Cantor digraphs and abbreviations of formulas

Abstract

A digraph $D=\langle V,E\rangle$ ($E\subset V\times V$) is Cantor if Cantor's theorem - for no set there is a surjection from it to its power set - holds in $D$, in the sense we explain. We construct a ZF formula $\varphi$ with length $494$ such that $D\models\varphi$ iff $D$ is Cantor. In order to obtain $\varphi$, which is a word over the alphabet $$ \{x_1,\,x_2,\,\dots\}\cup \{\in,\,=,\,\neg, \,\to,\,\leftrightarrow,\,\wedge,\, \vee,\,\exists,\,\forall,\,(,\,)\}\,, $$ we devise abbreviation schemes of ZF formulas. We introduce extensive and strongly extensive digraphs and show, by the standard argument, that they are Cantor. We construct a countable strongly extensive digraph with arbitrarily large finite in-degrees.