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{{Unreferenced stub|auto=yes|date=December 2009}} | |||
In [[metalogic]] and [[metamathematics]], '''Frege's theorem''' is a [[metatheorem]] which states that the [[Peano axiom]]s 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'' ([[Frege's Foundations of Arithmetic|Foundations of Arithmetic]]), published in 1884, and proven more formally in his ''Grundgesetze der Arithmetik'' ([[Frege's Grundgesetze|Basic Laws of Arithmetic]]), 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 == | |||
In [[Propositional calculus|propositional logic]], Frege's theorems refers to this [[tautology (logic)|tautology]]: | |||
:<math>(P \to (Q \to R)) \to ((P \to Q) \to (P \to R))</math> | |||
==External links== | |||
* [[Stanford Encyclopedia of Philosophy]]: | |||
** "[http://plato.stanford.edu/entries/frege-logic/ Frege's Theorem and Foundations for Arithmetic]" — by [[Edward Zalta]]. | |||
{{DEFAULTSORT:Frege's Theorem}} | |||
[[Category:Theorems in the foundations of mathematics]] | |||
[[Category:Theorems in propositional logic]] | |||
[[Category:Metatheorems]] | |||
{{Logic-stub}} | |||
Revision as of 00:50, 7 March 2013
Template:Unreferenced stub In metalogic and metamathematics, Frege's theorem is a metatheorem which 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 (Foundations of Arithmetic), published in 1884, and proven more formally in his Grundgesetze der Arithmetik (Basic Laws of Arithmetic), 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
In propositional logic, Frege's theorems refers to this tautology: