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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
2
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Excellent uses of induction and recursion
There are infinifely many positive integers $n$ such that $2^n+1$ is divisible by $n$. One can show this by proving that if $n$ divides $2^n+1$, then $2^n+1$ divides $2^{2^n+1}+1$.