I'm self-studying the book Incompleteness and Computability but I'm having troubles understanding the relation between consistency and completeness.
I'm working with the following definitions:
- A theory $\Gamma$ is consistent when no $A$ exists such that $\Gamma \vdash A$ $\land$ $\Gamma \vdash \neg A$.
- A theory $\Gamma$ is complete when for each $A$ either $\Gamma \vdash A$ or $\Gamma \vdash \neg A$ (but not both)
Assuming these definitions are correct (please correct me if they are not!), is it correct to say that if the theory $\Gamma$ is inconsistent then it is also incomplete?
My reasoning is that if $\Gamma$ is inconsistent then both $\Gamma \vdash A$ and $\Gamma \vdash \neg A$ for some $A$; hence, $A$ is a counterexample for the completeness of $\Gamma$, and so $\Gamma$ is incomplete.
If this is correct, however, it would imply that the contrapositive "$\Gamma$ complete $\implies$ $\Gamma$ consistent" is true, which feels a bit unintuitive to me.
Is something wrong with my line of reasoning, or maybe with my definitions?