A remedy for Berry’s ParadoxMathworldPlanetmath and related paradoxes is to separate the languagePlanetmathPlanetmath used to formulate a particular mathematical theory from the language used for its discourse.

The language used to formulate a mathematical theory is called the object language to contrast it from the metalanguage used for the discourse.

The most widely used object language is the first-order logic. The metalanguage could be English or other natural languages plus mathematical symbols such as .


  1. 1.

    The object language speaks of (¬An), but we speak of (,¬,An,) in the metalanguage. [Recall that a formulaMathworldPlanetmathPlanetmath is some finite sequencePlanetmathPlanetmath of the symbols. Cf. First Order Logic or Propositional LogicPlanetmathPlanetmath.]

  2. 2.

    In inductionMathworldPlanetmath proofs, one might encounter “the first symbol in the formula φ is (;” we know that the first symbol is indeed ( and not because is a symbol in our metalanguage. Similarly, “the third symbol is An” and not , because , is a symbol in our metalanguage.

  3. 3.

    and are members of the metalanguage, not of object language.

  4. 4.

    Parallel with the notion of metalanguage is metatheoremMathworldPlanetmath. “Γ(φψ) if Γ{φ}ψ,Γ0,φ,ψ0” is a metatheorem.

  5. 5.

    Examples from Set TheoryMathworldPlanetmath. Let “Con” denote consistency. Then Con(ZF) and Con(ZF+AC+GCH) are metamathematical statements; they are statements in the metalanguage.


  • 1 Schechter, E., Handbook of Analysis and Its Foundations, 1st ed., Academic Press, 1997.
Title metalanguage
Canonical name Metalanguage
Date of creation 2013-03-22 18:06:05
Last modified on 2013-03-22 18:06:05
Owner yesitis (13730)
Last modified by yesitis (13730)
Numerical id 6
Author yesitis (13730)
Entry type Definition
Classification msc 03B99