help with Godel's

hi

I'm reading about Godel's from "http://plus.maths.org/issue39/features/
dawson/"

"Now Gödel's completeness theorem states that whatever propositions
are taken as axioms, one can prove all (and only) those statements
that hold in all structures satisfying the axioms. But if some
statement is true of the natural numbers but is not true of another
system of entities that also satisfies the axioms, then it cannot be
proved. At first, that did not seem to be a serious problem, because
mathematicians hoped that entities that masqueraded as numbers but
were essentially different from them did not exist. So Gödel's next
theorem came as a shock."

Can somone help clarify "if some statement is true of the natural
numbers but is not true of another system of entities that also
satisfies the axioms, then it cannot be proved."

Question: if the statement is known not to be true, what is there left
to be proved? The sentence bothers me.

.

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