Why do we need sometimes other structures than mentioned in the theorem to prove theorems?

Question

For example when one proves that the elementary theory of finite fields is decidable, one uses pseudo-finite fields which are not in generally finite fields. Why do we need such a larger fields to prove a statement for finite fields.

Answer 1

I hope I am interpreting your question the way you intended.

When things are phrased in the right kind of generality they often become easier to prove. When something is too specific there are lots of irrelevant details with too many properties that have nothing to do with the question and thus only serve to confuse and obscure the right path to a solution. When something is too general some key properties required to prove the result may be missing, and thus the result no longer follows. But when something is phrased in just the right generality then only those properties required for the proof are visible, and thus the proof is more easy to follow/find.

Strictly speaking, we don't really need to phrase things more generally in order to solve them, since the proof could have been found without generalizing. Often this is the case. The more general and elegant proofs are typically found after the theory matured some more.