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] and proven more formally in his 1893 Grundgesetze der Arithmetik I (Basic Laws of Arithmetic I).[2] 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).

  1. ^ Gottlob Frege, Die Grundlagen der Arithmetik, Breslau: Verlag von Wilhelm Koebner, 1884, §63.
  2. ^ Gottlob Frege, Grundgesetze der Arithmetik I, Jena: Verlag Hermann Pohle, 1893, §§20 and 47.