Nonfirstorderizability

Nonfirstorderizability is a concept in mathematical logic and model theory that refers to the inability of certain mathematical structures to be characterized up to isomorphism by a first-order sentence or set of sentences. This concept is closely related to the Löwenheim–Skolem theorem, which states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ, it has a model of size κ.

Definition

A class of structures is said to be first-orderizable if there is a first-order sentence or set of sentences whose models are exactly the structures in that class, up to isomorphism. A class of structures is nonfirstorderizable if it is not first-orderizable.

Examples

An example of a nonfirstorderizable class of structures is the class of all infinite totally ordered sets. This is because any first-order sentence or set of sentences that is true in one infinite totally ordered set is also true in any other, due to the Löwenheim–Skolem theorem.

Another example is the class of all fields of a given characteristic. This is because any first-order sentence or set of sentences that is true in one field of a given characteristic is also true in any other field of the same characteristic, regardless of the size of the field.

Implications

The concept of nonfirstorderizability has important implications for the study of mathematical structures. It shows that first-order logic is not powerful enough to capture all properties of mathematical structures. This has led to the development of higher-order logics and other extensions of first-order logic.

See also

• First-order logic
• Model theory
• Löwenheim–Skolem theorem
• Field (mathematics)
• Total order