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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, ...