 We will not discuss the above research further in the present entry, for none of these alternatives have achieved a clear consensus. Instead, we focus on the theoretical accomplishment revealed by Frege’s work in Gg. As noted in the Introduction, Frege validly proved a rather deep fact about the natural numbers notwithstanding the inconsistency of Basic Law V. He derived the Dedekind/Peano axioms for number theory in second-order logic from Hume’s Principle (which was briefly mentioned above and which will be discussed in the next section). But this fact went unnoticed for many years. Though Geach (1955) claimed such a derivation was possible, C. Parsons (1965) was the first to note that Hume’s Principle was powerful enough for the derivation of the Dedekind/Peano axioms. Though Wright (1983) actually carried out most of the derivation, Heck (1993) showed that although Frege did use Basic Law V to derive Hume’s principle, his (Frege’s) subsequent derivations of the Dedekind/Peano axioms of number theory from Hume’s Principle never made an essential appeal to Basic Law V. Since Hume’s Principle can be consistently added to second-order logic, we may conclude that Frege himself validly derived the basic laws of number theory. It will be the task of the next few sections to explain Frege’s accomplishments in this regard. We will do this in two stages. In §3 we study Frege’s attempt to derive Hume’s Principle from Basic Law V by analyzing cardinal numbers as extensions. Then, we put this aside in §4 and §5 to examine how Frege was able to derive the Dedekind/Peano axioms of number theory from Hume’s Principle alone.