together describe exactly (1) and are valid (all together) for no other structure (up to isomorphism). Approach 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 and without constants: say that there are at least elements: say that there are at most elements: state every element of the relation : state every non-element of the relation : all for the same tuple , yielding the FO sentence . Extension to a fixed number of structures The method of describing a single structure by means of a first-order sentence can easily be extended for any fixed number of structures. A unique description can be obtained by the disjunction of the descriptions for each structure. For instance, for two structures and with defining sentences and this would be Extension to an infinite structure By definition, a set containing an infinite structure falls outside the area that FMT deals with. Note that infinite structures can never be discriminated in FO, because of the Löwenheim–Skolem theorem, which implies that no first-order theory with an infinite model can have a unique model up to isomorphism. The most famous example is probably Skolem's theorem, that there is a countable non-standard model of arithmetic. Characterisation of a class of structures Is a language L expressive enough to describe exactly (up to isomorphism) those finite structures that have certain property P? Problem The descriptions given so far all specify the number of elements of the universe. Unfortunately most interesting sets of structures are not restricted to a certain size, like all graphs that are trees, are connected or are acyclic. Thus to discriminate a finite number of structures is of special importance. Approach Instead of a general statement, the following is a sketch of a methodology to differentiate between structures that can and cannot be discriminated. 1. The core idea is that whenever one wants to see if a property P can be expressed in FO, one chooses structures A and B, where A does have P and B doesn't. If for A and B the same FO sentences hold, then P cannot be expressed in FO. In short: and where is shorthand for for all FO-sentences α, and P represents the class of structures with property P. 2. The methodology considers countably many subsets of