In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, a metatheorem is a statement about a
formal system
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...
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 and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...
(
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 or ...
s and
rules of inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the c ...
). 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 Set (mathematics), 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 mathema ...
(especially in
model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
) and
primitive recursive arithmetic (especially in
proof theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof The ...
). 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
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A \to B, it is sufficient to assume A ...
for
first-order logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
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
Class, Classes, or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used d ...
consisting of the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
s satisfying the formula.
*
Consistency proof
In deductive logic, a consistent theory (mathematical logic), theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no Formula (mathematical logic), formula \varphi such that both \varphi and its negat ...
s of systems such as
Peano arithmetic.
*
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantics, semantic truth and syntactic Provability logic, provability in first-order logic.
The completeness theorem applies ...
states that
first-order logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
is
complete.
See also
*
Metamathematics
*
Use–mention distinction
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