FINITELY GENERATED GROUPS AND FIRST-ORDER LOGIC
Andrey S. Morozov, André Nies · Journal of the London Mathematical Society · 2005
It is proved that the following classes of finitely generated groups have Π 1 1 -complete first-order theories: all finitely generated groups, the n-generated groups, and the strictly n-generated groups (n⩾2). Moreover, all those theories are distinct. Similar techniques show that quasi-finitely axiomatizable groups have a hyperarithmetical word problem, where a finitely generated group is quasi-finitely axiomatizable if it is the only finitely generated group satisfying an appropriate first-order sentence. The Turing degrees of word problems of quasi-finitely axiomatizable groups form a cofinal set in the Turing degrees of hyperarithmetical sets.