Special Sons, Morley Sequences

Abstract

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.

Un chant mystérieux tombe des astres d’or