Is the following true?
Let T be a complete theory in some elementary language. Let $n$ be a natural number and suppose $\Gamma$ is a non-principal $n$-type of T. Let $\Delta$ ne an $n+1$-type of T containing $\Gamma$. Then $\Delta$ too is non-principal. Give a proof or a counterexample.
During the lecture we got the following theorem: Given an elementary language and T a complete theory in this language with at least one model. Let $n$ be a natural number: T has infinitely $n$-types iff T has a non-prinical $n$-type. So i thought we can prove by contradiction: Suppose $\Delta$ is wel principal, then T has only finitely many $n+1$-types, but $\Gamma\subset\Delta$ and $\Gamma$ non-principal, thus it contains infinitely many n-types, thus contradiction.
Is this argumentation good? And so not why not and how can i solve it?
In general: How can i decide whether something is prinicpal or not?
Thank you for help.