# Domain of individuals (in logic)

*universe*

A term in model theory denoting the domain of variation of individual (object) variables of a given formal language of first-order predicate calculus. Each such language is completely described by the set

$$L=\{P_0,\ldots,P_n,\ldots,F_0,\ldots,F_m,\ldots\}$$

where $P_0,\ldots,P_n,\ldots,$ are predicate symbols and $F_0,\ldots,F_m,\ldots,$ are function symbols for each of which a number of argument places is given. A model $\mathfrak M$ (or an algebraic system) of $L$ is given by a non-empty set $M$ and an interpreting function $I$, defined on $L$ and assigning an $n$-place predicate to an $n$-place predicate symbol, i.e. a subset of the Cartesian power $M^n$ of $M$, and an $n$-place function $M^n\to M$ to an $n$-place function symbol. The set $M$ is called the domain of individuals (or universe) of the model $\mathfrak M$.

#### References

[1] | S.C. Kleene, "Mathematical logic" , Wiley (1967) |

[2] | C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1973) |

[3] | Yu.L. Ershov, E.A. Palyutin, "Mathematical logic" , Moscow (1987) (In Russian) |

**How to Cite This Entry:**

Domain of individuals (in logic).

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Domain_of_individuals_(in_logic)&oldid=39432