I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah.
I have to do this excercise:
Assume that Γ is a theory satisfying the following:
Γ is a Henkin theory.
For any two constants c,d, either $\Gamma \vdash c=d$ or $\Gamma \vdash c \neq d$.
There are two constants a,b such that $\Gamma \vdash a\neq b$.
We have to prove:
Show that Γ is a complete theory.
They give me a hint: For any sentence $\varphi$, consider the sentence $\exists x[(\varphi \land x=a)\lor (\neg \varphi \land x=b)]$.
I start with this:
As Γ is Henkin, there is a constant c such that $\Gamma \vdash \exists x[(\varphi \land x=a)\lor (\neg \varphi \land x=b)] \to [(\varphi(x/c) \land c=a)\lor (\neg \varphi(x/c) \land c=b)]$.
I'm stuck... Thank you :)