# Ways to show two structures are elementary equivalent

Let $$\mathcal{L}$$ be a finite first-order language. When we say structure we mean $$\mathcal{L}$$-structure.

Question. Can someone lists different ways which we may use to show two given structures are elementarily equivalent?

For example:

1) doing induction on complexity of formulas,

2) using Ehrenfeucht-Fraisse games,

3) ....

2. Building a forcing extension of the set-theoretic universe in which the structures become isomorphic. (This actually proves $$L_{\infty,\omega}$$-equivalence, and it tends to be similar to an argument about infinitely long Ehrenfeucht-Fraïssé games.)