Unbounded Dense Linear Order

Let be a Language We define DLO to be the -Theory with axioms:

Theorem

DLO is -Categorical.

Proof

Back and forth argument. Fix countable Models . Let and . We inductively construct a sequence of order-preserving bijections . Set .

Forth

First construct an order preserving bijection . Enumerate such that

Now let . By induction hypothesis:

We define where is chosen as follows:

Back

We construct extending such that .

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We can thus define and check that it is an Isomorphism.

Corollary

DLO is a Complete Theory.

Proof

Clearly, there are no countable models. By previous theorem, any two countable models have: and thus they are Elementary Equivalent. We are done by Vaught’s Test.