Around Classification Theory of Models
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1182
Saharan Shelah
Around Cla.ssification Theory of Models
Sprinqer-verlaq Berlin Heidelberg New York Tokyo
Author
Saharon Shelah Institute of Mathematics and Computer Science The Hebrew University, Givat Ram 91904 Jerusalem, Israel
Mathematics Subject Classification (1980): 03Cxx, 03Exx ISBN 3-540-16448-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16448-0 Springer-Verlag New York Heidelberg Berlin Tokyo
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Introduction
Though our focus is extending classification theory in various directions, there is here material which, I think, will interest researchers in quite different directions: general topology, Boolean algebras, set theory (mainly on coding sets) monadic logic and the theory of modules.
The only paper dealing directly with first order theories is a) of the Notes. A long time ago, in solving a problem of Keisler, we showed that if a .say countably first order) theory T, has only homogeneous models in one A > then this occurs in every JJ,
>
However this was done for sequence
homogeneous models and here we prove the parallel theorem for model homogeneous m odels.! In 2) we continue the classification of theories over a predicate. Here amalgamation properties over finite diagrams of models playa prominent part, and the combinatorics involving the non-structure theorems becomes much harder.
Hence we do it by forcing.
theories without two-cardinal models.
We' also restrict ourselves to
This will be continued and the
classification (for countable theories in a convenient set theory) will be completed in a paper together with Brad Hart. Another generalization is the classification of first-order T under any first-order definable quantifiers. We know much on this by previous works of 1 Some further light is thrown on the proof by the following theorem, Th.: Ii T is countable superstable unidimensional, then one of the following occurs; (1) T is categorical in every uncountable cardinal (2) T has the maximal of models in every uncountable cardinals (3) The number of models of T of power A is Minf2A,2 2
oJ
IV
Baldwin together with the author and of the author. An important case left open in the latter are the pairs (T, Qmon) (monadic logic) for T unstable; it was known that if some monadic expansion of T has the independence property the pair was "complicated" e.g. has Hanf numbers like second-order logic. Here we prove that the other pairs in this case, are all similar and have smaller Hanf numbers. A more basic qu
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