# 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. 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. 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. 3.

$\vdash$ and $\models$ are members of the metalanguage, not of object language.

4. 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. 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 Metalanguage 2013-03-22 18:06:05 2013-03-22 18:06:05 yesitis (13730) yesitis (13730) 6 yesitis (13730) Definition msc 03B99