Frege's two notions of "extension"

Our goal is to answer a question proposed by Richard Heck in the paper “Formal Arithmetic Before Grundgesetze”. Heck inquires as to the reasons why it took almost eight years for Frege to honor his promises of concluding his project of grounding mathematics in logic. Although Heck gave some answers, we think that a more adequate philosophical discussion can still be offered. This paper will try to fill in that gap by presenting what we understand was the central problem faced by Frege on Gl: the lack of a standard criterion to fix the meaning of identity propositions of mathematics. In our account, the German philosopher finally decided to fill in this gap by providing a new construal of “extension”, one which included some important refinements on his previous account of that notion. The new concept thus construed allowed Frege to unify his treatment of identity propositions by including in his system a universal criterion for deciding the truth of any identity proposition supported by his famous basic law V. So, our claim will be that Frege’s resistance and doubts about the inclusion of axiom V as a logical law in his system were the primary cause of that delay.