In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
, a metatheorem is a statement about a
formal system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A for ...
proven in a
metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a
metatheory, and may reference concepts that are present in the metatheory but not the object theory.
A formal system is determined by a formal language and a
deductive system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A fo ...
(
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...
s and
rules of inference). The formal system can be used to prove particular sentences of the formal language with that system. Metatheorems, however, are proved externally to the system in question, in its metatheory. Common metatheories used in logic are
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
(especially in
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
) and
primitive recursive arithmetic (especially in
proof theory). Rather than demonstrating particular sentences to be provable, metatheorems may show that each of a broad class of sentences can be proved, or show that certain sentences cannot be proved.
Examples
Examples of metatheorems include:
* The
deduction theorem for
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quanti ...
says that a sentence of the form φ→ψ is provable from a set of axioms ''A'' if and only if the sentence ψ is provable from the system whose axioms consist of φ and all the axioms of ''A''.
* The
class existence theorem of
von Neumann–Bernays–Gödel set theory states that for every formula whose
quantifiers range only over sets, there is a
class consisting of the
sets satisfying the formula.
*
Consistency proofs of systems such as
Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
.
See also
*
Metamathematics
*
Use–mention distinction
The use–mention distinction is a foundational concept of analytic philosophy, according to which it is necessary to make a distinction between a word (or phrase) and it.Devitt and Sterelny (1999) pp. 40–1W.V. Quine (1940) p. 24 Many philos ...
References
*
Geoffrey Hunter (1969), ''Metalogic''.
* Alasdair Urquhart (2002), "Metatheory", ''A companion to philosophical logic'', Dale Jacquette (ed.), p. 307
External links
''Meta-theorem'' at Encyclopaedia of Mathematics*
{{Metalogic
Metalogic
Mathematical terminology