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

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:

 3e+99 Set theory

### Keywords:

set theory; proper classes; Russell’s paradox
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