This article is about logical statements. For theories about theories, see Metatheory.
In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.[citation needed]
A formal system is determined by a formal language and a deductive system (axioms and rules of inference). The formal system can be used to prove particular sentences of the formal language with that system. Metatheorems, however, are proved externally to the system in question, in its metatheory. Common metatheories used in logic are set theory (especially in model theory) and primitive recursive arithmetic (especially in proof theory). Rather than demonstrating particular sentences to be provable, metatheorems may show that each of a broad class of sentences can be proved, or show that certain sentences cannot be proved.[citation needed]
The deduction theorem for first-order logic says that a sentence of the form φ→ψ is provable from a set of axioms A if and only if the sentence ψ is provable from the system whose axioms consist of φ and all the axioms of A.
metatheorem, this, article, about, logical, statements, theories, about, theories, metatheory, logic, metatheorem, statement, about, formal, system, proven, metalanguage, unlike, theorems, proved, within, given, formal, system, metatheorem, proved, within, met. This article is about logical statements For theories about theories see Metatheory In logic a metatheorem is a statement about a formal system proven in a metalanguage Unlike theorems proved within a given formal system a metatheorem is proved within a metatheory and may reference concepts that are present in the metatheory but not the object theory citation needed A formal system is determined by a formal language and a deductive system axioms and rules of inference The formal system can be used to prove particular sentences of the formal language with that system Metatheorems however are proved externally to the system in question in its metatheory Common metatheories used in logic are set theory especially in model theory and primitive recursive arithmetic especially in proof theory Rather than demonstrating particular sentences to be provable metatheorems may show that each of a broad class of sentences can be proved or show that certain sentences cannot be proved citation needed Contents 1 Examples 2 See also 3 References 4 External linksExamples editExamples of metatheorems include The deduction theorem for first order logic says that a sentence of the form f ps is provable from a set of axioms A if and only if the sentence ps is provable from the system whose axioms consist of f and all the axioms of A The class existence theorem of von Neumann Bernays Godel set theory states that for every formula whose quantifiers range only over sets there is a class consisting of the sets satisfying the formula Consistency proofs of systems such as Peano arithmetic See also editMetamathematics Use mention distinctionReferences editGeoffrey Hunter 1969 Metalogic Alasdair Urquhart 2002 Metatheory A companion to philosophical logic Dale Jacquette ed p 307External links editMeta theorem at Encyclopaedia of Mathematics Barile Margherita Metatheorem MathWorld Retrieved from https en wikipedia org w index php title Metatheorem amp oldid 1148740543, wikipedia, wiki, book, books, library,
