Distinguishing mathematical structures by their theories

With first-order logic we can often express subtle features that distinguish a favored structure from similar alternatives.

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How are we to distinguish our mathematical structures from one another—how can we tell them apart? Often even very similar-seeming mathematical structures can be distinguished by subtle features differing amongst them, features expressible in the underlying formal language of the structures.

Consider, for an easy initial example, the following three order structures:

\(⟨ℕ,⩽ ⟩\qquad ⟨ℤ,⩽ ⟩\qquad ⟨ℚ,⩽ ⟩.\)

We have the familiar order relations on the domain of natural numbers, the integers, or on the set of rational numbers. Although we have used the same symbol ⩽ in each case, we interpret this symbol differently in the three structures as the usual order relation intended for the domain of that structure. These relations are each reflexive, transitive, and anti-symmetric, and so we have three partial orders here.

But these orders differ in their order-theoretic properties—let me challenge you to find statements expressible in the language of orders that distinguish them. For each structure, find a statement expressed in the language of ⩽ that is true in that order structure and false in the other two structures. Think about it on your own before continuing further.

Interlude