World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0005106151
Reproduction Date:

Title: Metatheorem  
Author: World Heritage Encyclopedia
Language: English
Subject: Foundations of mathematics, Outline of logic, Judgment (mathematical logic), Metalogic, Independence (mathematical logic)
Collection: Mathematical Terminology, Metalogic, Metatheorems
Publisher: World Heritage Encyclopedia


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.

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.


  • Examples 1
  • See also 2
  • References 3
  • External links 4


Examples of metatheorems include:

  • 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.
  • Consistency proofs of systems such as Peano arithmetic

See also


  • Geoffrey Hunter (1969), Metalogic.
  • Alasdair Urquhart (2002), "Metatheory", A companion to philosophical logic, Dale Jacquette (ed.), p. 307

External links

  • at Encyclopaedia of MathematicsMeta-theorem
  • Barile, Margherita, "Metatheorem", MathWorld.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.