metalanguage

A remedy for Berry’s Paradox and related paradoxes is to separate the language 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 $\Rightarrow$.

Examples

1. The object language speaks of $(\mathrm{\neg }{A}_n)$, but we speak of $⟨(,\mathrm{\neg },A_n,)⟩$ in the metalanguage. [Recall that a formula is some finite sequence of the symbols. Cf. First Order Logic or Propositional Logic.]

2. In induction proofs, one might encounter “the first symbol in the formula $\phi$ is $($;” we know that the first symbol is indeed $($ and not $⟨$ because $⟨$ is a symbol in our metalanguage. Similarly, “the third symbol is $A_n$” and not $,$ because $,$ is a symbol in our metalanguage.

3. $⊢$ and $\vDash$ are members of the metalanguage, not of object language.

4. Parallel with the notion of metalanguage is metatheorem. “$\mathrm{\Gamma }⊢(\phi \to \psi )$ if $\mathrm{\Gamma }\cup \{\phi \}⊢\psi ,\mathrm{\Gamma }\subseteq {ℒ}_0,\phi ,\psi \in {ℒ}_0$” is a metatheorem.

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

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