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''universe''
 
''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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033800/d0338001.png" /></td> </tr></table>
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$$L=\{P_0,\ldots,P_n,\ldots,F_0,\ldots,F_m,\ldots\}$$
  
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" />.
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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====
 
====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>
 
<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>

Revision as of 11:01, 7 August 2014

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=32761
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article