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 firstorder 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 $\u27e8(,\mathrm{\neg},{A}_{n},)\u27e9$ 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 $\u27e8$ because $\u27e8$ is a symbol in our metalanguage. Similarly, “the third symbol is ${A}_{n}$” and not $,$ because $,$ is a symbol in our metalanguage.

3.
$\u22a2$ and $\vDash $ are members of the metalanguage, not of object language.

4.
Parallel with the notion of metalanguage is metatheorem^{}. “$\mathrm{\Gamma}\u22a2(\phi \to \psi )$ if $\mathrm{\Gamma}\cup \{\phi \}\u22a2\psi ,\mathrm{\Gamma}\subseteq {\mathcal{L}}_{0},\phi ,\psi \in {\mathcal{L}}_{0}$” 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.
References
 1 Schechter, E., Handbook of Analysis and Its Foundations, 1st ed., Academic Press, 1997.
Title  metalanguage 

Canonical name  Metalanguage 
Date of creation  20130322 18:06:05 
Last modified on  20130322 18:06:05 
Owner  yesitis (13730) 
Last modified by  yesitis (13730) 
Numerical id  6 
Author  yesitis (13730) 
Entry type  Definition 
Classification  msc 03B99 