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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u0957701.png" /> which is closed under the formation of unions, singletons, subelements, power sets, and pairs; more precisely:
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A set $\mathcal{U}$ which is closed under the formation of unions, singletons, subelements, power sets, and pairs; more precisely:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u0957702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u0957703.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u0957704.png" />;
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1) $i \in \mathcal{U}$, $X_i \in \mathcal{U}$ implies $\cup_{i\in I}X_i \in \mathcal{U}$;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u0957705.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u0957706.png" />;
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2) $x \in \mathcal{U}$ implies $\{x\} \in \mathcal{U}$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u0957707.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u0957708.png" />;
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3) $x \in X \in \mathcal{U}$ implies $x \in \mathcal{U}$;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u0957709.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u09577010.png" />;
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4) $X \in \mathcal{U}$ implies $\mathcal{P}X \in \mathcal{U}$;
  
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u09577011.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u09577012.png" />.
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5) $(x,y) \in \mathcal{U}$ if and only if $x,y \in \mathcal{U}$.
  
The existence of infinite universes in [[Axiomatic set theory|axiomatic set theory]] is equivalent to the existence of strongly inaccessible cardinals (cf. [[Cardinal number|Cardinal number]]). A universe is a model for Zermelo–Fraenkel set theory. Universes were introduced by A. Grothendieck in the context of [[Category|category]] theory in order to introduce the  "set"  of natural transformations of functors between (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095770/u09577013.png" />-) categories, and in order to admit other  "large"  category-theoretic constructions.
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The existence of infinite universes in [[axiomatic set theory]] is equivalent to the existence of strongly inaccessible cardinals (cf. [[Cardinal number]]). A universe is a model for Zermelo–Fraenkel set theory. Universes were introduced by A. Grothendieck in the context of [[category]] theory in order to introduce the  "set"  of natural transformations of functors between ($\mathcal{U}$-) categories, and in order to admit other  "large"  category-theoretic constructions.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Barwise (ed.) , ''Handbook of mathematical logic'' , North-Holland  (1977)  ((especially the article of D.A. Martin on Descriptive set theory))</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Gabriel,  "Des catégories abéliennes"  ''Bull. Soc. Math. France'' , '''90'''  (1962)  pp. 323–448</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Kunen,  "Set theory" , North-Holland  (1980)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Barwise (ed.) , ''Handbook of mathematical logic'' , North-Holland  (1977)  ((especially the article of D.A. Martin on Descriptive set theory))</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Gabriel,  "Des catégories abéliennes"  ''Bull. Soc. Math. France'' , '''90'''  (1962)  pp. 323–448</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Kunen,  "Set theory" , North-Holland  (1980)</TD></TR>
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</table>
 +
 
 +
{{TEX|done}}

Revision as of 18:20, 12 October 2017

A set $\mathcal{U}$ which is closed under the formation of unions, singletons, subelements, power sets, and pairs; more precisely:

1) $i \in \mathcal{U}$, $X_i \in \mathcal{U}$ implies $\cup_{i\in I}X_i \in \mathcal{U}$;

2) $x \in \mathcal{U}$ implies $\{x\} \in \mathcal{U}$;

3) $x \in X \in \mathcal{U}$ implies $x \in \mathcal{U}$;

4) $X \in \mathcal{U}$ implies $\mathcal{P}X \in \mathcal{U}$;

5) $(x,y) \in \mathcal{U}$ if and only if $x,y \in \mathcal{U}$.

The existence of infinite universes in axiomatic set theory is equivalent to the existence of strongly inaccessible cardinals (cf. Cardinal number). A universe is a model for Zermelo–Fraenkel set theory. Universes were introduced by A. Grothendieck in the context of category theory in order to introduce the "set" of natural transformations of functors between ($\mathcal{U}$-) categories, and in order to admit other "large" category-theoretic constructions.

References

[a1] J. Barwise (ed.) , Handbook of mathematical logic , North-Holland (1977) ((especially the article of D.A. Martin on Descriptive set theory))
[a2] P. Gabriel, "Des catégories abéliennes" Bull. Soc. Math. France , 90 (1962) pp. 323–448
[a3] K. Kunen, "Set theory" , North-Holland (1980)
How to Cite This Entry:
Universe. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universe&oldid=42050
This article was adapted from an original article by B. Pareigis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article