  Using this definition as a basis, Frege later derived many important theorems of number theory. Philosophers appreciated the importance of this work only relatively recently (C. Parsons 1965, Smiley 1981, Wright 1983, and Boolos 1987, 1990, 1995). Wright 1983 in particular showed how the Dedekind/Peano axioms for number might be derived from one of the consistent principles that Frege discussed in 1884, now known as Hume’s Principle (“The number of \(F\)s is equal to the number of \(G\)s if and only if there is a one-to-one correspondence between the \(F\)s and the \(G\)s”). It was recently shown by R. Heck [1993] that, despite the logical inconsistency in the system of Frege 1893/1903, Frege himself validly derived the Dedekind/Peano axioms from Hume’s Principle. Although Frege used Basic Law V (which yields an inconsistency when added to his second-order logic) to establish Hume’s Principle, once Hume’s Principle was established, the subsequent derivations of the Dedekind/Peano axioms make no further essential appeals to Basic Law V.  Following the lead of George Boolos, philosophers today call the derivation of the Dedekind/Peano Axioms from Hume’s Principle ‘Frege’s Theorem’. For a comprehensive introduction to the subtle and complex logical reasoning involved in this theorem, see the entry  Frege’s theorem and foundations for  arithmetic.