I'm not sure whether this is more "down-to-earth" but take 0-1 laws of random graphs. The probability of a first order property to be satisfied by a random graph distributed $G(n,p)$ tends to 0 or 1.

The proof is by showing that limit theory satisfied by these graphs has only one countable model, and is therefore complete.

I'm not sure whether this is more "down-to-earth" but take 0-1 laws of random graphs. The probability of a first order property to be satisfied by a random graph distributed $G(n,p)$ tends to 0 or 1.

The proof is by showing that the limit theory satisfied by these graphs has only one countable model, and is therefore complete.