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
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article