Special Sons, Morley Sequences
Let M be a model of T and p a type over M. Let N be an elementary extension of M that realizes all types of S n (M) for all n. A son q of p over N is called special if for every formula \( f(x,\overrightarrow y ) \), if \( a \) and \( b \) are in N and have the same type over M, and if \( q \vDash f(x,\overrightarrow a ) \), then \( q \vDash f(x,\overrightarrow b ) \). In other words, the fact that \( q \vDash f(x,\overrightarrow a ) \) depends only on the type of \( a \) over M. We also call q M-special to say that it is a special son of its restriction to M; in this case, the function that sends a formula \( f(x,\overrightarrow y ) \) to the set of all types over M of tuples a of N such that \( q \vDash f(x,\overrightarrow a ) \) is called an infinitary definition of q over M.
KeywordsUltrametric Space Stable Type Independence Property Infinite Subset Elementary Extension
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