Atomic model (mathematical logic)
An atomic model in model theory is a model where every finite tuple of elements is described by a single formula. Such a formula is called complete. A complete type p(x1, ..., xn) is principal (relative to a theory T) if you can find one formula φ(x1, ..., xn) in p that axiomatizes p together with T; in other words, T ∪ {φ} implies exactly the formulas in p. A formula φ is complete in T if, for every ψ(x1, ..., xn), T ∪ {φ} decides ψ or ¬ψ. Therefore, a complete type is principal iff it contains a complete formula.
A model M is atomic if every n-tuple from M satisfies a formula that is complete in Th(M), the theory of M. The back-and-forth method shows that any two countable atomic models of the same theory that are elementarily equivalent are isomorphic.
This page was last edited on 3 February 2026, at 05:43 (CET).