Geodesic
$\in $-representation and set-prolongations
Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 4, pp. 661-666 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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By an $\in $-representation of a relation we mean its isomorphic embedding to $\Bbb E = \{\langle x,y\rangle;\,x\in y\}$. Some theorems on such a representation are presented. Especially, we prove a version of the well-known theorem on isomorphic representation of extensional and well-founded relations in $\Bbb E$, which holds in Zermelo-Fraenkel set theory. This our version is in Zermelo-Fraenkel set theory false. A general theorem on a set-prolongation is proved; it enables us to solve the task of the representation in question.
By an $\in $-representation of a relation we mean its isomorphic embedding to $\Bbb E = \{\langle x,y\rangle;\,x\in y\}$. Some theorems on such a representation are presented. Especially, we prove a version of the well-known theorem on isomorphic representation of extensional and well-founded relations in $\Bbb E$, which holds in Zermelo-Fraenkel set theory. This our version is in Zermelo-Fraenkel set theory false. A general theorem on a set-prolongation is proved; it enables us to solve the task of the representation in question.
Classification : 03E70, 04A99
Keywords: isomorphic representation; extensional relation; well-founded relation; set-pro\-lon\-gation 
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     author = {Ml\v{c}ek, Josef},
     title = {$\in $-representation and set-prolongations},
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     year = {1992},
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     url = {http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a10/}
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Mlček, Josef. $\in $-representation and set-prolongations. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 4, pp. 661-666. http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a10/

[V] Vopěnka P.: Mathematics in the Alternative Set Theory. TEUBNER TEXTE Leipzig (1979). | MR