohn, Symmetric elements in free powers of rings, J. London Math. Soc. (2) 1 (1969) 525–534. MR 40 #4301.
G. R. Brightwell, S. Felsner and W. T. Trotter, Balancing pairs and the cross product conjecture, Order 12 (1995), 327–349. MR 96k :06004.
Frank Harary, A survey of the reconstruction conjecture, in Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973), pp. 18–28. Lecture Notes in Math., Vol, 406, Springer, 1974. MR 50 #12818.
Wilfrid Hodges, Six impossible rings, J. Algebra 31 (1974) 218–244. MR 50 #315.
G. R. Krause and T. H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, Research Notes in Mathematics Series, v. 116, Pitman, 1985. MR 86g :16001.
Marcin Peczarski, The Gold Partition Conjecture, Order 23 (2006) 89–95. MR 2007i :06005.
Jean-Xavier Rampon, What is reconstruction for ordered sets? Discrete Math. 291 (2005) 191–233. MR 2005k:06010.
Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Diff. Geom. 2 (1968) 421–446. MR 40 #1939.
W. Hugh Woodin, The continuum hypothesis. I, Notices AMS 48 (2001) 567–576. MR 2002j :03056.
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Bergman, G.M. (2015). Ordered Sets, Induction, and the Axiom of Choice. In: An Invitation to General Algebra and Universal Constructions. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-11478-1_5
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DOI: https://doi.org/10.1007/978-3-319-11478-1_5
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