User blog comment:Vel!/Formal logic challenge/@comment-1605058-20140710070627

Let \(A\) be infinite with infinite complement. Let \(\phi_1:\mathbb{N}\rightarrow A\) and \(\phi_2:\mathbb{N}\rightarrow A'\) be a bijection. Now define \(\phi(2n)=\phi_1(n),\phi(2n+1)=\phi_2(n)\). \(\phi\) satisfies requirements, is nontrivial and \(A\) is decidable by checking if \(2|k\).

Rice's theorem requires \(\phi\) to be computable surjection, to exclude counterexamples as above.