Extension by definitions explained
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol
for the
set that has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant
and the new
axiom \forallx(x\notin\emptyset)
, meaning "for all
x,
x is not a member of
". It can then be proved that doing so adds essentially nothing to the old theory, as should be expected from a definition. More precisely, the new theory is a
conservative extension of the old one.
Definition of relation symbols
Let
be a first-order theory and
a
formula of
such that
, ...,
are distinct and include the variables free in
. Form a new first-order theory
from
by adding a new
-ary relation symbol
, the logical axioms featuring the symbol
and the new axiom
\forallx1...\forallxn(R(x1,...,xn)\leftrightarrow\phi(x1,...,xn))
,called the
defining axiom of
.
If
is a formula of
, let
be the formula of
obtained from
by replacing any occurrence of
by
(changing the
bound variables in
if necessary so that the variables occurring in the
are not bound in
). Then the following hold:
\psi\leftrightarrow\psi\ast
is provable in
, and
is a
conservative extension of
.
The fact that
is a conservative extension of
shows that the defining axiom of
cannot be used to prove new theorems. The formula
is called a
translation of
into
. Semantically, the formula
has the same meaning as
, but the defined symbol
has been eliminated.
Definition of function symbols
Let
be a first-order theory (with equality) and
a formula of
such that
,
, ...,
are distinct and include the variables free in
. Assume that we can prove
\forallx1...\forallxn\exists!y\phi(y,x1,...,xn)
in
, i.e. for all
, ...,
, there exists a unique
y such that
. Form a new first-order theory
from
by adding a new
-ary function symbol
, the logical axioms featuring the symbol
and the new axiom
\forallx1...\forallxn\phi(f(x1,...,xn),x1,...,xn)
,called the
defining axiom of
.
Let
be any atomic formula of
. We define formula
of
recursively as follows. If the new symbol
does not occur in
, let
be
. Otherwise, choose an occurrence of
in
such that
does not occur in the terms
, and let
be obtained from
by replacing that occurrence by a new variable
. Then since
occurs in
one less time than in
, the formula
has already been defined, and we let
be
(changing the bound variables in
if necessary so that the variables occurring in the
are not bound in
). For a general formula
, the formula
is formed by replacing every occurrence of an atomic subformula
by
. Then the following hold:
\psi\leftrightarrow\psi\ast
is provable in
, and
is a
conservative extension of
.
The formula
is called a
translation of
into
. As in the case of relation symbols, the formula
has the same meaning as
, but the new symbol
has been eliminated.
The construction of this paragraph also works for constants, which can be viewed as 0-ary function symbols.
Extensions by definitions
A first-order theory
obtained from
by successive introductions of relation symbols and function symbols as above is called an
extension by definitions of
. Then
is a conservative extension of
, and for any formula
of
we can form a formula
of
, called a
translation of
into
, such that
\psi\leftrightarrow\psi\ast
is provable in
. Such a formula is not unique, but any two of them can be proved to be equivalent in
T.
In practice, an extension by definitions
of
T is not distinguished from the original theory
T. In fact, the formulas of
can be thought of as
abbreviating their translations into
T. The manipulation of these abbreviations as actual formulas is then justified by the fact that extensions by definitions are conservative.
Examples
- Traditionally, the first-order set theory ZF has
(equality) and
(membership) as its only primitive relation symbols, and no function symbols. In everyday mathematics, however, many other symbols are used such as the binary relation symbol
, the constant
, the unary function symbol
P (the
power set operation), etc. All of these symbols belong in fact to extensions by definitions of ZF.
be a first-order theory for
groups in which the only primitive symbol is the binary product ×. In
T, we can prove that there exists a unique element
y such that
x×
y =
y×
x =
x for every
x. Therefore we can add to
T a new constant
e and the axiom
\forallx(x x e=xande x x=x)
,
and what we obtain is an extension by definitions
of
. Then in
we can prove that for every
x, there exists a unique
y such that
x×
y=
y×
x=
e. Consequently, the first-order theory
obtained from
by adding a unary function symbol
and the axiom
\forallx(x x f(x)=eandf(x) x x=e)
is an extension by definitions of
. Usually,
is denoted
.
See also
Bibliography
- S. C. Kleene (1952), Introduction to Metamathematics, D. Van Nostrand
- E. Mendelson (1997). Introduction to Mathematical Logic (4th ed.), Chapman & Hall.
- J. R. Shoenfield (1967). Mathematical Logic, Addison-Wesley Publishing Company (reprinted in 2001 by AK Peters)
