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Model theory without choice? Categoricity

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah*
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus. Givat Ram, Jerusalem 91904, Israel Department of Mathematics, Hill Center-Busch Campus, Rutgers., The State University of New Jersey, 110 Frelinghuysen Road, Piscataway. Nj 08854-8019, USA, E-mail:


We prove Los conjecture = Morley theorem in ZF. with the same characterization, i.e., of first order countable theories categorical in ℵα for some (eqiuvalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality ℵα is either ≥ ∣α∣ for every α or it has a small upper bound (independent of α close to ⊐2).

Research Article
Copyright © Association for Symbolic Logic 2009

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