4 problems

A group is finitely generated if it has a finite generating set. A Scott sentence is an infinitary sentence whose countable models are exactly the structures isomorphic to the give…

Let $G$ be a finitely presented computable group. A computable $d$-$\Sigma_2$ Scott sentence is a Scott sentence for $G$ that is the conjunction of a computable $\Sigma_2$ sentence…

Let a typical group mean a randomly chosen finitely generated group with a single relator, in the sense of Kapovich and Schupp. A $d$-$\Sigma_2$ Scott sentence is a Scott sentence…

Index-set and optimal-Scott-sentence complexity conjecture. The complexity of the index set of $G$ should equal the complexity of an optimal Scott sentence for $G$.