Does Gödel's first incompleteness theorem apply to quantifier-free arithmetics? 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 $\phi$?
What you are referring to is called Baby Arithmetic ($\mathsf{BA}$) in Peter Smith's book An introduction to Gödel's Theorems.


  $\mathsf{BA}$'s language [has] one single individual constant $0$, the one-place function symbol $S$, and the two-place function symbols $+$ and $\times$.  [...] it lacks quantifiers and variables.


$\mathsf{BA}$ is complete, negation complete, and decidable.