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Nice infinitary logics

Author: Saharon Shelah
Journal: J. Amer. Math. Soc. 25 (2012), 395-427
MSC (2010): Primary 03C95; Secondary 03C80, 03C55
Published electronically: August 26, 2011
MathSciNet review: 2869022
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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'').

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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

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

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