Frege's theorem

The theorem was re-discovered by Crispin Wright in the early 1980s and has since been the focus of significant work. It is at the core of the philosophy of mathematics known as neo-logicism (at least of the Scottish School variety). In The Foundations of Arithmetic (1884), and later, in Basic Laws of Arithmetic (vol. 1, 1893; vol. 2, 1903), Frege attempted to derive all of the laws of arithmetic from axioms he asserted as logical (see logicism). Most of these axioms were carried over from his Begriffsschrift; the one truly new principle was one he called the Basic Law V (now known as the axiom schema of unrestricted comprehension): the "value-range" of the function f(x) is the same as the "value-range" of the function g(x) if and only if ∀x[f(x) = g(x)]. However, not only did Basic Law V fail to be a logical proposition, but the resulting system proved to be inconsistent, because it was subject to Russell's paradox. The inconsistency in Frege's Grundgesetze overshadowed Frege's achievement: according to Edward Zalta, the Grundgesetze "contains all the essential steps of a valid proof (in second-order logic) of the fundamental propositions of arithmetic from a single consistent principle." This achievement has become known as Frege's theorem. In propositional logic, Frege's theorem refers to this tautology: The theorem already holds in one of the weakest logics imaginable, the constructive implicational calculus.