Dense linear orderings categorical

[ky2345]

Homework Statement

Prove that the theory of dense linear orderings with no endpoints is not categorical in the cardinality of the reals.

Homework Equations

A theory is categorical in the cardinality of the reals (denoted c) if every c-model is ismorphic.

Isomorphic means that there is an isomorphism between the two models that is onto, one to one, and preserves order.

If two models A and B are elementarily equivalent, this means that A logically implies a formula a iff B logically implies a.

Isomorphic => elementarily equivalent

The Attempt at a Solution

Basically, I need to find two dense linear orderings without endpoints with cardinality=c that are not isomorphic. It would be great if I could get two dense linear orderings without endpoints with cardinality=c that are not elementarily equivalent, because then I would just have to list the two models and the sentence that is true in one but not true int eh other. I'm thinking of using A=(R,<) and B=(I,<), where I is the set of irrationals, as my two models. But, I'm having trouble proving why they are not ismorphic/elementarily equivalent. Obviously, A has the least upper bound property while B does not, but I'm having trouble saying that in first order logic.

 

[AKG]

Homework Statement

Prove that the theory of dense linear orderings with no endpoints is not categorical in the cardinality of the reals.

Homework Equations

A theory is categorical in the cardinality of the reals (denoted c) if every c-model is ismorphic.

Isomorphic means that there is an isomorphism between the two models that is onto, one to one, and preserves order.

If two models A and B are elementarily equivalent, this means that A logically implies a formula a iff B logically implies a.

Isomorphic => elementarily equivalent

The Attempt at a Solution

Basically, I need to find two dense linear orderings without endpoints with cardinality=c that are not isomorphic. It would be great if I could get two dense linear orderings without endpoints with cardinality=c that are not elementarily equivalent, because then I would just have to list the two models and the sentence that is true in one but not true int eh other. I'm thinking of using A=(R,<) and B=(I,<), where I is the set of irrationals, as my two models. But, I'm having trouble proving why they are not ismorphic/elementarily equivalent. Obviously, A has the least upper bound property while B does not, but I'm having trouble saying that in first order logic.

The theory of dense linear orders is $\omega$-categorical, hence complete. Thus any two dense linear orders are elementarily equivalent, so you won't be able to show the irrationals and reals aren't elementarily equivalent. They are, however, not isomorphic, and you can prove this using the idea you had about the least upper bound property. So think about this a little more.