I am trying to solve the following problem:
Let DLO be the set of formulas that assert that $<$ is a dense linear order without endpoints. Show that for every sentence $\varphi$ in the language of $<$, either $DLO \vdash \varphi$ or $DLO \vdash \neg\varphi$.
I am not sure what is DLO. According to the description of the problem, I would say $DLO = \{ \forall x \forall y[ (x \neq y) \implies (x < y \vee y < x)] (\text{linear order}), \forall x \exists y [x < y], \forall x \exists y [y < x](\text{without endpoints}), \forall x \forall y[ (x < y) \implies \exists z[x < z < y] ] (\text{dense})\}$.
In my opinion, these are the formulas that assert that $<$ is a dense linear order without endpoints, but I am not sure if I am missing something. Please if you can help me, I will appreciate it. Thanks!