A formula that maps elements to proper classes in an arbitrary $\in$-universe
Esser, Olivier
Bull. Belg. Math. Soc. Simon Stevin, Tome 17 (2010) no. 1, p. 479-483 / Harvested from Project Euclid
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In this paper we construct a formula $\varphi(x,a)$ on the language of set theory $\mathcal{L}\mathbin{:}(\in,=)$ such that $\{x\ | \ \varphi(x,a)\}$ is a proper class for each element $a$ and such that if $a\neq a'$, the classes $\{x\ | \ \varphi(x,a)\}$ and $\{x\ | \ \varphi(x,a')\}$ are different. This formula works for any structure with the exception of 2 structures with two elements each. This formula ``maps'' elements to proper classes injectively.
Publié le : 2010-08-15
Classification:  Set theory,  proper classes,  Russell's paradox,  03E99 
@article{1284570733,
     author = {Esser, Olivier},
     title = {A formula that maps elements to proper classes in an arbitrary $\in$-universe},
     journal = {Bull. Belg. Math. Soc. Simon Stevin},
     volume = {17},
     number = {1},
     year = {2010},
     pages = { 479-483},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1284570733}
}

Esser, Olivier. A formula that maps elements to proper classes in an arbitrary $\in$-universe. Bull. Belg. Math. Soc. Simon Stevin, Tome 17 (2010) no. 1, pp.  479-483. http://gdmltest.u-ga.fr/item/1284570733/