The various subdisciplines of mathematical logic connect in interesting ways to separable profinite groups. We present three examples. 1. The complexity of the isomorphism problem can be classified using tools of descriptive set theory: isomorphism is smooth for finitely generated profinite groups, but as complicated as isomorphism of countable graphs in general (with Kechris and Tent). 2. Work of Jarden and Lubotzky can be connected to algorithmic randomness. For instance, every algorithmically random e-tuple in the absolute Galois group of Q generates a free profinite group (with Fouche). 3. Finally we ask which profinite groups can be uniquely identified by a first-order sentence. We give a first example, the Heisenberg group over the p-adic integers.