# Let $T$ be the theory of linear, dense order, without minimum or maximum. Is $T\cup\{c_{i}<c_{j}\mid i<j\}$ complete?

Let $T$ be the theory of linear, dense order, without minimum or maximum in the language $\mathscr{L}$ . Expend the language by adding it countable amount of constants: $\mathscr{L}^{*}=\mathscr{L}\cup\{c_{0},c_{1},c_{2},...\}$ and let $T^{*}=T\cup\{c_{i}<c_{j}\mid i<j\}$ .

Prove that $T^{*}$ is complete.

I already know that $T$ is complete since $T$ is $\aleph_{0}$ -categorical. Two countable models of $T$ are isomorphic.

• You can show that $T^*$ is not $\aleph_0$-categorical (in one model the constants are bounded, in another they are not). So that direction won't work here. – Asaf Karagila Nov 29 '14 at 17:02
• I know. So what do you suggest? – Astro Nauft Nov 29 '14 at 17:05

Hint: $T$ has quantifier elimination. Since we don't add any non-constant symbols, $T^*$ has it as well.