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 $(\neg A_n)$ , but we speak of $\langle (, \neg, A_n, ) \rangle$ 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 $\varphi$ is $($ ;'' we know that the first symbol is indeed $($ and not $\langle$ because $\langle$ is a symbol in our metalanguage. Similarly, ``the third symbol is $A_n$ '' and not $,$ because $,$ is a symbol in our metalanguage.
3. $\vdash$ and $\models$ are members of the metalanguage, not of object language.
4. Parallel with the notion of metalanguage is metatheorem. ``$\Gamma\vdash(\varphi\rightarrow\psi)$ if $\Gamma\cup\{\varphi\}\vdash\psi, \Gamma\subseteq\mathcal{L}_0, \varphi, \psi\in\mathcal{L}_0$ " 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.

Bibliography

1

Schechter, E., Handbook of Analysis and Its Foundations, 1st ed., Academic Press, 1997.