$Т$ - theory of signature $\{\le\}$, defined by the axioms А1-А5:
- А1: $\forall x(x=x)$
- А2: $\forall x,y\big((x\le y\land y\le x)\to x=y\big)$
- A3: $\forall x,y,z\big((x\le y\land y\le z)\to x\le z\big)$
- A4: $\forall x,y(x\le y\lor y\le x)$
- A5: $\forall x,y\left((x\le y\land\neg(x=y)\to\exists z\big(x\le z\land z\le y\land\neg(x=z)\land\neg(z=y)\big)\right)$
How many complete theory extend theory $T$ are there?
I think about 4 theories: dense linear order with endpoints, without them, and with one (left or right edge). But i don't khow, how can i prove completeness of them and lack of other theories(if there aren't they).
There are exactly four completions of this theory (for infinite models), and you've got them all. The theory is known as the theory of a dense linear order, and Cantor's back-and-forth argument shows that any two countable dense linear orders, with the same endpoint situation, are isomorphic. It follows that each of those four theories is complete, since otherwise they would have non-isomorphic models. Any complete theory extending your axioms must decide the endpoint situation in one of those four ways, and so it will be the same as one of your four theories.
But as noted in the comments, there are also two finite models: the model consisting of just one point and the empty model. Any model with at least two points must be infinite. So this makes for 6 distinct completions of the theory (or 5 if one's formal system does not allow empty models).