metalanguage

A remedy for Berry’s Paradox and related paradoxes is toseparate the language used to formulate a particular mathematicaltheory from the language used for its discourse.

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

The most widely used object language is the first-order logic. Themetalanguage could be English or other natural languages plusmathematical symbols such as $\Rightarrow$.

Examples

1. The object language speaks of $(\mathrm{\neg }{A}_n)$, but we speakof $⟨(,\mathrm{\neg },A_n,)⟩$ 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 firstsymbol in the formula $\phi$ is $($;” we know that the firstsymbol is indeed $($ and not $⟨$ because $⟨$ is asymbol 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” denoteconsistency. Then Con(ZF) and Con(ZF+AC+GCH) are metamathematicalstatements; they are statements in the metalanguage.