\b[No.] The map $s(x)=-x$ is an <automorphism of a first-order structure> for the additive integers, but it sends one to minus one. By <automorphism invariance of parameter-free definability>, every parameter-free definable singleton is fixed by every <structure automorphism>. The singleton containing one is not fixed, so it is not definable.

Solution

Past exam of the model theory course of the University of Oxford / 2011 / 1 / b / i / β / Solution

 Solution (source code)