  Despite Frege’s failure to provide a coherent systematization of the notion of an extension, we shall make use of the notion in what follows to explain Frege’s theory of numbers and analysis of number statements, without assuming Basic Law V. It suffices to use our informal understanding of the notion, though extensions can be rehabilitated in various ways, either axiomatically as in modern set theory or as in the various ways logicians have found to weaken Frege’s system. (See T. Parsons 1987, Burgess 1998, Heck 1996, Wehmeier 1999, Ferreira &amp; Wehmeier 2002, Fine 2002, Anderson &amp; Zalta 2004, Ferreira 2005, and Antonelli &amp; May 2005.)