 In Gl, Frege then moves from this realization, in which statements of numbers are analyzed as predicating second-level numerical concepts of first-level concepts, to develop an account of the cardinal and natural numbers as ‘self-subsistent’ objects. He introduces a ‘cardinality operator’ on concepts, namely, ‘the number belonging to the concept \( F\)’, which designates the cardinal number which numbers the objects falling under \(F\). In what follows, we say this more simply as ‘the number of \(F\)s’ and use the simple notation ‘\(\#F\)’. Note that the operator # behaves like the \(\epsilon\) operator – when it is prefixed to a concept name like planet (\(P\)), then \(\#P\) (“the number of planets”) denotes an object; when it is prefixed to a variable like \(F\), then \(\# F\) ranges over the domain of objects (for each concept that \(F\) can take as a value, \(\#F\) denotes an object relative to that concept). Frege offers both an implicit (i.e., contextual) and an explicit definition of this operator in Gl. Both of these definitions require a preliminary definition of when two concepts \(F\) and \(G\) are in one-to-one correspondence or ‘equinumerous’. The notion of equinumerosity plays an important and fundamental role in the development of Frege’s Theorem. After developing the definition of equinumerosity, we then discuss Frege’s implicit and explicit definition of the number of \(F\)s. Only the former is needed for the proof of Frege’s Theorem, however.