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There is a problem I'm working on which ass for an example of a
language L, two L-structures, and a complete type that is realized by
exactly three elements in one of the structures and not realized by
exactly three structures in the other structure.
The solution that I came up with was a language with a countably
infinite number of constant symbols and no other non-logical symbols.
And the type is an element not being equal to any of these constants.
The two structures will both have the universe of omega. And the
constant symbols are interpreted as follows:
In the first structure constant symbol c_i is interpreted as i+3 (so
the 0th constant symbol goes to 3, the 1st to 4, etc...)
In the second structure constant symbol c_i in interpreted as i+4.
So in the first structure this type is realized by 3 elements, and in
the second structure its realized by 4 elements.
The problem I have with this is "How do I show this type is complete?"
(If it is indeed a complete type).
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