https://en.wikipedia.org/wiki/Frege%27s_theorem#Overview Frege's theorem In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, informally, by Gottlob Frege in his 1884 Die Grundlagen der Arithmetik (The Foundations of Arithmetic)[1]
Overview the one truly new principle was one he called the Basic Law V[2] (now known as the axiom schema of unrestricted comprehension):[3] 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.[4]
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."[4] This achievement has become known as Frege's theorem.[4][5]
