I read a result of Vaught(a little down the page) that says that there cannot be any first order theory which has exactly two countable models upto isomorphism. Is this not a counter example:
The theory of dense linear orders(DLO) has 4 countable models upto isomorphism. They are the intervals $[0,1]$, $(0,1)$, $[0,1)$, and $(0,1]$ (restricted to the rationals). If to DLO we add the statement that there is a lower bound, do we not end up with exactly two of these 4 theories?
Am I misunderstanding Vaught's result or making a mistake with the DLO part?