Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Nice infinitary logics


Author: Saharon Shelah
Journal: J. Amer. Math. Soc. 25 (2012), 395-427
MSC (2010): Primary 03C95; Secondary 03C80, 03C55
DOI: https://doi.org/10.1090/S0894-0347-2011-00712-1
Published electronically: August 26, 2011
MathSciNet review: 2869022
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We deal with soft model theory of infinitary logics. We find a logic between $ \mathbb{L}_{\infty,\aleph_0}$ and $ \mathbb{L}_{\infty,\infty}$ which has some striking properties. First, it has interpolations (it was known that each of those logics fails interpolation though the pair has interpolation). Second, well ordering is not characterized in a strong way. Third, it can be characterized as the maximal such nice logic (in fact, it is the maximal logic stronger than $ \mathbb{L}_{\infty,\aleph_0}$ and which satisfies ``well ordering is not characterized in a strong way'').


References [Enhancements On Off] (What's this?)

  • [Be85] Jon Barwise and Solomon Feferman (editors), Model-theoretic logics, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg-New York, 1985. MR 819531 (87g:03033)
  • [Dic85] M. A. Dickman, Larger infinitary languages, Model Theoretic Logics (J. Barwise and S. Feferman, eds.), Perspectives in Mathematical Logic, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985, pp. 317-364. MR 819540
  • [EM02] Paul C. Eklof and Alan Mekler, Almost free modules: Set theoretic methods, North-Holland Mathematical Library, vol. 65, North-Holland Publishing Co., Amsterdam, 2002, Revised Edition. MR 1914985 (2003e:20002)
  • [FV59] S. Feferman and R.L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57-103. MR 0108455 (21:7171)
  • [Kop85] Sabine Koppelberg, Homogeneous Boolean algebras may have nonsimple automorphism groups, Topology and its Applications 21 (1985), 103-120. MR 813282 (87a:06030)
  • [Mak85] Johann A. Makowsky, Compactnes, embeddings and definability, Model-Theoretic Logics (J. Barwise and S. Feferman, eds.), Springer-Verlag, 1985, pp. 645-716. MR 819549
  • [Mos52] Andrzej Mostowski, On direct products of theories, J. Symbolic Logic 17 (1952), 1-31. MR 0047574 (13:897a)
  • [Rv89] Matatyahu Rubin and Petr Štěpánek, Homogeneous Boolean algebras, Handbook of Boolean algebras, vol. 2, North-Holland, Amsterdam, 1989, pp. 679-715. MR 991606
  • [Sh:b] Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, xxix+496 pp., 1982. MR 675955 (84h:03002)
  • [Sh:g] -, Cardinal Arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994. MR 1318912 (96e:03001)
  • [Sh:12] -, The number of non-isomorphic models of an unstable first-order theory, Israel Journal of Mathematics 9 (1971), 473-487. MR 0278926 (43:4652)
  • [Sh:18] -, On models with power-like orderings, Journal of Symbolic Logic 37 (1972), 247-267. MR 0446955 (56:5272)
  • [Sh:72] -, Models with second-order properties. I. Boolean algebras with no definable automorphisms, Annals of Mathematical Logic 14 (1978), 57-72. MR 501097 (80b:03047a)
  • [Sh:199] -, Remarks in abstract model theory, Annals of Pure and Applied Logic 29 (1985), 255-288. MR 808815 (87g:03040)
  • [Sh:384] -, Compact logics in ZFC : Complete embeddings of atomless Boolean rings, Non structure theory, Chapter X.
  • [FuSh:766] Laszlo Fuchs and Saharon Shelah, On a non-vanishing Ext, Rend. Sem. Mat. Univ. Padova 109 (2003), 235-239, math.LO/0405015. MR 1997989
  • [Sh:800] Saharon Shelah, On complicated models, Preprint.
  • [GbSh:880] Ruediger Goebel and Saharon Shelah, Absolutely Indecomposable Modules, Proceedings of the American Mathematical Society 135 (2007), 1641-1649, arXiv 0711.3011. MR 2286071 (2007k:13047)
  • [GbHeSh:948] Ruediger Goebel, Daniel Herden, and Saharon Shelah, Absolute $ E$-rings, Advances in Mathematics, accepted.
  • [Sh:F1046] Saharon Shelah, Nice infinitary logics II.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 03C95, 03C80, 03C55

Retrieve articles in all journals with MSC (2010): 03C95, 03C80, 03C55


Additional Information

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
Address at time of publication: Department of Mathematics, Hill Center - Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
Email: shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/S0894-0347-2011-00712-1
Keywords: Model theory, soft model theory, characterization theorems, Lindström theorem, interpolation, well ordering
Received by editor(s): May 16, 2010
Received by editor(s) in revised form: June 10, 2011
Published electronically: August 26, 2011
Additional Notes: The author thanks Alice Leonhardt for the beautiful typing. The author thanks the Israel Science Foundation for partial support of this research. Part of this work was done while the author was visiting Mittag-Leffler Institut, Djursholm, Sweden, in the fall of 2000 and the fall of 2009. We thank the Institut for hospitality and support. Publication No. 797 in the author list of publications.
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society