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 Die Grundlagen der Arithmetik (The Foundations of Arithmetic),[page needed] published in 1884, and proven more formally in his Grundgesetze der Arithmetik (The Basic Laws of Arithmetic),[page needed] published in two volumes, in 1893 and 1903. 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.
Frege's theorem in propositional logic
- (P → (Q→R)) → ((P→Q) → (P→R))
The truth table to the right gives a proof. For all possible assignments of false (✗) or true (✓) to P, Q, and R (columns 1, 3, 5), each subformula is evaluated according to the rules for material conditional, the result being shown below its main operator. Column 6 shows that the whole formula evaluates to true in every case, i.e. that it is a tautology. In fact, its antecedent (column 2) and its consequent (column 10) are even equivalent.
- Zalta, Edward (2013), "Frege's Theorem and Foundations for Arithmetic", Stanford Encyclopedia of Philosophy.
- Gottlob Frege (1884). Die Grundlagen der Arithmetik — eine logisch-mathematische Untersuchung über den Begriff der Zahl (PDF) (in German). Breslau: Verlage Wilhelm Koebner.
- Gottlob Frege (1893). Grundgesetze der Arithmetik (in German). 1. Jena: Verlag Hermann Pohle. — Edition in modern notation
- Gottlob Frege (1903). Grundgesetze der Arithmetik (in German). 2. Jena: Verlag Hermann Pohle. — Edition in modern notation
|This logic-related article is a stub. You can help Wikipedia by expanding it.|