If M and N are two models and the inclusion M ⊆ N is an embedding,then N is an extension of M.
An existential sentence is a sentence which consists of a string of existential quantifiers followed by a quantifier-free formula.
How can I show that if a first-order theory T is closed under extension, then it can be axiomatised using existential sentences?
I think the proof might involve the method of diagrams, the compactness theorem and even Loewenheim-Skolem theorem, but I don't know how to connect the dots.