   However if we try instead to prove that \(\{\{x\} : x \in X\}\) is not the same size as \(\wp(X)\) we find that the diagonal set is defined by a stratified condition and exists, so the proof succeeds. This tells us that we cannot prove that \(\lvert X\rvert = \lvert \{\{x\} : x \in X\}\rvert\) for arbitrary \(X\): graphs of restrictions of the singleton function tend not to exist. If they did, we would be able to prove Cantor’s theorem in full generality. This gives rise to an endomorphism \(T\) on cardinals, where \(T\lvert X\rvert =_{df} \lvert \{\{x\} : x \in X\}\rvert\). \(T\) misbehaves in connection with the sets that in NF studies we call big (as opposed to large, as in large cardinals (in ZF)). These are the collections like the universal set, and the set of all cardinals and the set of all ordinals: collections denoted by expressions which in ZF-like theories will pick out proper classes. If \(\lvert X\rvert = \lvert \{\{x\} : x \in X\}\rvert\) we say that \(X\) is cantorian. If the singleton function restricted to \(X\) exists, we say that \(X\) is strongly cantorian. Sets whose sizes are concrete natural numbers are strongly cantorian. IN (the set of Frege natural numbers) is cantorian, but the assertion that it is strongly cantorian implies the consistency of NF.