you mention completeness in the rest of your comment, so I'm not sure how you aren't aware of this, but the famous incompleteness theorem says that for a consistent set of axioms there will always be true statements you can't prove.[1]
[1] https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...
Truth is some sort of value judgment that is outside the scope of formal systems. And looking at how bizarre Gödel statements are, it's unclear if there's any particular justification for declaring them to be true or false.
In fact there is a simple way to do it -- add contradictory axioms and then you can use the principle of explosion to prove any statement as true. Is such a system inconsistent and thus useless? Yes, but it is complete.
>The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.