A formula that maps elements to proper classes in an arbitrary \(\in \)-universe. (English) Zbl 1207.03069

Esser, Olivier

Bull. Belg. Math. Soc. - Simon Stevin 17, No. 3, 479-483 (2010).

Summary: We construct a formula \(\varphi(x,a)\) of the language of set theory \(\mathcal{L} : (\in ,=)\) such that \(\{x \mid\varphi(x,a)\}\) is a proper class for each element \(a\) and such that if \(a\neq a^{\prime }\), the classes \(\{x \mid \varphi(x,a)\}\) and \(\{x\mid \varphi(x,a^{\prime })\}\) are different. This formula works for any structure with the exception of two structures with two elements each. This formula “maps” elements injectively to proper classes.

MSC:

03E99

Set theory

Keywords:

set theory; proper classes; Russell’s paradox