 Frege thus opted instead to define numbers explicitly in terms of extensions: the number of Fs = the extension of the concept ‘equal to the concept F’ (where F is equal to G just in case the Fs are in one-to-one correspondence with the Gs). These explicit definitions arguably allow Frege to maintain the logicist thesis that arithmetic is logic: they define numbers as logical objects in the sense that their existence can be justified on the basis of (Frege's) logic alone. But, as is well known, these explicit definitions foundered on the inconsistency of Frege's underlying theory of extensions, in particular, the naïve comprehension principle embodied in his Basic Law V.