Conservativity theorem
Encyclopedia
In mathematical logic
, the conservativity theorem states the following: Suppose that a closed formula
is a theorem of a first-order theory
. Let 
be a theory obtained from 
by extending its language
with new constants
and adding a new axiom
.
Then
is a conservative extension
of
, which means that the theory 
has the same set of theorems in the original language (i.e., without constants 
) as the theory 
.
In a more general setting, the conservativity theorem is formulated for extensions of a first-order theory by introducing a new functional symbol:
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, the conservativity theorem states the following: Suppose that a closed formula
is a theorem of a first-order theory
. Let be a theory obtained from by extending its languageFormal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
with new constants
and adding a new axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
.Then
is a conservative extensionConservative extension
In mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...
of
, which means that the theory has the same set of theorems in the original language (i.e., without constants ) as the theory .In a more general setting, the conservativity theorem is formulated for extensions of a first-order theory by introducing a new functional symbol:
- Suppose that a closed formula
is a theorem of a first-order theory 
, where we denote 
. Let 
be a theory obtained from 
by extending its language with new functional symbol 
(of arity 
) and adding a new axiom 
. Then 
is a conservative extensionConservative extensionIn mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...
of
, i.e. the theories 
and 
prove the same theorems not involving the functional symbol 
).