In propositional logic I believe we aim to show the system is "correct" by showing it is both sound and complete.
For the completeness piece is this the same as showing that the Boolean operator set $\Omega$ is functionally complete, i.e. Post's Lattice/Theorem? Or is this a different concept and completeness is proved some other way?
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No, these are completely (!) different concepts.
In ease case "complete" means something like "able to do everything we reasonably want". But even the kind of things we reasonably want them to do are entirely different between the two situations.
For a proof system what we reasonably want to prove all logically valid formulas.
For a set of connectives what we reasonably want is to write down formulas that express arbitrary truth functions.
You are misunderstanding the roles played by the language (and in particular the set of connectives) and the proof system.
When we say that a language is complete in some sense - e.g. that the usual set of Boolean operations is functionally complete - what we mean is that it can describe a lot of things. By contrast, when we say that a proof system is complete (and sound) for a given language (with respect to a notion of semantics), we mean that it can prove all the things expressible in that language which it should (and none that it shouldn't) (where "should(n't)" is understood via the semantics).
Note that these are completely different tasks. In particular, if I have a really really weak language, in some sense it's easier to construct a complete proof system for it, since there are fewer things it needs to prove (and fewer things it needs not to prove)!