View source for Domain of individuals (in logic)

''universe''

A term in [[Model theory|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

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d0338002.png" /> are predicate symbols and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d0338003.png" /> are function symbols for each of which a number of argument places is given. A model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d0338004.png" /> (or an algebraic system) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d0338005.png" /> is given by a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d0338006.png" /> and an interpreting function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d0338007.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d0338008.png" /> and assigning an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d0338009.png" />-place predicate to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d03380010.png" />-place predicate symbol, i.e. a subset of the Cartesian power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d03380011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d03380012.png" />, and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d03380013.png" />-place function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d03380014.png" /> to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d03380015.png" />-place function symbol. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d03380016.png" /> is called the domain of individuals (or universe) of the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d03380017.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,   "Mathematical logic" , Wiley  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.C. Chang,   H.J. Keisler,   "Model theory" , North-Holland  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.L. Ershov,   E.A. Palyutin,   "Mathematical logic" , Moscow  (1987)  (In Russian)</TD></TR></table>