I don't like the wording. How do we KNOW if a statement is true if we cannot prove it? I would prefer to say, instead, given a system of axioms, large enough to encompass the integers, that there exist some statements which can neither be proven nor disproven ("disproven" meaning that the negation can be proved). Yes, it follows that there must be an infinite number of statements that can neither be proven nor disproven. That's the case because, if there were only a finite number of such statements, we could add those statements themselves to the axioms, getting a new system of axioms in which all statements can be either proven or disproven, contradicting the original statement.