The general way to show completeness of a theory

Question

I have difficulty showing the completeness of a certain theory. Namely, we know the DLO (dense linear order) without endpoints is complete, but what is the clearest way to show that? Similar variance is DLO with one endpoint, with both side endpoints, etc...

Thanks.

Answer 1

There is no "general way" to show the completeness of an arbitrary theory. (I think this is most likely a provable statement: you should be able to reduce the halting problem to deciding the completeness of a recursively axiomatized theory, although I don't feel confident enough in recursion theory to say that with certainty.)

The usual way of proving completeness is via quantifier elimination in a language we understand (with the caveat that we can only expand the language by symbols definable in the theory). For instance, DLO has quantifier elimination in the usual language $<$. DLO with the left endpoint has q.e. once you add a constant symbol for the endpoint, etc.

Once you have quantifier elimination, completeness is typically immediate (it is enough to show that quantifier-free sentences are decided).