 Since the logic of identity guarantees that no object is non-self-identical, nothing falls under the concept being non-self-identical. Had Frege’s explicit definition of the \(\#F\) worked as he had intended, the number 0 would, in effect, be identified with the extension consisting of all those extensions of concepts under which nothing falls. However, for the present purposes, we may note that 0 is defined in terms of (a) the primitive notion ‘the number of \(F\)s’ and (b) a concept \(([\lambda x \, x\neq x])\) whose existence is guaranteed by our second-order logic with identity and comprehension. It is straightforward to prove the following Lemma Concerning Zero from this definition of 0: 