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 .
Examples
-
1.
The object language speaks of , but we speak of in the metalanguage. [Recall that a formula is some finite sequence of the symbols. Cf. First Order Logic or Propositional Logic.]
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2.
In induction proofs, one might encounter “the first symbol in the formula is ;” we know that the first symbol is indeed and not because is a symbol in our metalanguage. Similarly, “the third symbol is ” and not because is a symbol in our metalanguage.
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3.
and are members of the metalanguage, not of object language.
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4.
Parallel with the notion of metalanguage is metatheorem. “ if ” is a metatheorem.
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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 | 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 |