I've been musing over this problem over the past few days, and believe I have an answer. However, I am still a bit shaky with some of the definitions I am using, and would appreciate if anyone could either confirm or point out errors in what I have. The problem reads: Let $\delta$ be the conjunction of the following sentences:
• $(\forall x)\lnot Exx$;
• $(\forall x)(\forall y)(\forall z)\big((Exy \land Eyz) \to Exz\big)$;
• $(\forall x)(\forall y)\big(x\ne y\to(Exy \lor Eyx)\big)$;
• $(\forall x)(\exists y)(\exists z)(Exy \land Ezx)$;
• $(\forall x)(\forall y)\big(Exy \to (∃z)(Exz \land Ezy)\big)$.
Show that $\operatorname{Cn}(\delta)$ is complete.
My thought process was as follows: I believed $\delta$ to be the axiomatization of a dense linear order without endpoints. Enderton (p155) states that a set $T$ of sentences is a theory iff $T=\operatorname{Cn}(T)$. I then thought I must prove that $\operatorname{Cn}(\delta)$ is in fact the theory of dense linear order without endpoints. If this is the case, I do know how to use a back and forth argument to show the theory is $\aleph_0$ categorical, then apply the Łoś-Vaught test to get completeness.
Any guidance would be appreciated!