Elementary embeddings

Question

Im not quite sure I understand elementary embeddings right. Say M elementary embeds into N and $M \models\varphi$, does it follow that $N \models \varphi$ or is there a problem with existential formulas?

Answer 1

Short answer: yes that follows.

This is really direct from the definition of elementary embedding.

Definition. A function $f: M \to N$ is an elementary embedding if for every formula $\phi(x_1, \ldots, x_n)$ and all $a_1, \ldots, a_n \in M$ we have $$ M \models \phi(a_1, \ldots, a_n) \quad \Longleftrightarrow \quad N \models \phi(f(a_1), \ldots, f(a_n)).$$

So if $\phi$ is a sentence, i.e. contains no free variables, then this simplifies to $$ M \models \phi \quad \Longleftrightarrow \quad N \models \phi. $$ In particular this means that $M$ and $N$ are elementarily equivalent, which means that they satisfy exactly the same sentences.

You might be confused with embeddings. Let me recall that definition (well, one of the equivalent definitions).

Definition. A function $f: M \to N$ is an embedding if for every quantifier-free formula $\phi(x_1, \ldots, x_n)$ and all $a_1, \ldots, a_n \in M$ we have $$ M \models \phi(a_1, \ldots, a_n) \quad \Longleftrightarrow \quad N \models \phi(f(a_1), \ldots, f(a_n)).$$

Note that in this definition we only care about quantifier-free formulas. The following fact is where existential formulas and universal formulas come into play.

Fact. Let $f: M \to N$ be an embedding and let $\phi(x_1, \ldots, x_n)$ be a formula and let $a_1, \ldots, a_n \in M$.

1. If $\phi$ is an existential formula then $M \models \phi(a_1, \ldots, a_n) \implies N \models \phi(f(a_1), \ldots, f(a_n))$.
2. If $\phi$ is a universal formula then $M \models \phi(a_1, \ldots, a_n) \Longleftarrow N \models \phi(f(a_1), \ldots, f(a_n))$.

Proof. Exercise.