Barwise compactness and $\alpha$-recursion theory. The idea many properties of the following are captured by thinking of how to define analogs in $V_\omega$:
(1) Finite sets are elements of $V_{\omega}$.
(2) Computable sets can are $\Delta_1$ definable over $V_{\omega}$.
(3) Computable enumerable sets can are $\Sigma_1$ definable over $V_{\omega}$.
(4) First order logic is $L_{\infty, \omega} \cap V_\omega$.
Then, if we replace $V_\omega$ by a different countable admissible set $A$, many of the results relating these classes have analogs. E.g. Barwise compactness, completeness, the existence of an $A$-Turing jump, ...
