Finite model theory

Finite model theory (FMT) is a subarea of model theory (MT). MT is the branch of mathematical logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). FMT is a restriction of MT to interpretations on finite structures, which have a finite universe.

A single structure can always be axiomatized in first-order logic, where axiomatized in a language L means described uniquely up to isomorphism by a single L-sentence. Some finite sets of structures can also be axiomatized in FO. However, FO is not sufficient to axiomatize any set containing infinite structures.

Is a language L expressive enough to axiomatize a single finite structure S?

A structure like (1) in the figure can be described by FO sentences in the logic of graphs like

However, these properties do not axiomatize the structure, since for structure (1') the above properties hold as well, yet structures (1) and (1') are not isomorphic.

Informally the question is whether by adding enough properties, these properties together describe exactly (1) and are valid (all together) for no other structure (up to isomorphism).

For a single finite structure it is always possible to precisely describe the structure by a single FO sentence. The principle is illustrated here for a structure with one binary relation ${\displaystyle R}$ and without constants:

all for the same tuple ${\displaystyle x_{1}..x_{n}}$, yielding the FO sentence ${\displaystyle \exists _{x_{1}}\dots \exists _{x_{n}}(\varphi _{1}\land \varphi _{2}\land \varphi _{3}\land \varphi _{4})}$.

...
Wikipedia