This paper uses the tools of continuous first-order logic to study dynamical systems of the form \(\langle X,s\rangle\), where \(X\) is a Cantor set and \(s\) is a homeomorphism on \(X\). By the Gel’fand duality, the category of compact Hausdorff spaces and continuous maps is contravariantly equivalent to the category of commutative unital \(C^*\)-algebras and unit-preserving algebra homomorphisms. This allows continuous model theory to be applied to structures of the form \(\langle C,\sigma\rangle\), where \(C=C(X)\), is the \(C^*\)-algebra of all continuous complex-valued functions on \(X\) (the Gel’fand dual of \(X\)), and \(\sigma=C(s)\) is the isomorphism on \(C\) induced by \(S\). The main results are the following:

Theorem 1.

The theory of \(\langle C,\sigma\rangle\) does not have a model companion.

Theorem 2.

If two odometers on \(X\) are elementarily equivalent, then they are topologically conjugate.

Theorem 3.

Being a generic homeomorphism is not axiomatizable, but it is expressible as an omitting types property. If \(s\) is the generic homeomorphism of the Cantor set \(X\), then \(\langle C,\sigma\rangle\) is the prime model of its theory.