Abstract
This is part of the classification developed in Baldwin Shelah [BSh]. The paper is divided into two parts. In part I we show that (T ∞, 2nd)≨(T, mon) iff the Hanf number for the theory T in monadic logic is smaller than the Hanf number of second order logic.
For this we deal with partition relations for models of T. The main result is that if T does not have the independence property even after expanding by monadic predicates (or equivalently (T ∞, 2nd)≨(T, mon) then: ℶω+1(λ)+→s(λ) <ωT . In Part II we analyze such T getting a decomposition theorem like that in [BSh] (but weaker) (This is needed in part I.)
I thank Rami Gromberg for many corrections.
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References
J. Baldwin and S. Shelah. Second order quantifiers and the complexity of theories, Proc. of the 1980/1 model theory year in Jerusalem. Notre Dame J. of Formal Logic, 1985.
Y. Gurevich and S. Shelah. Monadic Logic and the next world. Proc. of the 1980/1 model theory year in Jerusalem; Israel J. Math, 1985.
S. Shelah. Classification theory, North Holland Publ. Co. 1978.
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S. Shelah. On the monadic theory of order. Annals of Math 102(1975) 379–419.
S. Shelah, More on monadic theories, in preparation.
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© 1986 Springer-Verlag
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Shelah, S. (1986). Monadic logic: Hanf Numbers. In: Around Classification Theory of Models. Lecture Notes in Mathematics, vol 1182. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098511
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DOI: https://doi.org/10.1007/BFb0098511
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