Gödel's first incompleteness theorem states roughly that "for any axiomatization of arithmetic, there are statements that can neither be proven to be false nor true."
Does this still hold when it comes to quantifier-free statements?
I.e. if we have the structure of arithmetic: $(\mathbb N, +,\cdot, 0,1)$, and we restrict to sentences $\Phi^{QF}$ about this structure that don't contain $\exists, \forall$, can we then for all $\phi\in \Phi^{QF}$ either prove or disprove $\phi$?