Uncountable Dense Linear Orders – Order Theory

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Is there an example of two uncountable equipollent dense linear orders without endpoints that don't satisfy the same first order properties? Or is it true that two uncountable equipollent dense linear orders will always satisfy the same first order properties?

Best Answer

Every two dense linear orders without endpoints, regardless to their cardinality, have the same first-order theory. The reason is simple. $\sf DLO$ is a complete theory.

To see that it's a complete theory first note that $\Bbb Q$ is the unique (up to isomorphism) countable model of the theory. Then prove the following theorem:

If $T$ is a theory without finite models, that for some $\kappa\geq\aleph_0$, $T$ is $\kappa$-categorical, then $T$ is complete (HINT: Lowenheim-Skolem theorems).

Then $\sf DLO$ satisfies this theorem for $\kappa=\aleph_0$.

On the other hand, you can easily find two dense linear orders without endpoints of cardinality $\aleph_1$ which are not isomorphic.

  1. Take $\omega_1$ and replace each point by a copy of $\Bbb Q$

  2. Let $A=\omega_1^*+\omega_1$ (namely a copy of $\omega_1$ reversed followed by a copy of $\omega_1$) now iterate the following, $A_0=A$, $A_{n+1}$ is obtained by taking $A_\alpha$ and replacing each node by a copy of $A$. Finally, $A_\omega$ is the direct limit of these models.

    Show that $A_\omega$ is dense and without endpoints, and that it is not isomorphic to the order from the first suggestion.

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