Indeed. Soundness and completeness are different things.
There are two well accepted definitions of soundness. One of them is the inability to prove true == false, that is, one cannot prove a contradiction from within that axiomatic system.
There are two well accepted definitions of soundness. One of them is the inability to prove true == false, that is, one cannot prove a contradiction from within that axiomatic system.
Indeed, as you allude, you cannot have both in an expressive enough system.