Nice infinitary logics

Shelah, Saharon

doi:10.1090/S0894-0347-2011-00712-1

Nice infinitary logics
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by Saharon Shelah

J. Amer. Math. Soc. 25 (2012), 395-427

DOI: https://doi.org/10.1090/S0894-0347-2011-00712-1

Published electronically: August 26, 2011

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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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